![Introduction
I restricted myself to characteristic zero: for a short time, the quantum
jump to p = 0 was beyond the range . . . but it did not take me too long
to make this jump.
— Oscar Zariski
The arithmetic of abelian varieties with complex multiplication over a number field
is fascinating. However this will not be our focus. We study the theory of complex
multiplication in mixed characteristic.
Abelian varieties over finite fields. In 1940 Deuring showed that an elliptic
curve over a finite field can have an endomorphism algebra of rank 4 [33, §2.10].
For an elliptic curve in characteristic zero with an endomorphism algebra of rank 2
(rather than rank 1, as in the “generic” case), the j-invariant is called a singular j-
invariant. For this reason elliptic curves with even more endomorphisms, in positive
characteristic, are called supersingular.1
Mumford observed as a consequence of results of Deuring that for any elliptic
curves E1 and E2 over a finite field κ of characteristic p 0 and any prime = p,
the natural map
Z ⊗Z Hom(E1, E2)−→ HomZ[Gal(κ/κ)](T(E1), T(E2))
(where on the left side we consider only homomorphisms “defined over κ”) is an
isomorphism [118, §1]. The interested reader might find it an instructive exercise
to reconstruct this (unpublished) proof by Mumford. Tate proved in [118] that
the analogous result holds for all abelian varieties over a finite field and he also
incorporated the case = p by using p-divisible groups. He generalized this result
into his influential conjecture [117]:
An -adic cohomology class2
that is fixed under the Galois group should be
a Q-linear combination of fundamental classes of algebraic cycles when
the ground field is finitely generated over its prime field.
Honda and Tate gave a classification of isogeny classes of simple abelian vari-
eties A over a finite field κ (see [50] and [121]), and Tate refined this by describing
1Of course, a supersingular elliptic curve isn’t singular. A purist perhaps would like to say “an
elliptic curve with supersingular j-value”. However we will adopt the generally used terminology
“supersingular elliptic curve” instead.
2The prime number is assumed to be invertible in the base field.
1](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-16-320.jpg)
![2 INTRODUCTION
the structure of the endomorphism algebra End0
(A) (working in the isogeny cate-
gory over κ) in terms of the Weil q-integer of A, with q = #κ; see [121, Thm. 1].
It follows from Tate’s work (see 1.6.2.5) that an abelian variety A over a finite field
κ admits sufficiently many complex multiplications in the sense that its endomor-
phism algebra End0
(A) contains a CM subalgebra3
L of rank 2 dim(A). We will
call such an abelian variety (in any characteristic) a CM abelian variety and the
embedding L → End0
(A) a CM structure on A.
Grothendieck showed that over any algebraically closed field K, an abelian
variety that admits sufficiently many complex multiplications is isogenous to an
abelian variety defined over a finite extension of the prime field [89]. This was
previously known in characteristic zero (by Shimura and Taniyama), and in that
case there is a number field K
⊂ K such that the abelian variety can be defined
over K
(in the sense of 1.7.1). However in positive characteristic such abelian
varieties can fail to be defined over a finite subfield of K; examples exist in every
dimension 1 (see Example 1.7.1.2).
Abelian varieties in mixed characteristic. In characteristic zero, an abelian
variety A gives a representation of the endomorphism algebra D = End0
(A) on the
Lie algebra Lie(A) of A. If A has complex multiplication by a CM algebra L of
degree 2 dim(A) then the isomorphism class of the representation of L on Lie(A) is
called the CM type of the CM structure L → End0
(A) on A (see Lemma 1.5.2 and
Definition 1.5.2.1).
As we noted above, every abelian variety over a finite field is a CM abelian vari-
ety. Thus, it is natural to ask whether every abelian variety over a finite field can be
“CM lifted” to characteristic zero (in various senses that are made precise in 1.8.5).
One of the obstacles4
in this question is that in characteristic zero there is the no-
tion of CM type that is invariant under isogenies, whereas in positive characteristic
whatever can be defined in an analogous way is not invariant under isogenies. For
this reason we will use the terminology “CM type” only in characteristic zero.
For instance, the action of the endomorphism ring R = End(A0) of an abelian
variety A0 on the Lie algebra of A0 in characteristic p 0 defines a representa-
tion of R/pR on Lie(A0). Given an isogeny f : A0 → B0 we get an identification
End0
(A0) = End0
(B0) of endomorphism algebras, but even if End(A0) = End(B0)
under this identification, the representations of this common endomorphism ring
on Lie(A0) and Lie(B0) may well be non-isomorphic since Lie(f) may not be an
isomorphism. Moreover, if we have a lifting A of A0 over a local domain of char-
acteristic 0, in general the inclusion End(A) ⊂ End(A0) is not an equality. If the
inclusion End0
(A) ⊂ End0
(A0) is an equality then the character of the representa-
tion of End(A0) on Lie(A0) is the reduction of the character of the representation of
End(A) on Lie(A). This relation can be viewed as an obstruction to the existence
of CM lifting with the full ring of integers of a CM algebra operating on the lift;
see 4.1.2, especially 4.1.2.3–4.1.2.4, for an illustration.
In the case when End(A0) contains the ring of integers OL of a CM algebra
L ⊂ End0
(A0) with [L : Q] = 2 dim(A0), the representation of OL/pOL on Lie(A0)
turns out to be quite useful, despite the fact that it is not an isogeny invariant. Its
class in a suitable K-group will be called the Lie type of (A0, OL → End(A0)).
3A CM algebra is a finite product of CM fields; see Definition 1.3.3.1.
4surely also part of the attraction](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-17-320.jpg)
![INTRODUCTION 3
The above discrepancy between the theories in characteristic zero and charac-
teristic p 0 is the basic phenomenon underlying this entire book. Before dis-
cussing its content, we recall the following theorem of Honda and Tate ([50, §2,
Thm. 1] and [121, Thm. 2]).
For an abelian variety A0 over a finite field κ there is a finite extension
κ
of κ and an isogeny (A0)κ → B0 such that B0 admits a CM lifting
over a local domain of characteristic zero with residue field κ
.
This result has been used in the study of Shimura varieties, for settings where the
ground field is an algebraic closure of Fp and isogeny classes (of structured abelian
varieties) are the objects of interest; see [135]. Our starting point comes from the
following questions which focus on controlling ground field extensions and isogenies.
For an abelian variety A0 over a finite field κ, to ensure the existence
of a CM lifting over a local domain with characteristic zero and residue
field κ
of finite degree over κ,
(a) may we choose κ
= κ?
(b) is an isogeny (A0)κ → B0 necessary?
These questions are formulated in various precise forms in 1.8.
An isogeny is necessary. Question (b) was answered in 1992 (see [93]) as follows.
There exist (many) abelian varieties over Fp that do not admit any CM
lifting to characteristic zero.
The main point of [93] is that a CM liftable abelian variety over Fp can be defined
over a small finite field. This idea is further pursued in Chapter 3, where the size,
or more accurately the minima5
of the size, of all possible fields of definition of the
p-divisible group of a given abelian variety over Fp is turned into an obstruction for
the existence of a CM lifting to characteristic 0. This is used to show (in 3.8.3) that
in “most” isogeny classes of non-ordinary abelian varieties of dimension ⩾ 2 over
finite fields there is a member that has no CM lift to characteristic 0. (In dimension
1 a CM lift to characteristic 0 always exists, over the valuation ring of the minimal
possible p-adic field, by Deuring Lifting Theorem; see 1.7.4.6.) We also provide
effectively computable examples of abelian varieties over explicit finite fields such
that there is no CM lift to characteristic 0.
A field extension might be necessary—depending on what you ask.
Bearing in mind the necessity to modify a given abelian variety over a finite field to
guarantee the existence of a CM lifting, we rephrase question (a) in a more precise
version (a)
below.
(a)
Given an abelian variety A0 over a finite field κ of characteris-
tic p, is it necessary to extend scalars to a strictly larger finite field
κ
⊃ κ (depending on A0) to ensure the existence of a κ
-rational isogeny
(A0)κ → B0 such that B0 admits a CM lifting over a characteristic 0
local domain R with residue field κ
?
It turns out there are two quite different answers to question (a)
, depending on
whether one requires the local domain R of characteristic 0 to be normal. The
subtle distinction between using normal or general local domains for the lifting
5The size of a finite field κ1 is smaller than the size of a finite field κ2 if κ1 is isomorphic to
a subfield of κ2, or equivalently if #κ1 | #κ2. Among the sizes of a family of finite fields there
may not be a unique minimal element.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-18-320.jpg)
![4 INTRODUCTION
went unnoticed for a long time. Once this distinction came in focus, answers to the
resulting questions became available.
If we ask for a CM lifting over a normal domain up to isogeny, in general a base
field extension before modification by an isogeny is necessary. This is explained in
2.1.2, where we formulate the “residual reflex obstruction”, the idea for which goes
as follows. Over an algebraically closed field K of characteristic zero, we know that
a simple CM abelian variety B with K-valued CM type Φ (for the action of a CM
field L) is defined over a number field in K containing the reflex field E(Φ) of Φ.
Suppose that for every K-valued CM type Φ of L, the residue field of E(Φ) at any
prime above p is not contained in the finite field κ with which we began in question
(a). In such cases, for every CM structure L → End0
(A0) on A0 and any abelian
variety B0 over κ which is κ-isogenous to A0, there is no L-linear CM lifting of
B0 over a normal local domain R of characteristic zero with residue field κ.6
In
2.3.1–2.3.3 we give such an example, a supersingular abelian surface A0 over Fp2
with End(A0) = Z[ζ5] for any p ≡ ±2 (mod 5). A much broader class of examples
is given in 2.3.5, consisting of absolutely simple abelian varieties (with arbitrarily
large dimension) over Fp for infinitely many p.
Note that passing to the normalization of a complete local noetherian domain
generally enlarges the residue field. Hence, if we drop the condition that the mixed
characteristic local domain R be normal then the obstruction in the preceding
consideration dissolves. And in fact we were put on the right track by mathematics
itself. The phenomenon is best illustrated in the example in 4.1.2, which is the
same as the example in 2.3.1 already mentioned: an abelian surface C0 over Fp2
with CM order Z[ζ5] that, even up to isogeny, is not CM liftable to a normal local
domain of characteristic zero. On the other hand, this abelian surface C0 is CM
liftable to an abelian scheme C over a mixed characteristic non-normal local domain
of characteristic zero, though the maximal subring of Z[ζ5] whose action lifts to C
is non-Dedekind locally at p; see 4.1.2.7
This example is easy to construct, and
the proof of the existence of a CM lifting, possibly after applying an Fp2 -rational
isogeny, is not difficult either.
In Chapter 4 we show that the general question of existence of a CM lifting
after an appropriate isogeny can be reduced to the same question for (a mild gen-
eralization of) the example in 4.1.2, enabling us to prove:
every abelian variety A0 over a finite field κ admits an isogeny A0 → B0
over κ such that B0 admits a CM lifting to a mixed characteristic local
domain with residue field κ.
There are refined lifting problems, such as specifying at the beginning which CM
structure on A0 is to be lifted, or even what its CM type should be on a geometric
fiber in characteristic 0. These matters will also be addressed.
6The source of obstructions is that the base field κ might be too small to contain at least
one characteristic p residue field of the reflex field E(Φ) for at least one CM type Φ on L. Thus,
the field of definition of the generic fiber of the hypothetical lift may be too big. Likewise, an
obstruction for question (b) is that the field of definition of the p-divisible group A0[p∞] may be
too big (in a sense that is made precise in 3.8.3 and illustrated in 3.8.4–3.8.5).
7No modification by isogeny is necessary in this example, but the universal deformation for
C0 with its Z[ζ5]-action is a non-algebraizable formal abelian scheme over W(Fp2 ).](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-19-320.jpg)
![INTRODUCTION 5
Our basic method is to “localize” various CM lifting problems to the corre-
sponding problems for p-divisible groups. Although global properties of abelian va-
rieties are often lost in this localization process, the non-rigid nature of p-divisible
groups can be an advantage. In Chapter 3 the size of fields of definition of a p-
divisible group in characteristic p appears as an obstruction to the existence of CM
lifting. The reduction steps in Chapter 4 rely on a classification and descent of
CM p-divisible groups in characteristic p with the help of their Lie types (see 4.2.2,
4.4.2). In addition, the “Serre tensor construction” is applied to p-divisible groups,
both in characteristic p and in mixed characteristic (0, p); see 1.7.4 and 4.3.1 for
this general construction.
Survey of the contents. In Chapter 1 we start with a survey of general facts
about CM abelian varieties and their endomorphism algebras. In particular, we
discuss the deformation theory of abelian varieties and p-divisible groups, and we
review results in Honda-Tate theory that describe isogeny classes and endomor-
phism algebras of abelian varieties over a finite field in terms of Weil integers. We
conclude by formulating various CM lifting questions in 1.8. These are studied in
the following chapters. We will see that the questions can be answered with some
precision.
In Chapter 2 we formulate and study the “residual reflex condition”. Using this
condition we construct several examples of abelian varieties over finite fields κ such
that, even after applying a κ-isogeny, there is no CM lifting to a normal local
domain with characteristic zero and residue field of finite degree over κ; see 2.3. It
is remarkable that many such examples exist, but we do not know whether we have
characterized all possible examples; see 2.3.7.
We then study algebraic Hecke characters and review part of the theory of com-
plex multiplication due to Shimura and Taniyama. Using the relationship between
algebraic Hecke characters for a CM field L and CM abelian varieties with CM by
L (the precise statement of which we review and prove), we use global methods to
show that the residual reflex condition is the only obstruction to the existence of
CM lifting up to isogeny over a normal local domain of characteristic zero. We also
give another proof by local methods (such as p-adic Hodge theory).
In Chapter 3 we take up methods described in [93]. In that paper classical CM
theory in characteristic zero was used. Here we use p-divisible groups instead of
abelian varieties and show that the size of fields of definition of a p-divisible group
in characteristic p is a non-trivial obstruction to the existence of a CM lifting. In 3.3
we study the notion of isogeny for p-divisible groups over a base scheme (including
its relation with duality). We show, in one case of the CM lifting problem left
open in [93, Question C], that an isogeny is necessary. Our methods also provide
effectively computed examples. Some facts about CM p-divisible groups explained
in 3.7 are used in 3.8 to get an upper bound of a field of definition for the closed
fiber of a CM p-divisible group.
In Appendix 3.9, we use the construction (in 3.7) of a p-divisible group with
any given p-adic CM type over the reflex field to produce a semisimple abelian
crystalline p-adic representation of the local Galois group such that its restriction to
the inertia group is “algebraic” with algebraic part that we may prescribe arbitrarily
in accordance with some necessary conditions (see 3.9.4 and 3.9.8).](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-20-320.jpg)
![6 INTRODUCTION
In Chapter 4 we show CM liftability after an isogeny over the finite ground field
(lifting over a characteristic zero local domain that need not be normal). That is,
every CM structure (A0, L → End0
(A0)) over a finite field κ has an
isogeny over κ to a CM structure (B0, L → End0
(B0)) that admits a
CM lifting;
(see 4.1.1). This statement is immediately reduced to the case when L is a CM
field (not just a CM algebra) and the whole ring OL of integers of L operates on
A0, which we assume.
Our motivation comes from the proof in 4.1.2 (using an algebraization argument
at the end of 4.1.3) that the counterexample in 2.3.1 to CM lifting over a normal
local domain satisfies this property. In general, after an easy reduction to the
isotypic case, we apply the Serre-Tate deformation theorem to localize the problem
at p-adic places v of the maximal totally real subfield L+
of a CM field L ⊆
End0
(A0) of degree 2 dim(A0). This reduces the existence of a CM lifting for the
abelian variety A0 to a corresponding problem for the CM p-divisible group A0[v∞
]
attached to v.8
We formulate several properties of v with respect to the CM field L; any one
of them ensures the existence of a CM lifting of A0[v∞
]κ after applying a κ-isogeny
to A0[v∞
] (see 4.1.6, 4.1.7, and 4.5.7). These properties involve the ramification
and residue fields of L and L+
relative to v. If v violates all of these properties
then we call it bad (with respect to L/L+
and κ). Let Lv := L ⊗L+ L+
v . After
applying a preliminary κ-isogeny to arrange that OL ⊂ End(A0), for v that are
not bad we apply an OL-linear κ-isogeny to arrange that the Lie type of the OL,v-
factor of Lie(A0) (i.e., its class in a certain K-group of (OL,v/(p)) ⊗ κ-modules) is
“self-dual”. Under the self-duality condition (defined in 4.4.3) we produce an OL,v-
linear CM lifting of A0[v∞
]κ by specializing a suitable OL,v-linear CM v-divisible
group in mixed characteristic; see 4.4.6. We use an argument with deformation
rings to eliminate the intervention of κ: if every p-adic place v of L+
is not bad
then there exists a κ-isogeny A0 → B0 such that OL ⊂ End(B0) and the pair
(B0, OL → End(B0)) admits a lift to characteristic 0 without increasing κ.
If some p-adic place v of the totally real field L+
is bad then the above argument
does not work because in that case no member of the OL,v-linear κ-isogeny class of
the p-divisible group A0[v∞
] has a self-dual Lie type. Instead we change A0[v∞
] by a
suitable OL,v-linear κ-isogeny so that its Lie type becomes as symmetric as possible,
a condition whose precise formulation is called “striped”. Such a p-divisible group
is shown to be isomorphic to the Serre tensor construction applied to a special class
of 2-dimensional p-divisible groups of height 4 that are similar to the ones arising
from the abelian surface counterexamples in 2.3.1; we call these toy models (see
4.1.3, especially 4.1.3.2).
These “toy models” are sufficiently special that we can analyze their CM lift-
ing properties directly; see 4.2.10 and 4.5.15(iii). After this key step we deduce
the existence of a CM lifting of A0[v∞
]κ from corresponding statements for (the
p-divisible group version of) toy models. In the final step, once again we use de-
formation theory to produce an abelian variety B0 isogenous to (the original) A0
over κ and a CM lifting of B0 over a possibly non-normal 1-dimensional complete
local noetherian domain of characteristic 0 with residue field κ. Although OL acts
8See 1.4.5.3 for the statement of the Serre–Tate deformation theorem, and 2.2.3 and 4.6.3.1
for a precise statement of the algebraization criterion that is used in this localization step.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-21-320.jpg)
![INTRODUCTION 7
on the closed fiber, we can only ensure that a subring of OL of finite index9
acts
on the lifted abelian scheme (see 4.6.4).
Appendix A. In Appendix A.1 we provide a self-contained development of the
proof of the p-part of Tate’s isogeny theorem for abelian varieties over finite fields of
characteristic p, as well as a proof of Tate’s formula for the local invariants at p-adic
places for endomorphism algebras of simple abelian varieties over such fields. (An
exposition of these results is also given in [79]; our treatment uses less input from
non-commutative algebra.) Appendices A.2 and A.3 provide purely algebraic proofs
of the Main Theorem of Complex Multiplication for abelian varieties, as well as a
converse result, both of which are used in essential ways in Chapter 2. In Appendix
A.4 we use Shimura’s method to show that an algebraic Hecke character with a
given algebraic part can be constructed over the field of moduli of the algebraic
part, with control over places of bad reduction.
In the special case of the reflex norm of a CM type (L, Φ), combining this
construction of algebraic Hecke characters with the converse to the Main Theorem
of CM in A.3 proves that over the associated field of moduli M ⊂ Q (a subfield of
the Hilbert class field of the reflex field E(L, Φ)) there exists a CM abelian variety
A with CM type (L, Φ) such that A has good reduction at all p-adic places of
M; see A.4.6.5. Since M is the smallest possible field of definition given (L, Φ),
this existence result is optimal in terms of its field of definition. Typically M =
E(L, Φ), and this is regarded as a “class group obstruction” to finding A with its
CM structure by L over E(L, Φ), a well-known phenomenon in the classical CM
theory of elliptic curves.
(In the “local” setting of CM p-divisible groups over p-adic integer rings there
are no class group problems and one gets a better result: in 3.7 we use the preceding
global construction over the field of moduli to prove that for any p-adic CM type
(F, Φ) and the associated p-adic reflex field E ⊂ Qp over Qp there exists a CM
p-divisible group over OE with p-adic CM type (F, Φ).)
Appendix B. In Appendices B.1 and B.2, we give two versions of a more di-
rect (but more complicated) proof of the existence of CM liftings for a higher-
dimensional generalization of the toy model.10
The first version uses Raynaud’s
theory of group schemes of type (p, . . . , p). The second version uses recent devel-
opments in p-adic Hodge theory. We hope that material described there will be
useful in the future. In Appendix B.3 we compare several Dieudonné theories over
a perfect base field of characteristic p 0. In Appendix B.4 we give a formula for
the Dieudonné module of the closed fiber of a finite flat commutative group scheme,
constructed using integral p-adic Hodge theory; this formula is used in B.2.
9This subring of finite index can be taken to be Z + pOL.
10In the original proof of our main CM lifting result in 4.1.1, the case of a bad place v|p of
L+ was reduced through the Serre tensor construction to this existence result. Both B.1 and B.2
are logically independent of results in Chapter 4. Readers who cannot wait to see a proof of the
existence of a CM lifting (without modification by any isogeny) for such a higher-dimensional toy
model may proceed directly to B.1 or B.2, after consulting 4.2 for the definition of the Lie type
of an O-linear p-divisible group and related notation.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-22-320.jpg)
![8 INTRODUCTION
References
(1) Abelian varieties. In Mumford’s book [82] the theory of abelian varieties
is developed over an algebraically closed base field, and we need the theory
over a general field; references addressing this extra generality are Milne’s
article [76] (which rests on [82]) and the forthcoming book [45]. Since [45]
is not yet in final form we do not refer to it in the main text, but the reader
should keep in mind that many results for which we refer to [82] and [76]
are also treated in [45]. We refer the reader to [83, Ch. 6, §1–§2] for a self-
contained development of the elementary properties of abelian schemes, which
we freely use. (For example, the group law is necessarily commutative and is
determined by the identity section, as in the theory over a field.)
(2) Semisimple algebras. We assume familiarity with the classical theory of
finite-dimensional semisimple algebras over fields (including the theory of their
splitting fields and maximal commutative subfields). A suitable reference for
this material is [53, §4.1–4.6]; another reference is [11]. In 1.2.2–1.2.4 we
review some of the facts that we need from that theory.
(3) Descent theory and formal schemes. In many places, we need to use the
techniques of descent theory and Grothendieck topologies (especially the fppf
topology, though in some situations we use the fpqc topology to perform de-
scent from a completion). This is required for arguments with group schemes,
even over a field, such as in considerations with short exact sequences. For
accounts of descent theory, we refer the reader to [10, §6.1–6.2], and to [39,
Part 1] for a more exhaustive discussion. These techniques are discussed in a
manner well-suited to group schemes in [98] and [30, Exp. IV–VIA].
Our arguments with deformation theory rest on the theory of formal
schemes, especially Grothendieck’s formal GAGA and algebraization theo-
rems. A succinct overview of these matters is given in [39, Part 4], and the
original references [34, I, §10; III1, §4–§5] are also highly recommended.
(4) Dieudonné theory and p-divisible groups. To handle p-torsion phenom-
ena in characteristic p 0 we use Dieudonné theory and p-divisible groups.
Brief surveys of some basic definitions and properties in this direction are
given in 1.4, 3.1.2–3.1.6, and B.3.5.1–B.3.5.5. We refer the reader to [119],
[71] and [110, §6] for more systematic discussions of basic facts concerning
p-divisible groups, and to [29] and [41, Ch. II–III] for self-contained devel-
opments of (contravariant) Dieudonné theory, with applications to p-divisible
groups. Contravariant Dieudonné theory is used in Chapters 1–4.
Covariant Dieudonné theory is used in Appendix B.1 because the alter-
native proof there of the main result of Chapter 4 uses a covariant version of
p-adic Hodge theory. A brief summary of covariant Dieudonné can be found
in B.3.5.6–B.3.6.7. We recommend [136] for Cartier theory; an older standard
reference is [69].
A very useful technique within the deformation theory of p-divisible groups
is Grothendieck–Messing theory, which is developed from scratch in [75]. Al-
though we do not provide an introduction to this topic, we hope that our
applications of it may inspire an interested reader who is not familiar with
this technique to learn more about it.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-23-320.jpg)
![NOTATION AND TERMINOLOGY 9
Notation and terminology
• Numerical labeling of text items and displayed expressions.
– We use “x.y.z”, “x.y.z.w”, etc. for text items (sub-subsections, results,
remarks, definitions, etc.), arranged lexicographically without repetition.
– Any labeling of displayed expressions (equations, commutative diagrams,
etc.) is indicated with parentheses, so “see (x.y.z)” means that one should
look at the zth displayed expression in subsection x.y. This convention
avoids confusion with the use of “x.y.z” to label a text item.
– Any label for a text item is uniquely assigned, so even though “see x.y.z”
does not indicate if it is a sub-subsection or theorem (or lemma, etc.),
there is no ambiguity for finding it in this book.
• Convention on notation.
– p denotes a prime number.
– CM fields are usually denoted by L.
– K often stands for an arbitrary field, κ is usually used to denote either a
residue field or a finite field of characteristic p.
– V ∨
denotes the dual of a finite-dimensional vector space V over a field.
– k denotes a perfect field, often of characteristic p 0. In 4.2–4.6, k is an
algebraically closed field of characteristic p.
– K0 is the fraction field of W(k), where k is a perfect field of characteristic
p 0 and W(k) is the ring of p-adic Witt vectors with entries in k.
– Abelian varieties are usually written as A, B, or C, and p-divisible groups
are often denoted as G or as X or Y .
– The p-divisible group attached to an abelian variety or an abelian scheme
A is denoted by A[p∞
]; its subgroup scheme of pn
-torsion points is A[pn
].
• Fields and their extensions.
– For a field K, we write K to denote an algebraic closure and Ks to denote
a separable closure.
– An extension of fields K
/K is primary if K is separably algebraically
closed in K
(i.e., the algebraic closure of K in K
is purely inseparable
over K).
– For a number field L we write OL to denote its ring of integers. Similar
notation is used for non-archimedean local fields.
– If q is a power of a prime p, Fq denotes a finite field with size q (sometimes
understood to be the unique subfield of order q in a fixed algebraically
closed field of characteristic p). If κ and κ
are abstract finite fields with
respective sizes q = pn
and q
= pn
for integers n, n
⩾ 1 then κ ∩ κ
denotes the unique subfield of either κ or κ
with size pgcd(n,n
)
; the
context will always make clear if this is being considered as a subfield of
either κ or κ
. Likewise, κκ
denotes κ ⊗κ∩κ κ
, a common extension of
κ and κ
with size plcm(n,n
)
.
• Base change.
– If T → S is a map of schemes and S
is an S-scheme, then TS denotes
the S
-scheme T ×S S
if S is understood from context.
– When S = Spec(R) and S
= Spec(R
) are affine, we may write TR
to denote T ⊗R R
:= T ×Spec(R) Spec(R
) when R is understood from
context.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-24-320.jpg)
 and
V(A) denotes Q ⊗Z
T(A).
– For any abelian varieties A and B over a field K, Hom(A, B) denotes
the group of homomorphisms A → B over K, and Hom0
(A, B) denotes
Q ⊗Z Hom(A, B).
(Since Hom(A, B) → Hom(AK, BK) is injective, Hom(A, B) is a finite
free Z-module since the same holds over K by [82, §19, Thm. 3].)
– When B = A we write End(A) and End0
(A) respectively, and call End0
(A)
the endomorphism algebra of A (over K). The endomorphism algebra
End0
(A) is an invariant which only depends on A up to isogeny over K,
in contrast with the endomorphism ring End(A).
– We write A ∼ B to denote that abelian varieties A and B over K are
K-isogenous.
– To avoid any possible confusion with notation found in the literature, we
emphasize that what we call Hom(A, B) and Hom0
(A, B) are sometimes
denoted by others as HomK(A, B) and Hom0
K(A, B).11
• Adeles and local fields.
– We write AL to denote the adele ring of a number field L, AL,f to denote
the factor ring of finite adeles, and A and Af in the case L = Q.
– If v is a place of a number field L then Lv denotes the completion of L
with respect to v; OL,v denotes the valuation ring OLv
of Lv in case v is
non-archimedean, with residue field κv whose size is denoted qv.
– For a place w of Q we define Lw := Qw ⊗Q L =
v|w Lv, and in case w
is the -adic place for a prime we define OL, := Z ⊗Z OL =
v| OL,v.
• Class field theory and reciprocity laws.
– The Artin maps of local and global class field theory are taken with
the arithmetic normalization, which is to say that local uniformizers are
carried to arithmetic Frobenius elements.12
– recL : A×
L /L×
→ Gal(Lab
/L) denotes the arithmetically normalized glob-
al reciprocity map for a number field L.
– The composition of A×
L A×
L /L×
with recL is denoted rL.
– For a non-archimedean local field F we write rF : F×
→ Gal(Fab
/F) to
denote the arithmetically normalized local reciprocity map.
• Frobenius and Verschiebung.
– For a commutative group scheme N over an Fp-scheme S, N(p)
denotes
the base change of N by the absolute Frobenius endomorphism of S. The
relative Frobenius homomorphism is denoted FrN/S : N → N(p)
, and the
11with the notation Hom(A, B) and Hom0
(A, B) then reserved to mean the analogues for
AK and BK over K, or equivalently for AKs and BKs over Ks (see Lemma 1.2.1.2).
12Recall that for a non-archimedean local field F with residue field of size q, an element of
Gal(Fs/F) is called an arithmetic (resp. geometric) Frobenius element if its effect on the residue
field of Fs is the automorphism x → xq (resp. x → x1/q); this automorphism of the residue
field is likewise called the arithmetic (resp. geometric) Frobenius automorphism. We choose the
arithmetic normalization of class field theory so that uniformizers correspond to Frobenius endo-
morphisms of abelian varieties in the Main Theorem of Complex Multiplication.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-25-320.jpg)
![NOTATION AND TERMINOLOGY 11
Verschiebung homomorphism for S-flat N of finite presentation denoted
VerN/S : N(p)
→ N see [30, VIIA, 4.2–4.3]). If S is understood from
context then we may denote these as FrN and VerN respectively.
For n ⩾ 1, the pn
-fold relative Frobenius and Verschiebung homomor-
phisms N → N(pn
)
and N(pn
)
→ N are respectively denoted FrN/S,pn
and VerN/S,pn .
– For a perfect field k with char(k) = p 0 and the unique lift σ : W(k) →
W(k) of the Frobenius automorphism y → yp
of k, a Dieudonné module
over k is a W(k)-module M equipped with additive endomorphisms F :
M → M and V : M → M such that F ◦ V = [p]M = V ◦ F, F(c m) =
σ(c) F(m), and c V(m) = V(σ(c) m) for all c ∈ W(k) and m ∈ M; these
are the left modules over the Dieudonné ring Dk (see 1.4.3.1).
– The semilinear operators F and V on a Dieudonné module M corre-
spond to respective W(k)-linear maps M(p)
→ M and M → M(p)
, where
M(p)
:= W(k) ⊗σ,W (k) M.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-26-320.jpg)
![CHAPTER 1
Algebraic theory of complex multiplication
The theory of complex multiplication. . . is not only the most beautiful
part of mathematics but also of all science.
— David Hilbert
1.1. Introduction
1.1.1. Lifting questions. A natural question early in the theory of abelian vari-
eties is whether every abelian variety in positive characteristic admits a lift to char-
acteristic 0. That is, given an abelian variety A0 over a field κ with char(κ) 0,
does there exist a local domain R of characteristic zero with residue field κ and
an abelian scheme A over R whose special fiber Aκ is isomorphic to A0? We may
also wish to demand that a specified polarization of A0 or subring of the endomor-
phism algebra of A0 (or both) also lifts to A. (The functor A Aκ from abelian
R-schemes to abelian varieties over κ is faithful, by consideration of finite étale
torsion levels; see the beginning of 1.4.4.)
Suppose there is an affirmative solution A to such a lifting problem over some
local domain R as above. Let’s see that we can arrange for a solution to be found
over a local noetherian domain (that is even complete). This rests on a direct limit
technique (that is very useful throughout algebraic geometry), as follows. Observe
that for the directed system of noetherian local subrings Ri with local inclusions
Ri → R, we have R = lim
−
→
Ri. In [34, IV3, §8–§12; IV4, §17] there is an exhaustive
development of the technique of descent through direct limits. The principle is
that if {Di} is a directed system of rings with limit D, and if we are given a
“finitely presented” algebro-geometric situation over D (a diagram of finitely many
D-schemes of finite presentation, equipped with with finitely many D-morphisms
among them and perhaps some finitely presented quasi-coherent sheaves on them,
some of which may be D-flat, etc.) then the entire structure descends to Di for
sufficiently large i. Moreover, if we increase i enough then we can also descend
“reasonable” properties (such as flatness for morphisms or sheaves, and properness,
surjectivity, smoothness, and having geometrically connected fibers for morphisms),
any two descents become isomorphic after increasing i some more, and so on.
The results of this direct limit formalism are intuitively plausible, but their
proofs can be rather non-obvious to the uninitiated (e.g., descending the properties
of flatness and surjectivity). We will often use this limit formalism without much
explanation, and we hope that the plausibility of such results is sufficient for a non-
expert reader to follow the ideas. Everything we need is completed proved in the
cited sections of [34]. As a basic example, since the condition of being an abelian
13](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-28-320.jpg)
![14 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
scheme amounts to a group scheme diagram for a smooth proper R-scheme having
geometrically connected fibers, the abelian scheme A over R descends to an abelian
scheme over Ri0
for some sufficiently large i0.
The residue field κi0
of Ri0
is merely a subfield of κ. By [34, 0III, 10.3.1],
there is a faithfully flat local extension Ri0
→ R
with R
noetherian and having
residue field κ over κi0
. By faithful flatness, every minimal prime of
R has residue
characteristic 0, so we can replace
R with its quotient by such a prime to obtain a
solution over a complete local noetherian domain with residue field κ.
Typically our liftings will be equipped with additional structure such as a po-
larization, and so the existence of an affirmative solution for our lifting problem
(for a given A0) often amounts to an appropriate deformation ring R for A0 (over a
Cohen ring for κ) admitting a generic point in characteristic 0; the coordinate ring
of the corresponding irreducible component of Spec(R) is such an R. If κ → κ
is
an extension of fields and W → W
is the associated extension of Cohen rings then
often there is a natural isomorphism R
W
⊗W R relating the corresponding de-
formation rings for A0 and (A0)κ (see 1.4.4.5, 1.4.4.13, and 1.4.4.14). Thus, if R
has a generic point of characteristic 0 then so does R. Hence, to prove an affirmative
answer to lifting questions as above it is usually enough to consider algebraically
closed κ. For example, the general lifting problem for polarized abelian varieties
(allowing polarizations for which the associated symmetric isogeny A0 → At
0 is not
separable) was solved affirmatively by Norman-Oort [85, Cor. 3.2] when κ = κ, and
the general case follows by deformation theory (via 1.4.4.14 with O = Z).
1.1.2. Refinements. When a lifting problem as above has an affirmative solution,
it is natural to ask if the (complete local noetherian) base ring R for the lifting
can be chosen to satisfy nice ring-theoretic properties, such as being normal or a
discrete valuation ring. Slicing methods allow one to find an R with dim(R) = 1
(see 2.1.1 for this argument), but normalization generally increases the residue field.
Hence, asking that the complete local noetherian domain R be normal or a discrete
valuation ring with a specified residue field κ is a non-trivial condition unless κ is
algebraically closed.
We are interested in versions of the lifting problem for finite κ when we lift not
only the abelian variety but also a large commutative subring of its endomorphism
algebra. To avoid counterexamples it is sometimes necessary to weaken the lifting
problem by permitting the initial abelian variety A0 to be replaced with another
in the same isogeny class over κ. In 1.8 we will precisely formulate several such
lifting problems involving complex multiplication, and the main result of our work
is a rather satisfactory solution to these lifting problems.
1.1.3. Purpose of this chapter. Much of the literature on complex multiplica-
tion involves either (i) working over an algebraically closed ground field, (ii) making
unspecified finite extensions of the ground field, or (iii) restricting attention to sim-
ple abelian varieties. To avoid any uncertainty about the degree of generality in
which various foundational results in the theory are valid, as well as to provide a
convenient reference for subsequent considerations, in this chapter we provide an
extensive review of the algebraic theory of complex multiplication over a general
base field. This includes special features of the theory over finite fields and over
fields of characteristic 0, and for some important proofs we refer to the original
literature (e.g., papers of Tate). Some arithmetic aspects (such as reflex fields and](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-29-320.jpg)
![1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 15
the Main Theorem of Complex Multiplication) are discussed in Chapter 2, and
Appendix A provides proofs of the Main Theorem of Complex Multiplication and
some results of Tate over finite fields.
1.2. Simplicity, isotypicity, and endomorphism algebras
1.2.1. Simple abelian varieties. An abelian variety A over a field K is simple
(over K) if it is non-zero and contains no non-zero proper abelian subvarieties.
Simplicity is not generally preserved under extension of the base field; see Example
1.6.3 for some two-dimensional examples over finite fields and over Q. An abelian
variety A over K is absolutely simple (over K) if AK is simple.
1.2.1.1. Lemma. If A is absolutely simple over a field K then for any field ex-
tension K
/K, the abelian variety AK over K
is simple.
Proof. This is an application of direct limit and specialization arguments, as we
now explain. Assume for some K
/K that there is a non-zero proper abelian sub-
variety B
⊂ AK . By replacing K
with an algebraic closure we may arrange that
K
and then especially K is algebraically closed. (The algebraically closed property
for K
is unimportant, but it is crucial that we have it for K.) By expressing K
as a direct limit of finitely generated K-subalgebras, there is a finitely generated
K-subalgebra R ⊂ K
such that B
= BK for an abelian scheme B → Spec(R)
that is a closed R-subgroup of AR.
The constant positive dimension of the fibers of B → Spec(R) is strictly less
than dim(A), as we may check using the K
-fiber B
⊂ AK . Since K is algebraically
closed we can choose a K-point x of Spec(R). The fiber Bx is a non-zero proper
abelian subvariety of A, contrary to the simplicity of A over K.
For a pair of abelian varieties A and B over a field K, Hom0
(AK , BK ) can be
strictly larger than Hom0
(A, B) for some separable algebraic extension K
/K. For
example, if E is an elliptic curve over Q then considerations with the tangent line
over Q force End0
(E) = Q, but it can happen that End0
(EL) = L for an imaginary
quadratic field L (e.g., E : y2
= x3
− x and L = Q(
√
−1)).
Scalar extension from number fields to C or from an imperfect field to its perfect
closure are useful techniques in the study of abelian varieties, so there is natural
interest in considering ground field extensions that are not separable algebraic (e.g.,
non-algebraic or purely inseparable). It is an important fact that allowing such
general extensions of the base field does not lead to more homomorphisms:
1.2.1.2. Lemma (Chow). Let K
/K be an extension of fields that is primary (i.e.,
K is separably algebraically closed in K
). For abelian varieties A and B over K,
the natural map Hom(A, B) → Hom(AK , BK ) is bijective.
Proof. See [23, Thm. 3.19] for a proof using faithfully flat descent (which is
reviewed at the beginning of [23, §3]). An alternative proof is to show that the
locally finite type Hom-scheme Hom(A, B) over K is étale.
We shall be interested in certain commutative rings acting faithfully on abelian
varieties, so we need non-trivial information about the structure of endomorphism
algebras of abelian varieties. The study of such rings rests on the following funda-
mental result.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-30-320.jpg)
![16 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
1.2.1.3. Theorem (Poincaré reducibility). Let A be an abelian variety over a field
K. For any abelian subvariety B ⊂ A, there is is abelian subvariety B
⊂ A such
that the multiplication map B × B
→ A is an isogeny.
In particular, if A = 0 then there exist pairwise non-isogenous simple abelian
varieties C1, . . . , Cs over K such that A is isogenous to
Cei
i for some ei ⩾ 1.
Proof. When K is algebraically closed this result is proved in [82, §19, Thm. 1].
The same method works for perfect K, as explained in [76, Prop. 12.1]. (Perfectness
is implicit in the property that the underlying reduced scheme of a finite type K-
group is a K-subgroup. For a counterexample over any imperfect field, see [25,
Ex. A.3.8].) The general case can be pulled down from the perfect closure via
Lemma 1.2.1.2; see the proof of [23, Cor. 3.20] for the argument.
1.2.1.4. Corollary. For a non-zero abelian variety A over a field K and a primary
extension of fields K
/K, every abelian subvariety B
of AK has the form BK for
a unique abelian subvariety B ⊂ A.
Proof. By the Poincaré reducibility theorem, abelian subvarieties of A are pre-
cisely the images of maps A → A, and similarly for AK . Since scalar extension
commutes with the formation of images, the assertion is reduced to the bijectivity
of End(A) → End(AK ), which follows from Lemma 1.2.1.2.
Since any map between simple abelian varieties over K is either 0 or an isogeny,
by general categorical arguments the collection of Ci’s (up to isogeny) in the
Poincaré reducibility theorem is unique up to rearrangement, and the multiplic-
ities ei are also uniquely determined.
1.2.1.5. Definition. The Ci’s in the Poincaré reducibility theorem (considered
up to isogeny) are the simple factors of A.
By the uniqueness of the simple factors up to isogeny, we deduce:
1.2.1.6. Corollary. Let A be a non-zero abelian variety over a field, with simple
factors C1, . . . , Cs. The non-zero abelian subvarieties of A are generated by the
images of maps Ci → A from the simple factors.
1.2.2. Central simple algebras. Using notation from the Poincaré reducibility
theorem, for a non-zero abelian variety A we have
End0
(A)
Matei
(End0
(Ci))
where {Ci} is the set of simple factors of A and the ei’s are the corresponding
multiplicities. Each End0
(Ci) is a division algebra, by simplicity of the Ci’s. Thus,
to understand the structure of endomorphism algebras of abelian varieties we need
to understand matrix algebras over division algebras, especially those of finite di-
mension over Q. We therefore next review some general facts about such rings.
Although we have used K to denote the ground field for abelian varieties above,
in what follows we will use K to denote the ground field for central simple algebras;
the two are certainly not to be confused, since for abelian varieties in positive
characteristic the endomorphism algebras are over fields of characteristic 0.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-31-320.jpg)
![1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 17
1.2.2.1. Definition. A central simple algebra over a field K is a non-zero asso-
ciative K-algebra of finite dimension such that K is the center and the underlying
ring is simple (i.e., has no non-trivial two-sided ideals).
A central division algebra over K is a central simple algebra over K whose
underlying ring is a division algebra.
Among the most basic examples of central simple algebras over a field K are
the matrix algebras Matn(K) for n ⩾ 1. The most general case is given by:
1.2.2.2. Proposition (Wedderburn’s Theorem). Every central simple algebra D
over a field K is isomorphic to Matn(Δ) = EndΔ(Δ⊕n
) for some n ⩾ 1 and some
central division algebra Δ over K (where Δ⊕n
is a left Δ-module). Moreover, n is
uniquely determined by D, and Δ is uniquely determined up to K-isomorphism.
Proof. This is a special case of a general structure theorem for simple rings; see
[53, Thm. 4.2] and [53, §4.4, Lemma 2].
In addition to matrix algebras, another way to make new central simple algebras
from old ones is to use tensor products:
1.2.2.3. Lemma. If D and D
are central simple algebras over a field K, then so
is D ⊗K D
. For any extension field K
/K, DK := K
⊗K D is a central simple
K
-algebra.
Proof. The first part is [53, §4.6, Cor. 3]; the second is [53, §4.6, Cor. 1, 2].
1.2.3. Splitting fields. It is a general fact that for any central division algebra
Δ over a field K, ΔKs
is a matrix algebra over Ks (so [Δ : K] is a square). In
other words, Δ is split by a finite separable extension of K. There is a refined
structure theory concerning splitting fields and maximal commutative subfields of
central simple algebras over fields; [53, §4.1–4.6] gives a self-contained development
of this material. An important result in this direction is:
1.2.3.1. Proposition. Let D be a central simple algebra over a field F, with
[D : F] = n2
. An extension field F
/F with degree n embeds as an F-subalgebra of
D if and only if F
splits D (i.e., DF Matn(F
)). Moreover, if D is a division
algebra then every maximal commutative subfield of D has degree n over F.
Proof. The first assertion is a special case of [53, Thm. 4.12]. Now assume that
D is a division algebra and consider a maximal commutative subfield F
. In such
cases F
splits D (by [53, §4.6, Cor. to Thm. 4.8]), so n|[F
: F] by [53, Thm. 4.12].
To establish the reverse divisibility it suffices to show that for any central simple
algebra D of dimension n2
over F, every commutative subfield of D has F-degree
at most n. If A is any simple F-subalgebra of D and its centralizer in D is denoted
ZD(A) then n2
= [A : F][ZD(A) : F] by [53, §4.6, Thm. 4.11]. Thus, if A is also
commutative (so A is contained in ZD(A)) then [A : F] ⩽ n.
The second assertion in Proposition 1.2.3.1 does not generalize to central simple
algebras; e.g., perhaps D = Matn(F) with F having no degree-n extension fields.
In general, for a splitting field F
/F of a central simple F-algebra D, the
choice of isomorphism DF Matn(F
) is ambiguous up to composition with the](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-32-320.jpg)
![18 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
action of AutF (Matn(F
)), so it is useful to determine this automorphism group.
The subgroup of inner automorphisms is GLn(F
)/F×
, arising from conjugation
against elements of Matn(F
)×
= GLn(F
). In general, the inner automorphisms
are the only ones:
1.2.3.2. Theorem (Skolem–Noether). For a central simple algebra D over a field
F, the inclusion D×
/F×
→ AutF (D) carrying u ∈ D×
to (d → udu−1
) is an
equality. That is, all automorphisms are inner.
Proof. This is [53, §4.6, Cor. to Thm. 4.9].
We finish our discussion of central simple algebras by using the Skolem–Noether
theorem to build the K-linear reduced trace map TrdD/K : D → K for a central
simple algebra D over a field K.
1.2.3.3. Construction. Let D be a central simple algebra over an arbitrary field
K. It splits over a separable closure Ks, which is to say that there is a Ks-algebra
isomorphism f : DKs
Matn(Ks) onto the n × n matrix algebra for some n ⩾ 1.
By the Skolem-Noether theorem, all automorphisms of a matrix algebra are given
by conjugation by an invertible matrix. Hence, f is well-defined up to composition
with an inner automorphism.
The matrix trace map Tr : Matn(Ks) → Ks is invariant under inner automor-
phisms and is equivariant for the natural action of Gal(Ks/K), so the composition of
the matrix trace with f is a Ks-linear map DKs
→ Ks that is independent of f and
Gal(Ks/K)-equivariant. Thus, this descends to a K-linear map TrdD/K : D → K
that is defined to be the reduced trace. In other words, the reduced trace map is
a twisted form of the usual matrix trace, just as D is a twisted form of a matrix
algebra. (For d ∈ D, the K-linear left multiplication map x → d · x on D has
trace
[D : K] TrdD/K(x), as we can see by scalar extension to Ks and a direct
computation for matrix algebras. The elimination of the coefficient
[D : K] is the
reason for the word “reduced”.)
1.2.4. Brauer groups. For applications to abelian varieties it is important to
classify division algebras of finite dimension over Q (such as the endomorphism
algebra of a simple abelian variety over a field). If Δ is such a ring then its center
Z is a number field and Δ is a central division algebra over Z. More generally, the
set of isomorphism classes of central division algebras over an arbitrary field has an
interesting abelian group structure. This comes out of the following definition.
1.2.4.1. Definition. Central simple algebras D and D
over a field K are similar
if there exist n, n
⩾ 1 such that the central simple K-algebras D ⊗K Matn(K) =
Matn(D) and D
⊗K Matn (K) = Matn (D
) are K-isomorphic.
The Brauer group Br(K) is the set of similarity classes of central simple algebras
over K, and [D] denotes the similarity class of D. For classes [D] and [D
], define
[D][D
] := [D ⊗K D
].
This composition law on Br(K) is well-defined and makes it into an abelian
group with inversion given by [D]−1
= [Dopp
], where Dopp
is the “opposite algebra”.
By Proposition 1.2.2.2, each element in Br(K) is represented (up to isomorphism)](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-33-320.jpg)
![1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 19
by a unique central division algebra over K. In this sense, Br(K) is an abelian
group structure on the set of isomorphism classes of such division algebras.
1.2.4.2. Example. The computation of the Brauer group of a number field in-
volves computing the Brauer groups of local fields, so we now clear up any possible
confusion concerning sign conventions in the description of Brauer groups for non-
archimedean local fields. Upon choosing a separable closure Ks of an arbitrary
field K, there are two natural procedures to define a functorial group isomorphism
Br(K) H2
(Ks/K, K×
s ): a conceptual method via non-abelian cohomology as in
[107, Ch. X, §5] and an explicit method via crossed-product algebras. By [107,
Ch. X, §5, Exer. 2], these procedures are negatives of each other. We use the
conceptual method of non-abelian cohomology, but we do not need to make that
method explicit here and so we refer the interested reader to [107] for the details.
Let K be a non-archimedean local field with residue field κ and let Kun
denote
its maximal unramified subextension in Ks (with κ the residue field of Kun
). It
is known from local class field theory that the natural map H2
(Kun
/K, Kun×
) →
H2
(Ks/K, K×
s ) is an isomorphism, and the normalized valuation mapping Kun×
→
Z induces an isomorphism
H2
(Kun
/K, Kun×
) H2
(Kun
/K, Z)
δ
H1
(Gal(Kun
/K), Q/Z)
= H1
(Gal(κ/κ), Q/Z).
There now arises the question of choice of topological generator for Gal(κ/κ): arith-
metic or geometric Frobenius? We choose to work with arithmetic Frobenius. (In
[103, §1.1] and [107, Ch. XIII, §3] the arithmetic Frobenius generator is also used.)
Via evaluation on the chosen topological generator, our conventions lead to a
composite isomorphism
invK : Br(K) Q/Z
for non-archimedean local fields K. If one uses the geometric Frobenius convention,
then by also adopting the crossed-product algebra method to define the isomor-
phism
Br(K) H2
(Ks/K, K×
s )
one would get the same composite isomorphism invK since the two sign differences
cancel out in the composite. (Beware that in [103] and [107] the Brauer group of a
general field K is defined to be H2
(Ks/K, K×
s ), and so the issue of choosing between
non-abelian cohomology or crossed-product algebras does not arise in the founda-
tional aspects of the theory. However, this issue implicitly arises in the relationship
of Brauer groups and central simple algebras, such as in [103, Appendix to §1]
where the details are omitted.)
Since Br(R) is cyclic of order 2 and Br(C) is trivial, for archimedean local fields
K there is a unique injective homomorphism invK : Br(K) → Q/Z.
By [103, §1.1, Thm. 3], for a finite extension K
/K of non-archimedean local
fields, composition with the natural map rK
K : Br(K) → Br(K
) satisfies
(1.2.4.1) invK ◦ rK
K = [K
: K] · invK.
By [107, Ch. XIII, §3, Cor. 3], invK(Δ) has order
[Δ : K] for any central division
algebra Δ over K. These assertions are trivially verified to hold for archimedean
local fields K as well.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-34-320.jpg)
![20 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
1.2.4.3. Theorem. Let L be a global field. There is an exact sequence
0 // Br(L) //
v Br(Lv)
invLv
// Q/Z // 0
where the direct sum is taken over all places of L and the first map is defined via
extension of scalars.
Proof. This is [120, §9.7, §11.2].
For a global field L and central division algebra Δ over L, invv(Δ) denotes
invLv
(ΔLv
). Theorem 1.2.4.3 says that a central division algebra Δ over a global
field L is uniquely determined up to isomorphism by its invariants invv(Δ), and
that these may be arbitrarily assigned subject to the conditions invv(Δ) = 0 for all
but finitely many v and
invv(Δ) = 0. Moreover, the order of [Δ] in Br(L) is the
least common “denominator” of the local invariants invv(Δ) ∈ Q/Z.
If K is any field then for a class c ∈ Br(K) its period is its order and its index is
[Δ : K] with Δ the unique central division algebra over K representing the class
c. It is a classical fact that the period divides that index and that these integers
have the same prime factors (see [107, X.5], especially Lemma 1 and Exercise 3),
but in general equality does not hold. For example, there are function fields of
complex 3-folds for which some order-2 elements in the Brauer group cannot be
represented by a quaternion algebra; examples are given in [61, §4], and there are
examples with less interesting fields as first discovered by Brauer. We have noted
above that over local fields there is equality of period and index (the archimedean
case being trivial). The following deep result is an analogue over global fields.
1.2.4.4. Theorem. For a central division algebra Δ over a global field L, the order
of [Δ] in Br(L) is
[Δ : L].
As a special (and very important) case, elements of order 2 in Br(L) are pre-
cisely the Brauer classes of quaternion division algebras for a global field L; as noted
above, this fails for more general fields. Since Theorem 1.2.4.4 does not seem to
be explicitly stated in any of the standard modern references on class field theory
(though there is an allusion to it at the end of [4, Ch. X, §2]), and the structure
theory of endomorphism algebras of abelian varieties rests on it, here is a proof.
Proof. Let Δ have degree n2
over L and let d be the order of [Δ] in Br(L), so d|n.
Note that d is the least common multiple of the local orders dv of [ΔLv
] ∈ Br(Lv)
for each place v of L, with dv = 1 for complex v, dv|2 for real v, and dv = 1 for all
but finitely many v. Using these formal properties of the dv’s, we may call upon
the full power of global class field theory via Theorem 6 in [4, Ch. X] to infer the
existence of a cyclic extension L
/L of degree d such that [L
v : Lv] is a multiple of
dv for every place v of L (here, v
is any place on L
over v, and the constraint on the
local degree is only non-trivial when dv 1). In the special case d = 2 (the only
case we will require) one only needs weak approximation and Krasner’s Lemma
rather than class field theory: take L
to split a separable quadratic polynomial
over L that closely approximates ones that define quadratic separable extensions of
Lv for each v such that dv = 2.
By (1.2.4.1), restriction maps on local Brauer groups induce multiplication by
the local degree on the local invariants, so ΔL is locally split at all places of L
.
Thus, by the injectivity of the map from the global Brauer group into the direct](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-35-320.jpg)
![1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 21
sum of the local ones (for L
) we conclude that the Galois extension L
/L of degree
d splits Δ. (The existence of cyclic splitting fields for all Brauer classes is proved
for number fields in [120] and is proved for all global fields in [128], but neither
reference seems to control the degree of the global cyclic extension.) It is a general
fact for Brauer groups of arbitrary fields [107, Ch. X, §5, Lemma 1] that every
Brauer class split by a Galois extension of degree r is represented by a central
simple algebra with degree r2
. Applying this fact from algebra in our situation,
[Δ] = [D] for a central simple algebra D of degree d2
over L. But each Brauer class
is represented by a unique central division algebra, and so D must be L-isomorphic
to a matrix algebra over Δ. Since [D : L] = d2
and [Δ : L] = n2
with d|n, this
forces d = n as desired.
1.2.5. Homomorphisms and isotypicity. The study of maps between abelian
varieties over a field rests on the following useful injectivity result.
1.2.5.1. Proposition. Let A and B be abelian varieties over a field K. For any
prime (allowing = char(K)), the natural map
Z ⊗Z Hom(A, B) → Hom(A[∞
], B[∞
])
is injective, where the target is the Z-module of maps of -divisible groups over K
(i.e., compatible systems of K-group maps A[n
] → B[n
] for all n ⩾ 1).
Proof. Without loss of generality, K is algebraically closed (and hence perfect).
When = char(K) the assertion is a reformulation of the well-known analogous
injectivity with -adic Tate modules (and such injectivity in turn underlies the
proof of Z-module finiteness of Hom(A, B)). The proof in terms of Tate modules
is given in [82, §19, Thm. 3] for = char(K), and when phrased in terms of -
divisible groups it works even when = p = char(K) 0. For the convenience
of the reader, we now provide the argument for = p in such terms. We will use
that the torsion-free Z-module Hom(A, B) is finitely generated, and our argument
works for any (especially = char(K)).
Choose a Z-basis {f1, . . . , fn} of Hom(A, B). For c1, . . . , cn ∈ Z it suffices to
show that if
cifi kills A[] then |ci for all i. Indeed, if we can prove this then
consider the case when
cifi kills A[∞
]. Certainly ci = c
i for some c
i ∈ Z,
and (
c
ifi) · kills A[n
] for all n 0. But the map A[n
] → A[n−1
] induced by
-multiplication is faithfully flat since it is the pullback along A[n−1
] → A of the
faithfully flat map : A → A, so
c
ifi kills A[n−1
] for all n 0. In other words,
the kernel of the map in the Proposition would be -divisible, yet this kernel is a
finitely generated Z-module, so it would vanish as desired.
Now consider c1, . . . , cn ∈ Z such that
cifi kills A[]. For the purpose of
proving ci ∈ Z for all i, it is harmless to add to each ci any element of Z. Hence,
we may and do assume ci ∈ Z for all i, so
cifi : A → B makes sense and kills
A[]. Since : A → A is a faithfully flat homomorphism with kernel A[], by fppf
descent theory any K-group scheme homomorphism A → G that kills A[] factors
through : A → A (see [30, IV, 5.1.7.1] and [98]). Thus,
cifi = · h for some
h ∈ Hom(A, B). Writing h =
mifi with mi ∈ Z, we get
ci ⊗fi = ·
1⊗mifi
in Z ⊗Z Hom(A, B). This implies ci = mi for all i, so we are done.
A weakening of simplicity that is sometimes convenient is:](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-36-320.jpg)
![22 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
1.2.5.2. Definition. An abelian variety A over a field K is isotypic if it is isogenous
to Ce
for a simple abelian variety C over K with e ⩾ 1; that is, up to isogeny, A
has a unique simple factor. For a simple factor C of an abelian variety A over K,
the C-isotypic part of A is the isotypic subvariety of A generated by the images of
all maps C → A. An isotypic part of A is a C-isotypic part for some such C.
Clearly End0
(A) is a semisimple Q-algebra. It is simple if and only if A is
isotypic, and it is a division algebra if and only if A is simple.
By the Poincaré reducibility theorem, every non-zero abelian variety A over a
field K is naturally isogenous to the product of its distinct isotypic parts, and these
distinct parts admit no non-zero maps between them. Hence, if {Bi} is the set
of isotypic parts of A then End0
(A) =
End0
(Bi) with each End0
(Bi) a simple
algebra of finite dimension over Q. Explicitly, if Ci is the unique simple factor of
Bi then a choice of isogeny Bi → Cei
i defines an isomorphism from End0
(Bi) onto
the matrix algebra Matei
(End0
(Ci)) over the division algebra End0
(Ci). Beware
that the composite “diagonal” ring map End0
(Ci) → Matei
(End0
(Ci)) End0
(Bi)
is canonical only when End0
(Ci) is commutative.
In general isotypicity is not preserved by extension of the ground field. To
make examples illustrating this possibility, as well as other examples in the theory
of abelian varieties, we need the operation of Weil restriction of scalars. For a
field K and finite K-algebra K
, the Weil restriction functor ResK/K from quasi-
projective K
-schemes to separated (even quasi-projective) K-schemes of finite type
is characterized by the functorial identity ResK/K(X
)(A) = X
(K
⊗K A) for
K-algebras A. Informally, Weil restriction is an algebraic analogue of viewing
a complex manifold as a real manifold with twice the dimension. In particular, if
K
/K is an extension of fields then ResK/K(X
) is K
-smooth and equidimensional
when X
is K-smooth and equidimensional, with
dim(ResK/K(X
)) = [K
: K] · dim(X
).
We refer the reader to [10, §7.6] for a self-contained development of the con-
struction and properties of Weil restriction (replacing K with more general rings),
and to [25, A.5] for a discussion of further properties (especially of interest for
group schemes). In general the formation of Weil restriction naturally commutes
with any extension of the base field, and for K
equal to the product ring Kn
we
have that ResK/K carries a disjoint union
n
i=1 Si of quasi-projective K-schemes
(viewed as a K
-scheme) to the product
Si. Thus, the natural isomorphism
ResK/K(X
)Ks
Res(K⊗K Ks)/Ks
(X
K⊗K Ks
)
implies that if K
is a field separable over K then ResK/K(A
) is an abelian variety
over K of dimension [K
: K]dim(A
) for any abelian variety A
over K
(since
K
⊗K Ks K
[K
:K]
s ). If K
/K is a field extension of finite degree that is not
separable then ResK/K(X
) is never proper when X
is smooth and proper of
positive dimension [25, Ex. A.5.6].
1.2.6. Example. Consider a separable quadratic extension of fields K
/K and a
simple abelian variety A
over K
. Let σ ∈ Gal(K
/K) be the non-trivial element,
so K
⊗K K
K
× K
via x ⊗ y → (xy, σ(x)y). Thus, the Weil restriction
A := ResK/K(A
) satisfies AK A
× σ∗
(A
), so AK is not isotypic if and only
if A
is not isogenous to its σ-twist. Hence, for K = R examples of non-isotypic](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-37-320.jpg)
![1.3. COMPLEX MULTIPLICATION 23
AK are obtained by taking A
to be an elliptic curve over C with analytic model
C/(Z ⊕ Zτ) for τ ∈ C − R such that 1, τ, τ, ττ are Q-linearly independent. (In
Example 1.6.4 we give examples with K = Q.)
In cases when AK is non-isotypic, A is necessarily simple. Indeed, if A is not
simple then a simple factor of A would be a K-descent of a member of the isogeny
class of A
, contradicting that A
and σ∗
(A
) are not isogenous. Thus, we have
exhibited examples in characteristic 0 for which isotypicity is lost after a ground
field extension.
The failure of isotypicity to be preserved after a ground field extension does
not occur over finite fields:
1.2.6.1. Proposition. If A is an isotypic abelian variety over a finite field K then
AK is isotypic for any extension field K
/K.
Proof. By the Poincaré reducibility theorem, it is equivalent to show that
End0
(AK ) is a simple Q-algebra, so by Lemma 1.2.1.2 we may replace K
with
the algebraic closure of K in K
. That is, we can assume that K
/K is algebraic.
Writing K
= lim
−
→
K
i with {K
i} denoting the directed system of subfields of finite
degree over K, we have End(AK ) = lim
−
→
End(AK
i
). But End(AK ) is finitely gen-
erated as a Z-module, so for large enough i we have End0
(AK ) = End0
(AK
i
). We
may therefore replace K
with K
i for sufficiently large i to reduce to the case when
K
/K is of finite degree. Let q = #K.
The key point is to show that for any abelian variety B
over K
and any
g ∈ Gal(K
/K), B
and g∗
(B
) are isogenous. Since Gal(K
/K) is generated by
the q-Frobenius σq, it suffices to show that B
and B(q)
:= σ∗
q (B
) are isogenous.
The purely inseparable relative q-Frobenius morphism B
→ B(q)
(arising from
the absolute q-Frobenius map B
→ B
over the q-Frobenius of Spec(K
)) is such
an isogeny. Hence, the Weil restriction ResK/K(B
) satisfies ResK/K(B
)K
g g∗
(B
) ∼ B[K
:K]
.
Take B
to be a simple factor of AK (up to isogeny), so ResK/K(B
) is an
isogeny factor of ResK/K (AK ) ∼ A[K
:K]
. By the simplicity of A and the Poincaré
reducibility theorem, it follows that ResK/K(B
) is isogenous to a power of A.
Extending scalars, ResK/K(B
)K is therefore isogenous to a power of AK . But
ResK/K(B
)K ∼ B[K
:K]
, so non-trivial powers of AK and B
are isogenous. By
the simplicity of B
and Poincaré reducibility, this forces B
to be the only simple
factor of AK (up to isogeny), so AK is isotypic.
1.3. Complex multiplication
1.3.1. Commutative subrings of endomorphism algebras. The following
fact motivates the study of complex multiplication in the sense that we shall con-
sider.
1.3.1.1. Theorem. Let A be an abelian variety over a field K with g := dim(A)
0, and let P ⊂ End0
(A) be a commutative semisimple Q-subalgebra. Then [P : Q] ⩽
2g, and if equality holds then P is its own centralizer in End0
(A). If equality holds](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-38-320.jpg)
![24 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
and moreover P is a field of degree 2g over Q then A is isotypic and P is a maximal
commutative subfield of End0
(A).
Proof. Consider the decomposition P =
Li into a product of fields. Using
the primitive idempotents of P, we get a corresponding decomposition
Ai of A
in the isogeny category of abelian varieties over K, with each Ai = 0 and each Li
a commutative subfield of End0
(Ai) compatibly with the inclusion
End0
(Ai) ⊂
End0
(A) and the equality
Li = P. Since dim(A) =
dim(Ai), to prove that
[P : Q] ⩽ 2g it suffices to treat the Ai’s separately, which is to say that we may
and do assume that P = L is a field.
Since D = End0
(A) is of finite rank over Q, clearly [L : Q] is finite. Choose
a prime different from char(K). Recall that V(A) denotes Q ⊗Z
T(A) for
T(A) := lim
←
−
A[n
](Ks). The injectivity of the natural map
L := Q ⊗Q L → EndQ
(V(A))
(see Proposition 1.2.5.1) implies that L acts faithfully on the Q-vector space V(A)
of rank 2g. But L =
w| Lw, where w runs over all -adic places of L, so each
corresponding factor module V(A)w over Lw is non-zero as a vector space over Lw.
Hence,
2g = dimQ
V(A) =
w|
dimQ
V(A)w ⩾
w|
[Lw : Q] = [L : Q]
with equality if and only if V(A) is free of rank 1 over L.
Assume that equality holds, so V(A) is free of rank 1 over L. If A is not isotypic
then by passing to an isogenous abelian variety we may arrange that A = B × B
with B and B
non-zero abelian varieties such that Hom(B, B
) = 0 = Hom(B
, B).
Hence, End0
(A) = End0
(B) × End0
(B
) and so L embeds into End0
(B). But
2 dim(B) 2 dim(A) = [L : Q], so we have a contradiction (since B = 0).
It remains to prove, without assuming P is a field, that if [P : Q] = 2g then
P is its own centralizer in End0
(A). (In case P is a field, so A is isotypic and
hence End0
(A) is simple, such a centralizer property would imply that P is a
maximal commutative subfield of End0
(A), as desired.) Consider once again the
ring decomposition P =
Li and the corresponding isogeny decomposition
Ai
of A as at the beginning of this proof. We have [Li : Q] ⩽ 2 dim(Ai) for all i,
and these inequalities add up to an equality when summed over all i, so in fact
[Li : Q] = 2 dim(Ai) for all i. The preceding analysis shows that each V(Ai) is
free of rank 1 over Li, := Q ⊗Q Li, and so likewise V(A) is free of rank 1 over
P. Hence, EndP
(V(A)) = P, so if Z(P) denotes the centralizer of P in End0
(A)
then the P-algebra map
Z(P) = Q ⊗Q Z(P) → EndQ
(V(A))
is injective (Proposition 1.2.5.1) and lands inside EndP
(V(A)) = P. In other
words, the inclusion P ⊂ Z(P) of Q-algebras becomes an equality after scalar
extension to Q, so P = Z(P) as desired.
The preceding theorem justifies the interest in the following concept.
1.3.1.2. Definition. An abelian variety A of dimension g 0 over a field K
admits sufficiently many complex multiplications (over K) if there exists a commu-
tative semisimple Q-subalgebra P in End0
(A) with rank 2g over Q.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-39-320.jpg)
![1.3. COMPLEX MULTIPLICATION 25
The reason for the terminology in Definition 1.3.1.2 is due to certain examples
with K = C and P a number field such that the analytic uniformization of A(C)
expresses the P-action in terms of multiplication of complex numbers; see Example
1.5.3. The classical theory of complex multiplication focused on the case of Defini-
tion 1.3.1.2 in which P is a field, but it is useful to allow P to be a product of several
fields (i.e., a commutative semisimple Q-algebra). For example, by Theorem 1.3.1.1
this is necessary if we wish to consider the theory of complex multiplication with A
that is not isotypic, or more generally if we want Definition 1.3.1.2 to be preserved
under the formation of products. The theory of Shimura varieties provides further
reasons not to require P to be a field.
Note that we do not consider A to admit sufficiently many complex multipli-
cations merely if it does so after an extension of the base field K.
1.3.2. Example. The elliptic curve y2
= x3
−x admits sufficiently many complex
multiplications over Q(
√
−1) but not over Q. More generally, End0
(E) = Q for
every elliptic curve E over Q (since the tangent line at the origin is too small to
support a Q-linear action by an imaginary quadratic field), so in our terminology
an elliptic curve over Q does not admit sufficiently many complex multiplications.
1.3.2.1. Proposition. Let A be a non-zero abelian variety over a field K. The
following are equivalent.
(1) The abelian variety A admits sufficiently many complex multiplications.
(2) Each isotypic part of A admits sufficiently many complex multiplications.
(3) Each simple factor of A admits sufficiently many complex multiplications.
See Definition 1.2.5.2 for the terminology used in (2).
Proof. Let {Bi} be the set of isotypic parts of A, so End0
(Bi) Matei
(End0
(Ci))
where Ci is the unique simple factor of Bi and ei ⩾ 1 is its multiplicity as such.
Since End0
(A) =
End0
(Bi), (2) implies (1). It is clear that (3) implies (2).
Conversely, assume that End0
(A) contains a Q-algebra P satisfying [P : Q] =
2 dim(A). There is a unique decomposition P =
Li with fields L1, . . . , Ls, and
[Li : Q] = 2 dim(A). We saw in the proof of Theorem 1.3.1.1 that by replacing
A with an isogenous abelian variety we may arrange that A =
Ai with each Ai
a non-zero abelian variety having Li ⊂ End0
(Ai) compatibly with the embedding
End0
(Ai) ⊂ End0
(A) and the equality
Li = P. Thus, [Li : Q] ⩽ 2 dim(Ai) for
all i (by Theorem 1.3.1.1), and adding this up over all i yields an equality, so each
Ai admits sufficiently many complex multiplications using Li. Since each simple
factor of A is a simple factor of some Ai, to prove (3) we are therefore reduced to
the case when P = L is a field.
Applying Theorem 1.3.1.1 once again, L is its own centralizer in End0
(A) and
A is isotypic, say with unique simple factor C appearing with multiplicity e. In
particular, End0
(A) = Mate(D) for the division algebra D = End0
(C) of finite
rank over Q. If Z denotes the center of D then D is a central division algebra
over Z, and L contains Z since L is its own centralizer in End0
(A) = Mate(D).
Letting d = dim(C), Mate(D) contains the maximal commutative subfield L of
degree 2g/[Z : Q] = (2d/[Z : Q])e over Z.
As we noted in the proof of Proposition 1.2.3.1 (parts of which are carried
out for central simple algebras that may not be division algebras), the Z-degree of](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-40-320.jpg)
![26 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
Mate(D) is the product of the Z-degrees of L and the centralizer of L in Mate(D).
But L is its own centralizer, so
e2
[D : Z] = dimZMate(D) = [L : Z]2
= e2
(2d/[Z : Q])2
.
We conclude that 2d/[Z : Q] =
[D : Z], so (by Proposition 1.2.3.1) 2d/[Z : Q] is
the common Z-degree of all maximal commutative subfields of the central division
algebra D = End0
(C) over Z, or equivalently 2d is the Q-degree of all such fields.
But 2d = 2 dim(C), so choosing any maximal commutative subfield of D shows
that C admits sufficiently many complex multiplications.
1.3.3. CM algebras and CM abelian varieties. The following three conditions
on a number field L are equivalent:
(1) L has no real embeddings but is quadratic over a totally real subfield,
(2) for every embedding j : L → C, the subfield j(L) ⊂ C is stable under complex
conjugation and the involution x → j−1
(j(x)) in Aut(L) is non-trivial and
independent of j,
(3) there is a non-trivial involution τ ∈ Aut(L) such that for every embedding
j : L → C we have j(τ(x)) = j(x) for all x ∈ L.
The proof of the equivalence is easy. When these conditions hold, τ in (3) is unique
and its fixed field is the maximal totally real subfield L+
⊂ L (over which L is
quadratic). The case L+
= Q corresponds to the case when L is an imaginary
quadratic field.
1.3.3.1. Definition. A CM field is a number field L satisfying the equivalent
conditions (1), (2), and (3) above. A CM algebra is a product L1 × · · · × Ls of
finitely many CM fields (with s ⩾ 1).
The reason for this terminology is due to the following important result (along
with Example 1.5.3).
1.3.4. Theorem (Tate). Let A be an abelian variety of dimension g 0 over a
field K. Suppose A admits sufficiently many complex multiplications. Then there
exists a CM algebra P ⊂ End0
(A) with [P : Q] = 2 dim(A). In case A is isotypic
we can take P to be a CM field.
The proof of this theorem (which ends with the proof of Lemma 1.3.7.1) will
require some effort, especially since we consider an arbitrary base field K. Before
we start the proof, it is instructive to consider an example.
1.3.4.1. Example. Consider A = E2
with an elliptic curve E over K = C such
that L := End0
(E) is an imaginary quadratic field. The endomorphism algebra
End0
(A) = Mat2(L) is simple and contains as its maximal commutative subfields
all quadratic extensions of L. Those extensions which are biquadratic over Q are
CM fields, and the rest are not CM fields. Hence, in the setup of Theorem 1.3.4, even
when A is isotypic and char(K) = 0 there can be maximal commutative semisimple
subalgebras of End0
(A) that are not CM algebras. However, if char(K) = 0 and A
is simple (over K) then End0
(A) is a CM field; see Proposition 1.3.6.4.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-41-320.jpg)
![1.3. COMPLEX MULTIPLICATION 27
1.3.5. We will begin the proof of Theorem 1.3.4 now, but at a certain point we will
need to use deeper input concerning the fine structure of endomorphism algebras of
simple abelian varieties over general fields. At that point we will digress to review
the required structure theory, and then we will complete the argument.
By Proposition 1.3.2.1, every simple factor of A admits sufficiently many com-
plex multiplications. Thus, to prove the existence of the CM subalgebra P in Theo-
rem 1.3.4 it suffices to treat the case when A is simple. Note that in the simple case
such a CM subalgebra is automatically a field, since the endomorphism algebra is
a division algebra. Let us first show that the result in the simple case implies that
in the general isotypic case we can find P as a CM field. For isotypic A, by passing
to an isogenous abelian variety we can arrange that A = Am
for a simple abelian
variety A
over K and some m ⩾ 1. Thus, if g
= dimA
then g = mg
and End0
(A
)
contains a CM field P
of degree 2g
over Q. But End0
(A) Matm(End0
(A
)) and
this contains Matm(P
). To find a CM field P ⊂ End0
(A) of degree 2g = 2g
m over
Q it therefore suffices to construct a degree-m extension field P of P
such that P
is a CM field.
Let P+
be the maximal totally real subfield of P
, so for any totally real field
P+
of finite degree over P+
the ring P = P+
⊗P + P
is a field quadratic over P+
and it is totally complex, so it is a CM field and clearly [P : Q] = [P : P
][P
: Q] =
2g
[P+
: P+
]. Hence, to find the required CM field P in the isotypic case it suffices
to construct a degree-m totally real extension of P+
. To do this, first recall the
following basic fact from number theory [15, §6]:
1.3.5.1. Theorem (weak approximation). For any number field L and finite set
S of places of L, the map L →
v∈S Lv has dense image.
Proof. This is [15, §6].
Applying this to P+
, we can construct a monic polynomial f of degree m in
P+
[u] that is very close to a totally split monic polynomial of degree m at each
real place and is very close to an irreducible (e.g., Eisenstein) polynomial at a single
non-archimedean place. It follows that f is totally split at each real place of P+
and is irreducible over P+
, so the ring P+
= P+
[u]/(f) is a totally real field of
degree m over P+
as required.
1.3.5.2. We may and do assume for the remainder of the argument that A is
simple. In this case D = End0
(A) is a central division algebra over a number field
Z, so the commutative semisimple Q-subalgebra P ⊂ D is a field, and the proof of
Proposition 1.3.2.1 shows that the common Q-degree of all maximal commutative
subfields of D is 2g. Hence, our problem is to construct a maximal commutative
subfield of D that is a CM field.
Let TrdD/Q = TrZ/Q ◦ TrdD/Z, where TrdD/Z is the reduced trace. An abelian
variety over any field admits a polarization, so choose a polarization of A over K.
Let x → x∗
denote the associated Rosati involution on D (so (xy)∗
= y∗
x∗
and
x∗∗
= x).
1.3.5.3. Lemma. The quadratic form x → TrdD/Q(xx∗
) on D is positive-definite.
Proof. For any central simple algebra D over any field K whatsoever, let n =
[D : K] and define the variant TrmD/K : D → K of the reduced trace to be the](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-42-320.jpg)
![28 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
map that sends each y ∈ D to the trace of the K-linear map my : D → D defined by
d → yd. We have TrmD/K = n · TrdD/K, as may be checked by extending scalars
to Ks and directly computing with elementary matrices (see 1.2.3.3). Hence, in
the setting of interest with D = End0
(A) and K = Z we see that it is equivalent
to prove positive-definiteness for the quadratic form x → TrmD/Q(xx∗
), where
TrmD/Q = TrZ/Q ◦ TrmD/Z. The positive-definiteness for TrmD/Q can be verified
by replacing D with End0
K
(AK), to which [82, §21, Thm. 1] applies.
Lemma 1.3.5.3 says that x → x∗
is a positive involution of D (relative to the
linear form TrdD/Q). The existence of such an involution severely constrains the
possibilities for D. First we record the consequences for the center Z.
1.3.5.4. Lemma. The center Z of D = End0
(A) is either totally real or a CM
field, and in the latter case its canonical complex conjugation is induced by the
Rosati involution defined by any polarization of A over K.
Proof. Fix a polarization and consider the associated Rosati involution x → x∗
on the center Z of D. Clearly Z is stable under this involution. The positive-
definite TrdD/Q(xx∗
) on D restricts to
[D : Z] · TrZ/Q(xx∗
) on Z, so TrZ/Q(xx∗
)
is positive-definite on Z. If x∗
= x for all x ∈ Z then the rational quadratic
form TrZ/Q(x2
) is positive-definite on Z, so by extending scalars to R we see that
Tr(R⊗QZ)/R(x2
) is positive-definite. This forces the finite étale R-algebra R ⊗Q Z to
have no complex factors. Hence, Z is a totally real field in such cases.
It remains to show that if the involution x → x∗
is non-trivial on Z for some
choice of polarization then Z is a CM field (so the preceding argument would imply
that the Rosati involution arising from any polarization of A is non-trivial on Z)
and its intrinsic complex conjugation is equal to this involution on Z. Let Z+
be
the subfield of fixed points in Z for this involution, so [Z : Z+
] = 2 and 2 TrZ+/Q
is the restriction to Z+
of TrZ/Q. Hence, TrZ+/Q(x2
) is positive-definite on Z+
,
so Z+
is totally real. We aim to prove that Z has no real places, so we assume
otherwise and seek a contradiction.
Let v be a real place of Z. Since the involution x → x∗
is non-trivial on Z and
the field Zv R has no non-trivial field automorphisms, the real place v on Z is not
fixed by the involution x → x∗
. Thus, the real place v∗
obtained from v under the
involution is a real place of Z distinct from v, and so the positive-definiteness
of TrZ/Q(xx∗
) implies (after scalar extension to R) the positive-definiteness of
Tr(Zv×Zv∗ )/R(xx∗
), where x → x∗
on Zv × Zv∗ = R × R is the involution that
swaps the factors. In other words, this is the quadratic form (c, c
) → 2cc
, which
by inspection is not positive-definite.
1.3.6. Albert’s classification. To go further with the proof of Theorem 1.3.4,
we need to review properties of endomorphism algebras of simple abelian varieties
over arbitrary fields.
1.3.6.1. Definition. An Albert algebra is a pair consisting of a division algebra D
of finite dimension over Q and a positive involution x → x∗
on D.
For any Albert algebra D and any algebraically closed field K, there exists a
simple abelian variety A over K such that End0
(A) is Q-isomorphic to D (with the](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-43-320.jpg)
![1.3. COMPLEX MULTIPLICATION 29
given involution on D arising from a polarization on A); see [1], [2], [3], [112, §4.1,
Thm. 5], and [46, Thm. 13]. For a survey and further references on this topic, see
[92]. We will not need this result.
Instead, we are interested in the non-trivial constraints on the Albert algebras
that arise from polarized simple abelian varieties A over an arbitrary field K when
char(K) and dim A are fixed. Before listing these constraints, it is convenient to
record Albert’s classification of general Albert algebras (omitting a description of
the possibilities for the involution).
1.3.6.2. Theorem (Albert). Let (D, (·)∗
) be an Albert algebra. For any place v
of the center Z, let v∗
denote the pullback of v along x → x∗
. Exactly one of the
following occurs:
Type I: D = Z is a totally real field.
Type II: D is a central quaternion division algebra over a totally real field Z
such that D splits at each real place of Z.
Type III: D is a central quaternion division algebra over a totally real field Z
such that D is non-split at each real place of Z.
Type IV: D is a central division algebra over a CM field Z such that for all
finite places v of Z, invv(D) + invv∗ (D) = 0 in Q/Z and moreover D splits at such
a v if v = v∗
.
Proof. See [82, §21, Thm. 2] (which also records the possibilities for the involu-
tion).
1.3.6.3. Let A be a simple abelian variety over a field K, D = End0
(A), and Z the
center of D. Let Z+
be the maximal totally real subfield of Z, so either Z = Z+
or Z is a totally complex quadratic extension of Z+
. The invariants e = [Z : Q],
e0 = [Z+
: Q], d =
[D : Z], and g = dim(A) satisfy some divisibility restrictions:
• whenever char(K) = 0, the integer ed2
= [D : Q] divides 2g (proof: there
is a subfield K0 ⊆ K finitely generated over Q such that A descends to an
abelian variety A0 over K0 and the D-action on A in the isogeny category
over K descends to an action on A0 in the isogeny category over K0, so upon
choosing an embedding K0 → C we get a Q-linear action by the division
algebra D on the 2g-dimensional homology H1(A0(C), Q)),
• the action of D on V(A) with = char(K) implies (via Cor. to Thm. 4 of [82,
§19], whose proof is valid over any base field) that ed|2g,
• the structure of symmetric elements in
Q ⊗Z Hom(A, At
) Q ⊗Z Pic(A)/Pic0
(A)
(via [82, §20, Cor. to Thm. 3], whose proof is valid over any base field) yields
that [L : Q]|g for every commutative subfield L ⊂ D whose elements are
invariant under the involution.
• for Type II in any characteristic we have 2e|g (which coincides with the general
divisibility ed2
|2g when char(K) = 0 since d = 2 for Type II). To prove it
uniformly across all characteristics, first note that for Type II we have
R ⊗Q D = (R ⊗Q Z) ⊗Z D =
v|∞
Zv ⊗Z D Mat2(Zv)e
.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-44-320.jpg)
![30 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
Moreover, by [82, §21, Thm. 2] it can be arranged that under this composite
isomorphism the positive involution on D goes over to transpose on each factor
Mat2(Zv) = Mat2(R). Thus, for D of Type II the fixed part of the involution
on D has Q-dimension 2e and hence Z-degree 2. By centrality of Z in the
division algebra D, the condition x∗
= x for x in D of Type II therefore
defines a necessarily commutative quadratic extension Z
of Z inside D, so g
is divisible by [Z
: Q] = 2e as desired.
The preceding results are summarized in the following table, taken from the
end of [82, §21]. (As we have just seen, the hypothesis there that K is algebraically
closed is not necessary.) The invariants of D = End0
(A) are given in the first three
columns. In the last two columns we give some necessary divisibility restrictions
on these invariants.
Type e d char(K) = 0 char(K) 0
I e = e0 1 e | g e | g
II e = e0 2 2e | g 2e | g
III e = e0 2 2e | g e | g
IV e = 2e0 d e0d2
| g e0d | g
We refer the reader to [82, §21], and to [92] for further information on these
invariants. Using the above table, we can prove the following additional facts when
the simple A admits sufficiently many complex multiplications.
1.3.6.4. Proposition. Let A be a simple abelian variety of dimension g 0 over
a field K, and assume that A admits sufficiently many complex multiplications. Let
D = End0
(A).
(1) If char(K) = 0 then D is of Type IV with d = 1 and e = 2g (so D is a CM
field, by Theorem 1.3.6.2).
(2) If char(K) 0 then D is of Type III or Type IV.
Proof. By simplicity, D is a division algebra. Its center Z is a commutative field.
First suppose char(K) = 0. Let P ⊂ D be a commutative semisimple Q-
subalgebra with [P : Q] = 2g. Since D is a division algebra, P is a field. The above
table (or the discussion preceding it) says that the degree [D : Q] = ed2
divides
[P : Q] = 2g, so the inclusion P ⊂ D is an equality. Thus, D is commutative
(i.e., d = 1), so D = Z is a commutative field and hence e := [Z : Q] = 2g by
the complex multiplication hypothesis. The table shows that in characteristic 0 we
have e|g for Types I, II, and III, so D is of Type IV.
Suppose char(K) 0. In view of the divisibility relations in the table in positive
characteristic, D is not of Type I since in such cases D is a commutative field whose
Q-degree divides dim(A), contradicting the existence of sufficiently many complex
multiplications. For Type II we have 2e|g, yet the integer 2e = 2[Z : Q] is the Q-
degree of a maximal commutative subfield of the central quaternion division algebra
D over Z, so there are no such subfields with Q-degree 2g. Since a commutative
semisimple Q-subalgebra of D is a field (as D is a division algebra), Type II is not
possible if the simple A has sufficiently many complex multiplications.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-45-320.jpg)
![1.3. COMPLEX MULTIPLICATION 31
1.3.7. Returning to the proof of Theorem 1.3.4, recall that we reduced the proof
to the case of simple A. Proposition 1.3.6.4(1) settles the case of characteristic 0,
and Proposition 1.3.6.4(2) gives that D = End0
(A) is an Albert algebra of Type
III or IV when char(K) 0. If D is of Type III then the center Z is totally real
and d is even, whereas if D is of Type IV then Z is CM. Thus, we can apply the
following general lemma to conclude the proof.
1.3.7.1. Lemma (Tate). Let D be a central division algebra of degree d2
over a
number field Z that is totally real or CM. If Z is totally real then assume that d is
even. There exists a maximal commutative subfield L ⊂ D that is a CM field.
The parity condition on d is necessary when Z is totally real, since d = [L : Z]
by maximality of L in D.
Proof. By Proposition 1.2.3.1, any degree-d extension of Z that splits D is
a maximal commutative subfield of D. Hence, we just need to find a degree-d
extension L of Z that is a CM field and splits D. Let Σ be a finite non-empty
set of finite places of Z containing the finite places at which D is non-split. By
the structure of Brauer groups of local fields, for any v ∈ Σ the central simple
Zv-algebra
Dv := Zv ⊗Z D
of rank d2
over Zv is split by any extension of Zv of degree d.
First assume that Z is totally real, so d is even. By weak approximation
(Theorem 1.3.5.1), there is a monic polynomial f over Z of degree d/2 that is
close to a monic irreducible polynomial of degree d/2 over Zv for all v ∈ Σ (and in
particular f is irreducible over all such Zv, and hence over Z since Σ is non-empty).
We can also arrange that for each real place v of Z the polynomial f viewed over
Zv R is close to a totally split monic polynomial of degree d/2 and hence is
totally split over Zv. Thus, Z
:= Z[u]/(f) is a totally real extension field of Z
with degree d/2. By the same method, we can construct a quadratic extension
L/Z
that is unramified quadratic over each place v
over a place in Σ and is also
totally complex (by using approximations to irreducible quadratics over R at the
real places of Z
). This L is a CM field and it is designed so that Zv ⊗Z L is a
degree-d field extension of Zv for all v ∈ Σ. Hence, DL is split at all places of L
(the archimedean ones being obvious), so DL is split.
Assume next that Z is a CM field. Let Z+
⊂ Z be the maximal totally real
subfield. By the same weak approximation procedure as above (replacing d/2 with
d), we can construct a degree d totally real extension Z+
/Z+
such that for each
place v0 of Z+
beneath a place v ∈ Σ, the extension Z+
/Z+
has a unique place
v
0 over v0 and is totally ramified (resp. unramified) at v
0 when Z/Z+
is unramified
(resp. ramified) at v. Hence, (Z+
)v
0
and Zv are linearly disjoint over (Z+
)v0
. We
conclude that Z+
and Z are linearly disjoint over Z
, so L := Z+
⊗Z+ Z is a field
and each v ∈ Σ has a unique place w over it in L. Clearly [Lw : Zv] = d for all
such w, so L splits D. By construction, L is visibly CM. We have proved Lemma
1.3.7.1. This also finishes the proof of Theorem 1.3.4.
1.3.7.2. Corollary. An isotypic abelian variety A with sufficiently many complex
multiplications remains isotypic after any extension of the base field.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-46-320.jpg)
![32 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
Proof. By Theorem 1.3.4, the endomorphism algebra End0
(A) contains a com-
mutative field with Q-degree 2 dim(A). This property is preserved after any ground
field extension (even though the endomorphism algebra may get larger), so by the
final part of Theorem 1.3.1.1 isotypicity is preserved as well.
1.3.8. CM abelian varieties. It turns out to be convenient to view the CM
algebra P in Theorem 1.3.4 as an abstract ring in its own right, and to thereby
regard the embedding P → End0
(A) as additional structure on A. This is encoded
in the following concept.
1.3.8.1. Definition. Let A be an abelian variety over a field K, and assume that
A has sufficiently many complex multiplications. Let j : P → End0
(A) be an
embedding of a CM algebra P with [P : Q] = 2 dim(A). Such a pair (A, j) is called
a CM abelian variety (with complex multiplication by P).
Note that in this definition we are requiring P to be embedded in the endo-
morphism algebra of A over K (and not merely in the endomorphism algebra after
an extension of K). For example, according to this definition, no elliptic curve over
Q admits a CM structure (even if such a structure exists after an extension of the
base field).
As an application of Theorem 1.3.4, we establish the following result concerning
the possibilities for Z of Type III in Proposition 1.3.6.4(2). This will not be used
later.
1.3.8.2. Proposition. Let A, K, and D be as in Proposition 1.3.6.4(2) with p =
char(K) 0, and let Z be the center of D, g = dim(A), d =
[D : Z], and
e = [Z : Q]. We have ed = 2g, and if D is of Type III (so d = 2) then either Z = Q
or Z = Q(
√
p).
Note that in this proposition, K is an arbitrary field with char(K) 0; K is
not assumed to be finite.
Proof. We always have ed|2g, but ed =
[D : Q] and D contains a field P of
Q-degree 2g, so 2g|ed. Thus, ed = 2g. Now we can assume A is of Type III, so the
field Z is totally real.
Since A is of finite type over K and D is finite-dimensional over Q, by direct
limit considerations we can descend to the case when K is finitely generated over
Fp. Let S be a separated integral Fp-scheme of finite type whose function field is
K. Since A is an abelian variety over the direct limit K of the coordinate rings
of the non-empty affine open subschemes of S, by replacing S with a sufficiently
small non-empty affine open subscheme we can arrange that A is the generic fiber
of an abelian scheme A → S. Since S is connected, the fibers of the map A → S
all have the same dimension, and this common dimension is g (as we may compute
using the generic fiber A).
The Z-module End(A) is finitely generated, and each endomorphism of A ex-
tends uniquely to a U-endomorphism of AU for some non-empty open U in S
(with U perhaps depending on the chosen endomorphism). Using a finite set of
endomorphisms that spans End(A) allows us to shrink S so that all elements of
End(A) extend to S-endomorphisms of A , or in other words End(A) = End(A ).
We therefore have a specialization map D = End0
(A) → End0
(As) for every s ∈ S.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-47-320.jpg)
![1.4. DIEUDONNÉ THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 33
Fix a prime = p. Since S is connected and A [n
] is finite étale over S, an
S-endomorphism of A [n
] is uniquely determined by its effect on a single geometric
fiber over S. But maps between abelian varieties are uniquely determined by their
effect on -adic Tate modules when is a unit in the base field, so we conclude (via
consideration of -power torsion) that the specialization map D → End0
(As) is
injective for all s ∈ S. We can therefore speak of an element of End0
(As) “lifting”
over K in the sense that it is the image of a unique element of D = End0
(A) under
the specialization mapping at s. This will be of interest when s is a closed point
and we consider the qs-Frobenius endomorphism of As over the finite residue field
κ(s) at s (with qs = #κ(s)).
By Theorem 1.3.4, we can choose a CM field L ⊂ D with [L : Q] = 2g. In
particular, for each s ∈ S the field L embeds into End0
(As) with [L : Q] = 2g =
2 dim(As), so each As is isotypic. By Theorem 1.3.1.1, L is its own centralizer in
End0
(As). Take s to be a closed point of S, and let qs denote the size of the finite
residue field κ(s) at s. The qs-Frobenius endomorphism ϕs ∈ End0
(As) is central,
so it centralizes L and hence must lie in the image of L. In particular, ϕs lifts to
an element of End0
(A) = D that is necessarily central (as we may compute after
applying the injective specialization map D → End0
(As)). That is, ϕs ∈ Z ⊂ D
for all closed points s ∈ S.
Let Z
be the subfield of Z generated over Q by the lifts of the endomorphisms
ϕs as s varies through all closed points of S. Each Q[ϕs] is a totally real field
since Z is totally real. By Weil’s Riemann Hypothesis for abelian varieties over
finite fields (see the discussion following Definition 1.6.1.2), under any embedding
ι : Q[ϕs] → C we have each ι(ϕs)ι(ϕs) = qs for qs = #κ(s) ∈ pZ
, so the real number
ι(ϕs) is a power of
√
p. Hence, the subfield Q[ϕs] ⊂ Z is either Q or Q(
√
p), so the
subfield Z
⊂ Z is either Q or Q(
√
p).
Let η be a geometric generic point of S, and let Γ be the associated absolute
Galois group for the function field of S. Because each A [n
] is finite étale over S, the
representation of Γ on V(A) factors through the quotient π1(S, η). The Chebotarev
Density Theorem for π1(S, η) [97, App. B.9] says that the Frobenius elements at the
closed points of S generate a dense subgroup of the quotient π1(S, η) of Γ. Thus,
the image of Q[Γ] in EndQ
(V(A)) is equal to the subalgebra Z
:= Q ⊗Q Z
generated by the endomorphisms ϕs. We therefore have an injective map
Q ⊗Q D → EndQ[Γ](V(A)) = EndZ
(V(A)).
By Zarhin’s theorem [134] (see [80, XII, §2] for a proof valid for all p, especially
allowing p = 2) this injection is an isomorphism, so we conclude that Z is central
in EndZ
(V(A)). But the center of this latter matrix algebra is Z
, so the inclusion
Z
⊂ Z is an equality. Hence, the inclusion Z
⊂ Z is an equality as well. Since
Z
is either Q or Q(
√
p), we are done.
1.4. Dieudonné theory, p-divisible groups, and deformations
To solve problems involving lifts from characteristic p to characteristic 0, we
need a technique for handling p-torsion phenomena in characteristic p 0. The
two main tools for this purpose in what we shall do are Dieudonné theory and
p-divisible groups. For the convenience of the reader we review the basic facts in
this direction, and for additional details we refer to [119], [110, §6], and [75] for](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-48-320.jpg)
![34 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
p-divisible groups, and [41, Ch. II–III] for (contravariant) Dieudonné theory with
applications to p-divisible groups.
1.4.1. Exactness. We shall frequently use exact sequence arguments with abelian
varieties and finite group schemes over fields, as well as with their relative analogues
over more general base schemes. It is assumed that the reader has some familiarity
with these notions, but we now provide a review of this material.
1.4.1.1. Definition. Let S be a scheme, and let T be a Grothendieck topology
on the category of S-schemes. (For our purposes, only the étale, fppf, and fpqc
topologies will arise.) A diagram 1 → G
→ G
f
→ G
→ 1 of S-group schemes is
short exact for the topology T if G
→ G is an isomorphism onto ker(f) and the
map f is a T -covering.
By [30, Exp. IV, 5.1.7.1], in such cases G
represents the quotient sheaf G/G
for the chosen Grothendieck topology. By [31, Exp. V, Thm. 4.1(iv), Rem. 5.1], if
G is a quasi-projective group scheme over a noetherian ring R and if G
is a finite
flat R-subgroup of G then the fppf quotient sheaf G/G
is represented by a quasi-
projective R-group (also denoted G/G
), and the resulting map of group schemes
G → G/G
is an fppf G
-torsor (so G/G
is R-flat if G is).
1.4.1.2. The Cartier dual ND
of a commutative finite locally free group scheme
N over a base scheme S is the commutative finite locally free group scheme which
represents the fppf sheaf functor Hom(N, Gm) : S
HomS-gp(NS , Gm) on the
category of S-schemes. The structure sheaf OND of ND
is canonically isomorphic
to the OS-linear dual of the structure sheaf ON of N, and the co-multiplication
(respectively multiplication) map for OND is the OS-linear dual of the multiplication
(respectively co-multiplication) map for ON .
The functor N ND
on the category of commutative finite locally free group
schemes over S swaps closed immersions and quotient maps, preserves exactness,
and is an involution in the sense that there is a natural isomorphism fN : N
(ND
)D
satisfying (fN )D
= f−1
ND . See [87, Prop. 2.9] for further details.
As an application, if the S-homomorphism j : G
→ G is a closed immersion
between finite locally free commutative group schemes then we can use Cartier
duality to give a direct proof that the the fppf quotient sheaf G/G
is represented by
a finite locally free S-group (without needing to appeal to general existence results
for quotients by G
-actions on quasi-projective S-schemes). The key point is that
the Cartier dual map jD
: GD
→ GD
between finite flat S-schemes is faithfully flat,
as this holds on fibers over S (since injective maps between Hopf algebras over a
field are always faithfully flat [126, 14.1]). Such flatness implies that H := ker(jD
)
is a finite locally free commutative S-group, so HD
makes sense and the dual map
q : G (GD
)D
→ HD
is faithfully flat. It is clear that G
⊂ ker(q), and this
inclusion between finite locally free S-schemes is an isomorphism by comparison of
fibral degrees, so HD
represents G/G
.
The following result is useful for constructing commutative group schemes G →
S that are finite and fppf (equivalently, finite and locally free over S).](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-49-320.jpg)
![1.4. DIEUDONNÉ THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 35
1.4.1.3. Proposition. Let S be a scheme, and let G
and G
be finitely presented
separated S-group schemes with G
affine and flat over S. For any exact sequence
1 → G
→ G → G
→ 1
of group sheaves for the fppf topology on the category of S-schemes, G is represented
by a finitely presented S-group that is flat and affine over G
. Moreover, G
and
G
are S-finite if and only if G is S-finite.
See [87, 17.4] for a generalization (using the fpqc topology).
Proof. For any G
-scheme T viewed as an S-scheme, let g
∈ G
(T) correspond
to the given S-morphism T → G
. Consider the set Eg (T) that is the preimage
under G(T) → G
(T) of g
. This is a sheaf of sets on the category of G
-schemes
equipped with the fppf topology, and as such it is a left G
-torsor (strictly speaking,
a left torsor for the G
-group G
G ) due to the given exact sequence. In particular,
the fppf sheaves of sets Eg and G
G over G
are isomorphic fppf-locally over G
.
Since G
is fppf affine over S and fppf descent is effective for affine morphisms, it
follows that Eg as an fppf sheaf of sets over G
is represented by an affine fppf G
-
scheme (which is therefore affine fppf over S when G
is). It is elementary to check
that this affine G
-scheme viewed as an S-scheme has its functor of points naturally
identified with G (since for any S-scheme T and g ∈ G(T), visibly g ∈ Eg (T) for
the point g
∈ G
(T) arising from g), so G is represented by an S-group.
Separatedness of G
over S and exactness imply that G
is closed in G. More-
over, G → G
is a left G
G -torsor for the fppf topology over G
, so it is finite when
G
is S-finite. Thus, if G
and G
are S-finite then G is S-finite. Conversely, if G
is S-finite then its closed subscheme G
is S-finite, so the quotient G/G
exists as
an S-finite scheme. But G
represents this quotient, so G
is S-finite too.
1.4.1.4. Remark. If 1 → G
→ G → G
→ 1 is an exact sequence of separated
fppf S-groups with G
and G
abelian schemes then G is an abelian scheme. Indeed,
since G → G
is an fppf torsor for the G
-group G
G that is smooth and proper with
geometrically connected fibers, G → G
is smooth and proper with geometrically
connected fibers. The map G
→ S is also smooth and proper with geometrically
connected fibers, so G → S is as well. Hence, G is an abelian scheme.
It is also true that if G is an abelian scheme and G
is a closed S-subgroup of G
that is also an abelian scheme then the fppf quotient sheaf G/G
is represented by
an abelian scheme. We will give an elementary proof of this over fields in Lemma
1.7.4.4 using the Poincaré reducibility theorem (which is only available over fields).
In general the proof requires a detour through algebraic spaces.
1.4.2. Duality for abelian schemes. In [83, §6.1], duality is developed for pro-
jective abelian schemes, building on the case of an algebraically closed ground
field. Projectivity is imposed primarily due to the projectivity hypotheses in
Grothendieck’s work on Hilbert schemes. The projective case is sufficient for our
needs because any abelian scheme over a discrete valuation ring is projective (this
follows from Lemma 2.1.1.1, to which the interested reader may now turn). For
both technical and aesthetic reasons, it is convenient to avoid the projectivity hy-
pothesis. We now sketch Grothendieck’s results on duality in the projective case,
as well as Artin’s improvements that eliminated the projectivity assumption.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-50-320.jpg)
![36 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
1.4.2.1. Let A → S be an abelian scheme, and let PicA/S be the functor assigning
to any S-scheme T the group of isomorphism classes of pairs (L , i) consisting of
an invertible sheaf L on AT and a trivialization i : e∗
T (L ) OT along the identity
section eT of AT → T. This is an fppf group sheaf on the category of S-schemes,
and its restriction to the category of S
-schemes (for an S-scheme S
) is PicAS /S .
Let Pic0
A/S ⊂ PicA/S be the subfunctor classifying pairs (L , i) that lie in the
identity component of the Picard scheme on geometric fibers. By [7, Exp. XIII,
Thm. 4.7(i)] (see [39, §9.6] for the projective case), the inclusion j : Pic0
A/S →
PicA/S is an open subfunctor; i.e., it is relatively representable by open immersions.
This means that for any S-scheme T and (L , i) ∈ PicA/S(T), Pic0
A/S ×PicA/S
T as
a functor on T-schemes is represented by an open subscheme U ⊂ T; explicitly,
there is an open subscheme U ⊂ T such that a T-scheme T
lies over U if and only
if the T
-pullback of (L , i) lies in Pic0
on geometric fibers over T
.
By Grothendieck’s work on Picard schemes (see [39, Part 5]), if A → S is
projective Zariski-locally on S then PicA/S is represented by a locally finitely pre-
sented and separated S-scheme and the open subscheme At
representing Pic0
A/S
is quasi-projective Zariski-locally on S and finitely presented. For noetherian S,
functorial criteria show that At
is proper and smooth (see [83, §6.1]), hence an
abelian scheme; the case of general S (with A projective Zariski-locally on S) then
follows by descent to the noetherian case.
To drop the projectivity hypothesis, one has to use algebraic spaces. Infor-
mally, an algebraic space over S is an fppf sheaf on the category of S-schemes that
is “well-approximated” by a representable functor (relative to the étale topology),
so concepts from algebraic geometry such as smoothness, properness, and connect-
edness can be defined and behave as expected; see [60]. By Artin’s work on relative
Picard functors as algebraic spaces (see [5, Thm. 7.3]), PicA/S is always a separated
algebraic space locally of finite presentation, and by [7, Exp. XIII, Thm. 4.7(iii)]
the open algebraic subspace Pic0
A/S is finitely presented over S.
The functorial arguments that prove smoothness and properness for Pic0
A/S
when A is projective work without projectivity because the same criteria are avail-
able for algebraic spaces. Thus, Pic0
A/S is smooth and proper over S in the sense
of algebraic spaces. Consequently, by a theorem of Raynaud (see [38, Thm. 1.9]),
Pic0
A/S is represented by an S-scheme At
; this must be an abelian scheme, called
the dual abelian scheme. Its formation commutes with any base change on S, and
it is contravariant in A in an evident manner.
1.4.2.2. Over A × At
there is a Poincaré bundle PA/S provided by the universal
property of At
, exactly as in the theory of duality for abelian varieties over a field.
In particular, PA/S is canonically trivialized along e × idAt . Let e
∈ At
(S) be the
identity, so for any S-scheme T the point e
T ∈ At
(T) corresponds to OAT
equipped
with the canonical trivialization of e∗
T (OAT
). Thus, setting T = A gives that PA/S
is also canonically trivialized along idA × e
. Hence, the pullback of PA/S along
the flip At
× A A × At
defines a canonical S-morphism ιA/S : A → Att
. This
morphism carries the identity to the identity, so it is a homomorphism. By applying
the duality theory over fields to the fibers of A over S, it follows that ιA/S is an
isomorphism; in other words, the pullback of PA/S along the flip At
×A A×At
is
uniquely isomorphic to PAt/S respecting trivializations along the identity sections
of both factors. Such uniqueness implies that ιt
A/S is inverse to ιAt/S.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-51-320.jpg)
![1.4. DIEUDONNÉ THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 37
A homomorphism f : A → At
is symmetric when the map
ft
◦ ιA/S : A Att
→ At
is equal to f. Writing f†
:= ft
◦ιA/S, the equality ιt
A/S = ι−1
At/S and the functoriality
of ιA/S in A (applied with respect to f) implies f††
= f, so if we abuse notation by
writing ft
rather than f†
then (ft
)t
= f. We say f is symmetric when ft
= f (or
more accurately, f†
= f). This property holds if it does so on fibers over S, because
homomorphisms f, f
: A ⇒ B between abelian schemes coincide if fs = f
s for all
s ∈ S. Indeed, for noetherian S such rigidity is [83, Cor. 6.2], and the general case
reduces to this because equality on all fibers descends through direct limits (since
it says that the finitely presented ideal of (f, f
)−1
(ΔA/S) in OA is nilpotent).
A polarization of A is a homomorphism f : A → At
that is a polarization on
geometric fibers. Any such f is necessarily symmetric. The properties of polar-
izations are developed in [83, §6.2] for projective abelian schemes, but the only
purpose of imposing projectivity at the outset (even though it is a consequence of
the definition, due to [34, IV3, 9.6.4]) is to ensure the existence of the dual abelian
scheme, so such an assumption may be eliminated.
1.4.2.3. Definition. A homomorphism ϕ : A → B between abelian schemes over
a scheme S is an isogeny when it is surjective with finite fibers. (Equivalently, the
homomorphims ϕs are isogenies in the sense of abelian varieties for each s ∈ S.)
Since quasi-finite proper morphisms are finite by [34, IV4, 18.12.4] (or by [34,
IV3, 8.11.1] with finite presentation hypothesis, which suffices for us), any isogeny
between abelian schemes is a finite morphism. Moreover, by the fibral flatness
criterion [34, IV3, 11.3.11], such maps are flat. Hence, if ϕ as above is an isogeny
then it is finite locally free (and surjective), so the closed subgroup ker(ϕ) is a finite
locally free commutative S-group scheme. Thus, B represents the fppf quotient
sheaf A/ker(ϕ). For example, setting ϕ = [n]A for n ⩾ 1 gives A/A[n] A.
Turning this around, suppose we are given the abelian scheme A and a closed
S-subgroup N ⊂ A that is finite locally free over S. Consider the fppf quotient
sheaf A/N. We claim that this quotient is (represented by) an abelian scheme, so
the map A → A/N with kernel N is an isogeny. It suffices to work Zariski-locally
on S, so we may assume that N → S has all fibers with the same order n ⩾ 1. We
then have N ⊂ A[n], due to the following result (proved in [123, §1]):
1.4.2.4. Theorem (Deligne). Let S be a scheme and let H be a commutative S-
group scheme for which the structural morphism H → S is finite and locally free.
If the fibers Hs have rank n for all s ∈ S then H is killed by n.
The quotient sheaf A/N is an fppf torsor over A/A[n] A with fppf covering group
A[n]/N that is finite (and hence affine) over S. It then follows from effective fppf
descent for affine morphisms that the quotient A/N is represented by a scheme
finite over A/A[n] = A, and the map A → A/N is an fppf A[n]/N-torsor, so the
S-proper S-smooth A is finite locally free over A/N (as A[n]/N is finite locally free
over S). Hence, A/N is proper and smooth since A is, and likewise its fibers over
S are geometrically connected. Thus, A/N is an abelian scheme as desired.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-52-320.jpg)
![38 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
1.4.2.5. Theorem. Let ϕ : A → B be an isogeny between abelian schemes over a
scheme S, and let N = ker(ϕ). Duality applied to the exact sequence
0 → N −→ A
ϕ
−→ B → 0
functorially yields an exact sequence
0 → ND
−→ Bt ϕt
−→ At
→ 0.
That is, the map ϕt
is an isogeny whose kernel is canonically isomorphic to ND
.
Moreover, double duality for abelian schemes and for finite locally free commu-
tative group schemes are compatible up to a sign: if we identify ϕ and ϕtt
via ιA/S
and ιB/S then the natural isomorphism (ND
)D
ker((ϕt
)t
) = ker(ϕtt
) ker(ϕ) =
N is the negative of the canonical isomorphism provided by Cartier duality.
We refer the reader to [86, Thm. 1.1, Cor. 1.3] for a proof based on arguments
that relativize the ones over an algebraically closed field in [82]. (An alternative
approach, at least for the first part, is [87, Thm. 19.1], resting on the link between
dual abelian schemes and Ext-sheaves given in [87, Thm. 18.1].) The special case
ϕ = [n]A : A → A implies that naturally A[n]D
= At
[n] for every n ⩾ 1 because
[n]t
A = [n]At (by [87, 18.3]); this identification respects multiplicative change in n.
1.4.3. Constructions and definitions. Let us now focus on constructions spe-
cific to the theory of finite commutative group schemes over a perfect field k of
characteristic p 0. Let W = W(k) be the ring of Witt vectors of k; e.g., if k is
finite of size q = pr
then W is the ring of integers in an unramified extension of
Qp of degree r. Let σ be the unique automorphism of W that reduces to the map
x → xp
on the residue field k.
1.4.3.1. Definition. The Dieudonné ring Dk over k is W[F, V], where F and V
are indeterminates subject to the relations
(1) FV = VF = p,
(2) Fc = σ(c)F and cV = Vσ(c) for all c ∈ W.
Explicitly, elements of Dk have unique expressions as finite sums
a0 +
j0
ajFj
+
j0
bjVj
with coefficients in W (so the center of Dk is clearly Zp[Fr
, Vr
] if k has finite size
pr
and it is Zp otherwise; i.e., if k is infinite).
Some of the main conclusions in classical Dieudonné theory, as developed from
scratch in [41, Ch. I–III], are summarized in the following theorem.
1.4.3.2. Theorem. There is an additive anti-equivalence of categories G M∗
(G)
from the category of finite commutative k-group schemes of p-power order to the
category of left Dk-modules of finite W-length. Moreover, the following hold.
(1) A group scheme G has order pW (M∗
(G))
, where W (·) denotes W-length.
(2) If k → k
is an extension of perfect fields with associated extension W → W
of Witt rings (e.g., the absolute Frobenius automorphism of k) then the functor
W
⊗W (·) on Dieudonné modules is naturally identified with the base-change](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-53-320.jpg)
![1.4. DIEUDONNÉ THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 39
functor on finite commutative group schemes. In particular, M∗
(G(p)
)
σ∗
(M∗
(G)) as W-modules.
(3) Let FrG/k : G → G(p)
be the relative Frobenius morphism. The σ-semilinear
action on M∗
(G) induced by M∗
(FrG/k) through the isomorphism M∗
(G(p)
)
σ∗
(M∗
(G)) equals the action of F, and G is connected if and only if F is
nilpotent on M∗
(G).
(4) There is a natural k-linear isomorphism M∗
(G)/FM∗
(G) Lie(G)∨
respect-
ing extension of the perfect base field.
(5) For the Cartier dual GD
, naturally M∗
(GD
) HomW (M∗
(G), K/W) with
K = W[1/p], using the operators F() : m → σ((V(m))) and V() : m →
σ−1
((F(m))) on K/W-valued linear forms .
For an abelian scheme A → S with fibers of constant dimension g ⩾ 1 and
its finite commutative pn
-torsion subgroup scheme A[pn
] with order (pn
)2g
, the
directed system A[p∞
] := (A[pn
])n⩾1 satisfies the following definition (with h = 2g).
1.4.3.3. Definition. A p-divisible group of height h ⩾ 0 over a scheme S is a
directed system G = (Gn)n⩾1 of commutative S-groups Gn such that: Gn is killed
by pn
, each Gn → S is finite and locally free, [p]: Gn+1 → Gn is faithfully flat for
every n ⩾ 1, G1 → S has constant degree ph
, and Gn is identified with Gn+1[pn
]
for all n ⩾ 1.
The (Serre) dual p-divisible group Gt
is the directed system (GD
n ) of Cartier
dual group schemes GD
n with the transition maps GD
n → GD
n+1 that are Cartier
dual to the quotient maps [p] : Gn+1 → Gn.
As an illustration, if A → S is an abelian scheme with fibers of dimension
g ⩾ 1 then the isomorphisms A[n]D
At
[n] respecting multiplicative change in
n (as noted immediately below Theorem 1.4.2.5) yield a canonical isomorphism
between the Serre dual A[p∞
]t
and the p-divisible group At
[p∞
] of the dual abelian
scheme At
(see [86, Prop. 1.8] or [87, Thm. 19.1]).
1.4.3.4. Remark. In view of the sign discrepancy for comparisons of double du-
ality in Theorem 1.4.2.5, if ϕ : A → At
is an S-homomorphism and
f : A[p∞
] → At
[p∞
] A[p∞
]t
is the associated homomorphism between p-divisible groups then the dual homo-
morphism ϕt
: A → At
(strictly speaking, ϕt
◦ ιA/S via double duality for abelian
schemes) has as its associated homomorphism A[p∞
] → A[p∞
]t
the negative1
of ft
(using double duality for p-divisible groups).
It follows that if ϕ is symmetric with respect to double duality for abelian
schemes then f is skew-symmetric with respect to double duality for p-divisible
groups. The converse is also true: we can see immediately via skew-symmetry
of f that ϕ and ϕt
induce the same homomorphism between p-divisible groups,
and to conclude that ϕ = ϕt
it suffices to check on fibers due to the rigidity of
abelian schemes (as in 1.4.2.2). On fibers we can apply the faithfulness of passage
to p-divisible groups over fields via 1.2.5.1 with = p.
1A related sign issue in the double duality for commutative finite group schemes over perfect
fields is discussed in a footnote at the end of B.3.5.5.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-54-320.jpg)
![melancholious dremeth of sorrow, the Cholarike, of firy things, and
the Flematike, of Raine, Snow,' c.; id. lib. vi. c. 27.
4123. the humour of malencolye. 'The name (melancholy) is
imposed from the matter, and disease denominated from the
material cause, as Bruel observes, μελαγχολία quasi μελαιναχόλη,
from black choler.' Fracastorius, in his second book of Intellect, calls
those melancholy 'whom abundance of that same depraved humour
of black choler hath so misaffected, that they become mad thence,
and dote in most things or in all, belonging to election, will, or other
manifest operations of the understanding.'—Burton's Anat. of
Melancholy, p. 108, ed. 1805.
4128. 'That cause many a man in sleep to be very distressed.'
4130. Catoun. Dionysius Cato, de Moribus, l. ii. dist. 32: somnia ne
cures. 'I observe by the way, that this distich is quoted by John of
Salisbury, Polycrat. l. ii. c. 16, as a precept viri sapientis. In another
place, l. vii. c. 9, he introduces his quotation of the first verse of dist.
20 (l. iii.) in this manner:—Ait vel Cato vel alius, nam autor incertus
est.'—Tyrwhitt. Cf. note to G. 688.
4131. do no fors of = take no notice of, pay no heed to. Skelton, i.
118, has 'makyth so lytyll fors,' i. e. cares so little for.
4153. 'Wormwood, centaury, pennyroyal, are likewise magnified and
much prescribed, especially in hypochondrian melancholy, daily to be
used, sod in whey. And because the spleen and blood are often
misaffected in melancholy, I may not omit endive, succory,
dandelion, fumitory, c., which cleanse the blood.'—Burton's Anat. of
Mel. pp. 432, 433. See also p. 438, ed. 1845. 'Centauria abateth
wombe-ache, and cleereth sight, and vnstoppeth the splene and the
reines'; Batman upon Bartholomè, lib. xvii. c. 47. 'Fumus terre
[fumitory] cleanseth and purgeth Melancholia, fleme, and cholera';
id. lib. xvii. c. 69. 'Medicinal herbs were grown in every garden, and
were dried or made into decoctions, and kept for use'; Wright,
Domestic Manners, p. 279.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-56-320.jpg)
![4235. Harrow, a cry of distress; a cry for help. 'Harrow! alas! I swelt
here as I go.'—The Ordinary; see vol. iii. p. 150, of the Ancient
Drama. See F. haro in Godefroy and Littré; and note to A. 3286.
4237. outsterte (Elles., c.); upsterte (Hn., Harl.)
4242. A common proverb. Skelton, ed. Dyce, i. 50, has 'I drede
mordre wolde come oute.'
4274. And preyde him his viáge for to lette, And prayed him to
abandon his journey.
4275. to abyde, to stay where he was.
4279. my thinges, my business-matters.
4300. 'Kenelm succeeded his father Kenulph on the throne of the
Mercians in 821 [Haydn, Book of Dates, says 819] at the age of
seven years, and was murdered by order of his aunt, Quenedreda.
He was subsequently made a saint, and his legend will be found in
Capgrave, or in the Golden Legend.'—Wright.
St. Kenelm's day is Dec. 13. Alban Butler, in his Lives of the Saints,
says:—[Kenulph] 'dying in 819, left his son Kenelm, a child only
seven years old [see l. 4307] heir to his crown, under the tutelage of
his sister Quindride. This ambitious woman committed his person to
the care of one Ascobert, whom she had hired to make away with
him. The wicked minister decoyed the innocent child into an
unfrequented wood, cut off his head, and buried him under a thorn-
tree. His corpse is said to have been discovered by a heavenly ray of
light which shone over the place, and by the following inscription:—
In Clent cow-pasture, under a thorn,
Of head bereft, lies Kenelm, king born.'
Milton tells the story in his History of Britain, bk. iv. ed. 1695, p. 218,
and refers us to Matthew of Westminster. He adds that the](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-59-320.jpg)
![4350. This line is repeated from the Compleynt of Mars, l. 61.
4353-6. 'By way of quiet retaliation for Partlet's sarcasm, he cites a
Latin proverbial saying, in l. 344, 'Mulier est hominis confusio,' which
he turns into a pretended compliment by the false translation in ll.
345, 346.'—Marsh. Tyrwhitt quotes it from Vincent of Beauvais, Spec.
Hist. x. 71. Chaucer has already referred to this saying above; see p.
207, l. 2296. 'A woman, as saith the philosofre [i. e. Vincent], is the
confusion of man, insaciable, c.'; Dialogue of Creatures, cap. cxxi.
'Est damnum dulce mulier, confusio sponsi'; Adolphi Fabulae, x. 567;
pr. in Leyser, Hist. Poet. Med. Aevi, p. 2031. Cf. note to D. 1195.
4365. lay, for that lay. Chaucer omits the relative, as is frequently
done in Middle English poetry; see note to l. 4090.
4377. According to Beda, the creation took place at the vernal
equinox; see Morley, Eng. Writers, 1888, ii. 146. Cf. note to l. 4045.
4384. See note on l. 4045 above.
4395. Cf. Man of Lawes Tale, B. 421, and note. See Prov. xiv. 13.
4398. In the margin of MSS. E. and Hn. is written 'Petrus Comestor,'
who is probably here referred to.
4402. See the Squieres Tale, F. 287, and the note.
4405. col-fox; explained by Bailey as a 'coal-black fox'; and he seems
to have caught the right idea. Col- here represents M. E. col, coal;
and the reference is to the brant-fox, which is explained in the New
E. Dict. as borrowed from the G. brand-fuchs, 'the German name of
a variety of the fox, chiefly distinguished by a greater admixture of
black in its fur; according to Grimm, it has black feet, ears, and tail.'
Chaucer expressly refers to the black-tipped tail and ears in l. 4094
above. Mr. Bradley cites the G. kohlfuchs and Du. koolvos, similarly
formed; but the ordinary dictionaries do not give these names. The](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-61-320.jpg)
!['This is alluded to also by Camoẽs (Lusiad, x. 140). Burton's
translation has:—
But here, where earth spreads wider, ye shall claim
Realms by the ruddy dye-wood made renowned;
These of the 'Sacred Cross' shall win the name,
By your first navy shall that world be found.
'The medieval forms of brazil were many; in Italian, it is generally
verzi, verzino, or the like.'—Yule, Hobson-Jobson, p. 86.
Again—'Sappan, the wood of Cæsalpinia sappan; the baqqam of the
Arabs, and the Brazil-wood of medieval commerce. The tree appears
to be indigenous in Malabar, the Deccan, and the Malay peninsula.'—
id. p. 600. And in Yule's edition of Marco Polo, ii. 315, he tells us that
'it is extensively used by native dyers, chiefly for common and cheap
cloths, and for fine mats. The dye is precipitated dark-brown with
iron, and red with alum.'
Cf. Way's note on the word in the Prompt. Parv. p. 47.
Florio explains Ital. verzino as 'brazell woode, or fernanbucke
[Pernambuco] to dye red withall.'
The etymology is disputed, but I think brasil and Ital. verzino are
alike due to the Pers. wars, saffron; cf. Arab. warīs, dyed with saffron
or wars.
greyn of Portingale. Greyn, mod. E. grain, is the term applied to the
dye produced by the coccus insect, often termed, in commerce and
the arts, kermes; see Marsh, Lectures on the E. Language, Lect. III.
The colour thus produced was 'fast,' i. e. would not wash out; hence
the phrase to engrain, or to dye in grain, meaning to dye of a fast
colour. Various tones of red were thus produced, one of which was
crimson, and another carmine, both forms being derivatives of
kermes. Of Portingale means 'imported from Portugal.' In the Libell
of English Policy, cap. ii. (l. 132), it is said that, among 'the](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-67-320.jpg)
![commoditees of Portingale' are:—'oyl, wyn, osey [Alsace wine], wex,
and graine.'
4652. to another, to another of the pilgrims. This is so absurdly
indefinite that it can hardly be genuine. Ll. 4637-4649 are in
Chaucer's most characteristic manner, and are obviously genuine;
but there, I suspect, we must stop, viz. at the word Portingale. The
next three lines form a mere stop-gap, and are either spurious, or
were jotted down temporarily, to await the time of revision. The
former is more probable.
This Epilogue is only found in three MSS.; (see footnote, p. 289). In
Dd., Group G follows, beginning with the Second Nun's Tale. In the
other two MSS., Group H follows, i. e. the Manciple's Tale;
nevertheless, MS. Addit. absurdly puts the Nunne, in place of
another. The net result is, that, at this place, the gap is complete;
with no hint as to what Tale should follow.
It is worthy of note that this Epilogue is preserved in Thynne and the
old black-letter editions, in which it is followed immediately by the
Manciple's Prologue. This arrangement is obviously wrong, because
that Prologue is not introduced by the Host (as said in l. 4652).
In l. 4650, Thynne has But for Now; and his last line runs—'Sayd to
a nother man, as ye shal here.' I adopt his reading of to for unto (as
in the MSS.).
NOTES TO GROUP C.
The Phisiciens Tale.
For remarks on the spurious Prologues to this Tale, see vol. iii. p.
434. For further remarks on the Tale, see the same, p. 435, where its
original is printed in full.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-68-320.jpg)
![Nulla potest; operisque sui concepit amorem.'
In the margin of E. Hn. is the note—'Quere in Methamorphosios';
which supplies the reference; but cf. note to l. 16 below, shewing
that Chaucer also had in his mind Le Roman de la Rose, l. 16379. So
also the author of the Pearl, l. 750; see Morris, Allit. Poems.
16. In the margin of E. Hn. we find the note:—'Apelles fecit mirabile
opus in tumulo Darii; vide in Alexandri libro .1.º [Hn. has .6.º]; de
Zanze in libro Tullii.' This note is doubtless the poet's own; see
further, as to Apelles, in the note to D. 498.
Zanzis, Zeuxis. The corruption of the name was easy, owing to the
confusion in MSS. between n and u.[26] In the note above, we are
referred to Tullius, i. e. Cicero. Dr. Reid kindly tells me that Zeuxis is
mentioned, with Apelles, in Cicero's De Oratore, iii. § 26, and Brutus,
§ 70; also, with other artists, in Academia, ii. § 146; De Finibus, ii. §
115; and alone, in De Inventione, ii. § 52, where a long story is told
of him. Cf. note to Troil. iv. 414.
However, the fact is that Chaucer really derived his knowledge of
Zeuxis from Le Roman de la Rose (ed. Méon, l. 16387); for
comparison with the context of that line shews numerous points of
resemblance to the present passage in our author. Jean de Meun is
there speaking of Nature, and of the inability of artists to vie with
her, which is precisely Chaucer's argument here. The passage is too
long for quotation, but I may cite such lines as these:—
'Ne Pymalion entaillier' (l. 16379),
'voire Apelles
Que ge moult bon paintre appelles,
Biautés de li james descrive
Ne porroit,' c. (l. 16381).
'Zeuxis neis par son biau paindre
Ne porroit a tel forme ataindre,' c. (l. 16387).](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-70-320.jpg)
![79. See A. 476, and the note. Chaucer is here thinking of the same
passage in Le Roman de la Rose. I quote a few lines (3930-46):—
'Une vielle, que Diex honnisse!
Avoit o li por li guetier,
Qui ne fesoit autre mestier
Fors espier tant solement
Qu'il ne se maine folement....
Bel-Acueil se taist et escoute
Por la vielle que il redoute,
Et n'est si hardis qu'il se moeve,
Que la vielle en li n'aperçoeve
Aucune fole contenance,
Qu'el scet toute la vielle dance.'
See the English version in vol. i. p. 205, ll. 4285-4300.
82. See the footnote for another reading. The line there given may
also be genuine. It is deficient in the first foot.
85. This is like our proverb:—'Set a thief to catch [or take] a thief.'
An old poacher makes a good gamekeeper.
98. Cf. Prov. xiii. 24; P. Plowman, B. v. 41.
101. See a similar proverb in P. Plowman, C. x. 265, and my note on
the line. The Latin lines quoted in P. Plowman are from Alanus de
Insulis, Liber Parabolarum, cap. i. 31; they are printed in Leyser, Hist.
Poet. Med. Aevi, 1721, p. 1066, in the following form:—
'Sub molli pastore capit lanam lupus, et grex
Incustoditus dilaceratur eo.'
117. The doctour, i. e. the teacher; viz. St. Augustine. (There is here
no reference whatever to the 'Doctor' or 'Phisicien' who is supposed
to tell the tale.) In the margin of MSS. E. Hn. is written 'Augustinus';
and the matter is put beyond doubt by a passage in the Persones](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-73-320.jpg)
![Tale, l. 484:—'and, after the word of seint Augustin, it [Envye] is
sorwe of other mannes wele, and Ioye of othere mennes harm.' See
note to l. 484.
The same idea is exactly reproduced in P. Plowman, B. v. 112, 113.
Cf. 'Inuidus alterius macrescit rebus opimis'; Horace, Epist. i. 2. 57.
135. From Le Roman, l. 5620-3; see vol. iii. p. 436.
140. cherl, dependant. It is remarkable that, throughout the story,
MSS. E. Hn. and Cm. have cherl, but the rest have clerk. In ll. 140,
142, 153, 164, the Camb. MS. is deficient; but it at once gives the
reading cherl in l. 191, and subsequently.
Either reading might serve; in Le Roman, l. 5614, the dependant is
called 'son serjant'; and in l. 5623, he is called 'Li ribaus,' i. e. the
ribald, which Chaucer Englishes by cherl. But when we come to C.
289, the MSS. gives us the choice of 'fals cherl' and 'cursed theef';
very few have clerk (like MS. Sloane 1685). Cf. vol. iii. p. 437.
153, 154. The 'churl's' name was Marcus Claudius, and the 'judge'
was 'Appius Claudius.' Chaucer simply follows Jean de Meun, who
calls the judge Apius; and speaks of the churl as 'Claudius li
chalangieres' in l. 5675.
165. Cf. Le Roman, l. 5623-7; see vol. iii. p. 436.
168-9. From Le Roman, 5636-8, as above.
174. The first foot is defective; read—Thou | shalt have | al, c. al
right, complete justice. MS. Cm. has alle.
184. Cf. Le Roman, l. 5628-33.
203. From Le Roman, 5648-54.
207-253. The whole of this fine passage appears to be original.
There is no hint of it in Le Roman de la Rose, except as regards l.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-74-320.jpg)
![225, where Le Roman (l. 5659) has:—'Car il par amors, sans haïne.'
We may compare the farewell speech of Virginius to his daughter in
Webster's play of Appius and Virginia, Act iv. sc. 1.
240. Iepte, Jephtha; in the Vulgate, Jephte. See Judges, xi. 37, 38.
MSS. E. Hn. have in the margin—'fuit illo tempore Jephte Galaandes'
[error for Galaadites]. This reference by Virginia to the book of
Judges is rather startling; but such things are common enough in old
authors, especially in our dramatists.
255. Here Chaucer returns to Le Roman, 5660-82. The rendering is
pretty close down to l. 276.
280. Agryse of, shudder at; 'nor in what kind of way the worm of
conscience may shudder because of (the man's) wicked life'; cf. 'of
pitee gan agryse,' B. 614. When agryse is used with of, it is
commonly passive, not intransitive; see examples in Mätzner and in
the New E. Dictionary. Cf. been afered, i. e. be scared, in l. 284.
'Vermis conscientiae tripliciter lacerabit'; Innocent III., De Contemptu
Mundi, l. iii. c. 2.
286. Cf. Pers. Tale, I. 93:—'repentant folk, that stinte for to sinne,
and forlete [give up] sinne er that sinne forlete hem.'
Words of the Host.
In the Six-text Edition, pref. col. 58, Dr. Furnivall calls attention to
the curious variations in this passage, in the MSS., especially in ll.
289-292, and in 297-300; as well as in ll. 487, 488 in the Pardoneres
Tale. I note these variations below, in their due places.
287. wood, mad, frantic, furious; esp. applied to the transient
madness of anger. See Kn. Tale, A. 1301, 1329, 1578; also Mids. Nt.
Dr. ii. 1. 192. Cf. G. wüthend, raging.
288. Harrow! also spelt haro; a cry of astonishment; see A. 3286,
3825, B. 4235, c. 'Haro, the ancient Norman hue and cry; the](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-75-320.jpg)
![p. 551, in a passage which is a repetition of the former. The forms
Ronyan and Rinian are evidently corruptions of Ronan, a saint whose
name is well known to readers of 'St. Ronan's Well.' Of St. Ronan
scarcely anything is known. The fullest account that can easily be
found is the following:—
'Ronan, B. and C. Feb. 7.—Beyond the mere mention of his
commemoration as S. Ronan, bishop at Kilmaronen, in Levenax, in
the body of the Breviary of Aberdeen, there is nothing said about
this saint.... Camerarius (p. 86) makes this Ronanus the same as he
who is mentioned by Beda (Hist. Ecc. lib. iii. c. 25). This Ronan died
in A. D. 778. The Ulster annals give at [A. D.] 737 (736)—Mors Ronain
Abbatis Cinngaraid. Ængus places this saint at the 9th of February,'
c.; Kalendars of Scottish Saints, by Bp. A. P. Forbes, 1872, p. 441.
Kilmaronen is Kilmaronock, in the county and parish of Dumbarton.
There are traces of St. Ronan in about seven place-names in
Scotland, according to the same authority. Under the date of Feb. 7
(February vol. ii. 3 B), the Acta Sanctorum has a few lines about St.
Ronan, who, according to some, flourished under King Malduin, A. D.
664-684; or, according to others, about 603. The notice concludes
with the remark—'Maiorem lucem desideramus.' Beda says that
'Ronan, a Scot by nation, but instructed in ecclesiastical truth either
in France or Italy,' was mixed up in the controversy which arose
about the keeping of Easter, and was 'a most zealous defender of the
true Easter.' This controversy took place about A. D. 652, which does
not agree with the date above.
311. Tyrwhitt thinks that Shakespeare remembered this expression
of Chaucer, when he describes the Host of the Garter as frequently
repeating the phrase 'said I well': Merry Wives of Windsor, i. 3. 11; ii.
1. 226; ii. 3. 93, 99.
in terme, in learned terms; cf. Prol. A. 323.
312. erme, to grieve. For the explanation of unusual words, the
Glossary should, in general, be consulted; the Notes are intended,](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-79-320.jpg)
![highways, to the impeding of riders and others, and, by reason of
their excessive weight, to the great deterioration of the houses to
which they are fixed,... it was ordained,... that no one in future
should have a stake bearing either his sign or leaves [i. e. a bush]
extending or lying over the king's highway, of greater length than 7
feet at most,' c. And, at p. 292 of the same work, note 2, Mr. Riley
rightly defines an ale-stake to be 'the pole projecting from the
house, and supporting a bunch of leaves.'
The word ale-stake occurs in Chatterton's poem of Ælla, stanza 30,
where it is used in a manner which shews that the supposed 'Rowley'
did not know what it was like. See my note on this; Essay on the
Rowley Poems, p. xix; and cf. note to A. 667.
322. of a cake; we should now say, a bit of bread; the modern sense
of 'cake' is a little misleading. The old cakes were mostly made of
dough, whence the proverb 'my cake is dough,' i. e. is not properly
baked; Taming of the Shrew, v. 1. 145. Shakespeare also speaks of
'cakes and ale,' Tw. Nt. ii. 3. 124. The picture of the 'Simnel Cakes' in
Chambers' Book of Days, i. 336, illustrates Chaucer's use of the word
in the Prologue, l. 668.
324. The Pardoner was so ready to tell some 'mirth or japes' that the
more decent folks in the company try to repress him. It is a curious
comment on the popular estimate of his character. He has, moreover,
to refresh himself, and to think awhile before he can recollect 'some
honest (i. e. decent) thing.'
327, 328. The Harleian MS. has—
'But in the cuppe wil I me bethinke
Upon some honest tale, whil I drinke.'
The Pardoneres Prologue.
Title. The Latin text is copied from l. 334 below; it appears in the
Ellesmere and Hengwrt MSS. The A. V. has—'the love of money is](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-82-320.jpg)
![metal given in Batman upon Bartholomè; lib. xvi. c. 5. 'Of Laton.
Laton is called Auricalcum, and hath that name, for, though it be
brasse or copper, yet it shineth as gold without, as Isidore saith; for
brasse is calco in Greeke. Also laton is hard as brasse or copper; for
by medling of copper, of tinne, and of auripigment [orpiment] and
with other mettal, it is brought in the fire to the colour of gold, as
Isidore saith. Also it hath colour and likenesse of gold, but not the
value.'
351. The expression 'holy Jew' is remarkable, as the usual feeling in
the middle ages was to regard all Jews with abhorrence. It is
suggested, in a note to Bell's edition, that it 'must be understood of
some Jew before the Incarnation.' Perhaps the Pardoner wished it to
be understood that the sheep was once the property of Jacob; this
would help to give force to l. 365. Cp. Gen. xxx.
The best comment on the virtues of a sheep's shoulder-bone is
afforded by a passage in the Persones Tale (De Ira), I. 602, where
we find—'Sweringe sodeynly withoute avysement is eek a sinne. But
lat us go now to thilke horrible swering of adiuracioun and
coniuracioun, as doon thise false enchauntours or nigromanciens in
bacins ful of water, or in a bright swerd, in a cercle, or in a fyr, or in
a shulder-boon of a sheep'; c. Cf. also a curious passage in
Trevisa's tr. of Higden's Polychronicon, lib. i. cap. 60, which shews
that it was known among the Flemings who had settled in the west
of Wales. He tells us that, by help of a bone of a wether's right
shoulder, from which the flesh had been boiled (not roasted) away,
they could tell what was being done in far countries, 'tokens of pees
and of werre, the staat of the reeme, sleynge of men, and
spousebreche.' Selden, in his notes to song 5 of Drayton's Polyolbion,
gives a curious instance of such divination, taken from Giraldus, Itin.
i. cap. 11; and a writer in the Retrospective Review, Feb. 1854, p.
109, says it is 'similar to one described by Wm. de Rubruquis as
practised among the Tartars.' And see spade-bone in Nares. Cf.
Notes and Queries, 1 S. ii. 20.](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-85-320.jpg)
![commodity [i. e. the indulgences] in the churches, in the public
streets, in taverns and ale-houses. He paid over to his employers as
little as possible, pocketing the balance, as was subsequently proved
against him. The faith of the buyers diminishing, it became
necessary to exaggerate to the fullest extent the merit of the
specific.... The intrepid Tetzel stretched his rhetoric to the very
uttermost bounds of amplification. Daringly piling one lie upon
another, he set forth, in reckless display, the long list of evils which
this panacea could cure. He did not content himself with
enumerating known sins; he set his foul imagination to work, and
invented crimes, infamous atrocities, strange, unheard of, unthought
of; and when he saw his auditors stand aghast at each horrible
suggestion, he would calmly repeat the burden of his song:—Well, all
this is expiated the moment your money chinks in the pope's chest.'
This was in the year 1517.
390. An hundred mark. A mark was worth about 13s. 4d., and 100
marks about £66 13s. 4d. In order to make allowance for the
difference in the value of money in that age, we must at least
multiply by ten; or we may say in round numbers, that the Pardoner
made at least £700 a year. We may contrast this with Chaucer's own
pension of 20 marks, granted him in 1367, and afterwards increased
till, in the very last year of his life, he received in all, according to Sir
Harris Nicolas, as much as £61 13s. 4d. Even then his income did not
quite attain to the 100 marks which the Pardoner gained so easily.
397. dowve, a pigeon; lit. a dove. See a similar line in the Milleres
Tale, A. 3258.
402. namely, especially, in particular; cf. Kn. Ta. 410 (A. 1068).
406. blakeberied. The line means—'Though their souls go a-
blackberrying'; i. e. wander wherever they like. This is a well-known
crux, which all the editors have given up as unintelligible. I have
been so fortunate as to obtain the complete solution of it, which was
printed in Notes and Queries, 4 S. x. 222, xii. 45, and again in my](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-87-320.jpg)
![preface to the C-text of Piers the Plowman, p. lxxxvii. The simple
explanation is that, by a grammatical construction which was
probably due (as will be shewn) to an error, the verb go could be
combined with what was apparently a past participle, in such a
manner as to give the participle the force of a verbal substantive. In
other words, instead of saying 'he goes a-hunting,' our forefathers
sometimes said 'he goes a-hunted.' The examples of this use are at
least seven. The clearest is in Piers Plowman, C. ix. 138, where we
read of 'folk that gon a-begged,' i. e. folk that go a-begging. In
Chaucer, we not only have 'goon a-begged,' Frank. Tale, F. 1580, and
the instance in the present passage, but yet a third example in the
Wyf of Bath's Tale, Group D. 354, where we have 'goon a-
caterwawed,' with the sense of 'to go a-caterwauling'; and it is a
fortunate circumstance that in two of these cases the idiomatic forms
occur at the end of a line, so that the rime has preserved them from
being tampered with. Gower (Conf. Amant. bk. i. ed. Chalmers, pp.
32, 33, or ed. Pauli, i. 110) speaks of a king of Hungary riding out 'in
the month of May,' adding—
'This king with noble purueiance
Hath for him-selfe his chare [car] arayed,
Wherein he wolde ryde amayed,' c.
that is, wherein he wished to ride a-Maying. Again (in bk. v, ed.
Chalmers, p. 124, col. 2, or ed. Pauli, ii. 132) we read of a drunken
priest losing his way:—
'This prest was dronke, and goth a-strayed';
i. e. he goes a-straying, or goes astray.
The explanation of this construction I take to be this; the -ed was
not really a sign of the past participle, but a corruption of the ending
-eth (A. S.-að) which is sometimes found at the end of a verbal
substantive. Hence it is that, in the passage from Piers Plowman
above quoted, one of the best and earliest MSS. actually reads 'folk](https://image.slidesharecdn.com/2625758-250519201854-9d7f92b3/85/Complex-Multiplication-And-Lifting-Problems-Chingli-Chai-Brian-Conrad-88-320.jpg)
Complex Multiplication And Lifting Problems Chingli Chai Brian Conrad
- 1. Complex Multiplication And Lifting Problems Chingli Chai Brian Conrad download https://ebookbell.com/product/complex-multiplication-and-lifting- problems-chingli-chai-brian-conrad-5251516 Explore and download more ebooks at ebookbell.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Abelian Varieties With Complex Multiplication And Modular Functions Pms46 Goro Shimura https://ebookbell.com/product/abelian-varieties-with-complex- multiplication-and-modular-functions-pms46-goro-shimura-51949516 Cyclic Coverings Calabiyau Manifolds And Complex Multiplication 1st Edition Christian Rohde Auth https://ebookbell.com/product/cyclic-coverings-calabiyau-manifolds- and-complex-multiplication-1st-edition-christian-rohde-auth-1143756 Primes Of The Form X2 Ny2 Fermat Class Field Theory And Complex Multiplication With Solutions David A Cox https://ebookbell.com/product/primes-of-the-form-x2-ny2-fermat-class- field-theory-and-complex-multiplication-with-solutions-david-a- cox-47678966 Primes Of The Form X2ny2 Fermat Class Field Theory And Complex Multiplication 2nd Edition David A Cox https://ebookbell.com/product/primes-of-the-form-x2ny2-fermat-class- field-theory-and-complex-multiplication-2nd-edition-david-a- cox-33791454
- 3. Complex Multiplication Reinhard Schertz https://ebookbell.com/product/complex-multiplication-reinhard- schertz-2102422 Complex Multiplication Grundlehren Der Mathematischen Wissenschaften 255 Softcover Reprint Of The Original 1st Ed 1983 S Lang https://ebookbell.com/product/complex-multiplication-grundlehren-der- mathematischen-wissenschaften-255-softcover-reprint-of-the- original-1st-ed-1983-s-lang-51158786 Complex Systems Spanning Control And Computational Cybernetics Foundations Dedicated To Professor Georgi M Dimirovski On His Anniversary Peng Shi https://ebookbell.com/product/complex-systems-spanning-control-and- computational-cybernetics-foundations-dedicated-to-professor-georgi-m- dimirovski-on-his-anniversary-peng-shi-44888180 Complex Systems Smart Territories And Mobility Patricia Sajous https://ebookbell.com/product/complex-systems-smart-territories-and- mobility-patricia-sajous-46383262 Complex Variables Principles And Problem Sessions A K Kapoor https://ebookbell.com/product/complex-variables-principles-and- problem-sessions-a-k-kapoor-46469966
- 5. Mathematical Surveys and Monographs Volume 195 American Mathematical Society Complex Multiplication and Lifting Problems Ching-Li Chai "RIAN #ONRAD Frans Oort
- 6. Complex Multiplication and Lifting Problems
- 8. Mathematical Surveys and Monographs Volume 195 Complex Multiplication and Lifting Problems Ching-Li Chai Brian Conrad Frans Oort American Mathematical Society Providence, Rhode Island
- 9. EDITORIAL COMMITTEE Ralph L. Cohen, Chair Robert Guralnick Michael A. Singer Benjamin Sudakov Michael I. Weinstein 2010 Mathematics Subject Classification. Primary 11G15, 14K02, 14L05, 14K15, 14D15. For additional information and updates on this book, visit www.ams.org/bookpages/surv-195 Library of Congress Cataloging-in-Publication Data Chai, Ching-Li, author. Complex multiplication and lifting problems / Ching-Li Chai, Brian Conrad, Frans Oort. pages cm — (Mathematical surveys and monographs ; volume 195) Includes bibliographical references and index. ISBN 978-1-4704-1014-8 (alk. paper) 1. Multiplication, Complex. 2. Abelian varieties. I. Conrad, Brian, 1970– author. II. Oort, Frans, 1935– author. III. Title. QA564.C44 2014 516.353—dc23 2013036892 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to reprint-permission@ams.org. c 2014 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 19 18 17 16 15 14
- 10. This book is dedicated to John Tate for what he taught us, and for inspiring us
- 12. Contents Preface ix Introduction 1 References 8 Notation and terminology 9 Chapter 1. Algebraic theory of complex multiplication 13 1.1. Introduction 13 1.2. Simplicity, isotypicity, and endomorphism algebras 15 1.3. Complex multiplication 23 1.4. Dieudonné theory, p-divisible groups, and deformations 33 1.5. CM types 65 1.6. Abelian varieties over finite fields 70 1.7. A theorem of Grothendieck and a construction of Serre 76 1.8. CM lifting questions 86 Chapter 2. CM lifting over a discrete valuation ring 91 2.1. Introduction 91 2.2. Existence of CM lifting up to isogeny 102 2.3. CM lifting to a normal domain up to isogeny: counterexamples 109 2.4. Algebraic Hecke characters 117 2.5. Theory of complex multiplication 127 2.6. Local methods 130 Chapter 3. CM lifting of p-divisible groups 137 3.1. Motivation and background 137 3.2. Properties of a-numbers 143 3.3. Isogenies and duality 146 3.4. Some p-divisible groups with small a-number 156 3.5. Earlier non-liftability results and a new proof 161 3.6. A lower bound on the field of definition 164 3.7. Complex multiplication for p-divisible groups 166 3.8. An upper bound for a field of definition 182 3.9. Appendix: algebraic abelian p-adic representations of local fields 185 3.10. Appendix: questions and examples on extending isogenies 191 Chapter 4. CM lifting of abelian varieties up to isogeny 195 4.1. Introduction 195 4.2. Classification and Galois descent by Lie types 211 vii
- 13. viii CONTENTS 4.3. Tensor construction for p-divisible groups 224 4.4. Self-duality and CM lifting 228 4.5. Striped and supersingular Lie types 233 4.6. Complex conjugation and CM lifting 240 Appendix A. Some arithmetic results for abelian varieties 249 A.1. The p-part of Tate’s work 249 A.2. The Main Theorem of Complex Multiplication 257 A.3. A converse to the Main Theorem of Complex Multiplication 292 A.4. Existence of algebraic Hecke characters 296 Appendix B. CM lifting via p-adic Hodge theory 321 B.1. A generalization of the toy model 321 B.2. Construct CM lifting by p-adic Hodge theory 333 B.3. Dieudonné theories over a perfect field of characteristic p 343 B.4. p-adic Hodge theory and a formula for the closed fiber 359 Notes on Quotes 371 Glossary of Notations 373 Bibliography 379 Index 385
- 14. Preface During the Workshop on Abelian Varieties in Amsterdam in May 2006, the three authors of this book formulated two refined versions of a problem concerning lifting into characteristic 0 for abelian varieties over a finite field. These problems address the phenomenon of CM lifting: the lift into characteristic 0 is required to be a CM abelian variety (in the sense defined in 1.3.8.1). The precise formulations appear at the end of Chapter 1 (see 1.8.5), as problems (I) and (IN). Abelian surface counterexamples to (IN) were found at that time; see 2.3.1–2.3.3, and see 4.1.2 for a more thorough analysis. To our surprise, the same counterexam- ples (typical among toy models as defined in 4.1.3) play a crucial role in the general solution to problems (I) and (IN). This book is the story of our adventure guided by CM lifting problems. Ching-Li Chai thanks Hsiao-Ling for her love and support during all these years. He also thanks Utrecht University for hospitality during many visits, including the May 2006 Spring School on Abelian Varieties which concluded with the workshop in Amsterdam. Support by NSF grants DMS 0400482, DMS 0901163, and DMS120027 is gratefully acknowledged. Brian Conrad thanks the many participants in the “CM seminar” at the Univer- sity of Michigan for their enthusiasm on the topic of complex multiplication, as well as Columbia University for its hospitality during a sabbatical visit, and grate- fully acknowledges support by NSF grants DMS 0093542, DMS 0917686, and DMS 1100784. Frans Oort thanks the University of Pennsylvania for hospitality and stimulating environment during several visits. We are also grateful to Burcu Baran, Bas Edixhoven, Ofer Gabber, Johan de Jong, Bill Messing, Ben Moonen, James Parson, René Schoof, and Jonathan Wise for insightful and memorable discussions. ix
- 16. Introduction I restricted myself to characteristic zero: for a short time, the quantum jump to p = 0 was beyond the range . . . but it did not take me too long to make this jump. — Oscar Zariski The arithmetic of abelian varieties with complex multiplication over a number field is fascinating. However this will not be our focus. We study the theory of complex multiplication in mixed characteristic. Abelian varieties over finite fields. In 1940 Deuring showed that an elliptic curve over a finite field can have an endomorphism algebra of rank 4 [33, §2.10]. For an elliptic curve in characteristic zero with an endomorphism algebra of rank 2 (rather than rank 1, as in the “generic” case), the j-invariant is called a singular j- invariant. For this reason elliptic curves with even more endomorphisms, in positive characteristic, are called supersingular.1 Mumford observed as a consequence of results of Deuring that for any elliptic curves E1 and E2 over a finite field κ of characteristic p 0 and any prime = p, the natural map Z ⊗Z Hom(E1, E2)−→ HomZ[Gal(κ/κ)](T(E1), T(E2)) (where on the left side we consider only homomorphisms “defined over κ”) is an isomorphism [118, §1]. The interested reader might find it an instructive exercise to reconstruct this (unpublished) proof by Mumford. Tate proved in [118] that the analogous result holds for all abelian varieties over a finite field and he also incorporated the case = p by using p-divisible groups. He generalized this result into his influential conjecture [117]: An -adic cohomology class2 that is fixed under the Galois group should be a Q-linear combination of fundamental classes of algebraic cycles when the ground field is finitely generated over its prime field. Honda and Tate gave a classification of isogeny classes of simple abelian vari- eties A over a finite field κ (see [50] and [121]), and Tate refined this by describing 1Of course, a supersingular elliptic curve isn’t singular. A purist perhaps would like to say “an elliptic curve with supersingular j-value”. However we will adopt the generally used terminology “supersingular elliptic curve” instead. 2The prime number is assumed to be invertible in the base field. 1
- 17. 2 INTRODUCTION the structure of the endomorphism algebra End0 (A) (working in the isogeny cate- gory over κ) in terms of the Weil q-integer of A, with q = #κ; see [121, Thm. 1]. It follows from Tate’s work (see 1.6.2.5) that an abelian variety A over a finite field κ admits sufficiently many complex multiplications in the sense that its endomor- phism algebra End0 (A) contains a CM subalgebra3 L of rank 2 dim(A). We will call such an abelian variety (in any characteristic) a CM abelian variety and the embedding L → End0 (A) a CM structure on A. Grothendieck showed that over any algebraically closed field K, an abelian variety that admits sufficiently many complex multiplications is isogenous to an abelian variety defined over a finite extension of the prime field [89]. This was previously known in characteristic zero (by Shimura and Taniyama), and in that case there is a number field K ⊂ K such that the abelian variety can be defined over K (in the sense of 1.7.1). However in positive characteristic such abelian varieties can fail to be defined over a finite subfield of K; examples exist in every dimension 1 (see Example 1.7.1.2). Abelian varieties in mixed characteristic. In characteristic zero, an abelian variety A gives a representation of the endomorphism algebra D = End0 (A) on the Lie algebra Lie(A) of A. If A has complex multiplication by a CM algebra L of degree 2 dim(A) then the isomorphism class of the representation of L on Lie(A) is called the CM type of the CM structure L → End0 (A) on A (see Lemma 1.5.2 and Definition 1.5.2.1). As we noted above, every abelian variety over a finite field is a CM abelian vari- ety. Thus, it is natural to ask whether every abelian variety over a finite field can be “CM lifted” to characteristic zero (in various senses that are made precise in 1.8.5). One of the obstacles4 in this question is that in characteristic zero there is the no- tion of CM type that is invariant under isogenies, whereas in positive characteristic whatever can be defined in an analogous way is not invariant under isogenies. For this reason we will use the terminology “CM type” only in characteristic zero. For instance, the action of the endomorphism ring R = End(A0) of an abelian variety A0 on the Lie algebra of A0 in characteristic p 0 defines a representa- tion of R/pR on Lie(A0). Given an isogeny f : A0 → B0 we get an identification End0 (A0) = End0 (B0) of endomorphism algebras, but even if End(A0) = End(B0) under this identification, the representations of this common endomorphism ring on Lie(A0) and Lie(B0) may well be non-isomorphic since Lie(f) may not be an isomorphism. Moreover, if we have a lifting A of A0 over a local domain of char- acteristic 0, in general the inclusion End(A) ⊂ End(A0) is not an equality. If the inclusion End0 (A) ⊂ End0 (A0) is an equality then the character of the representa- tion of End(A0) on Lie(A0) is the reduction of the character of the representation of End(A) on Lie(A). This relation can be viewed as an obstruction to the existence of CM lifting with the full ring of integers of a CM algebra operating on the lift; see 4.1.2, especially 4.1.2.3–4.1.2.4, for an illustration. In the case when End(A0) contains the ring of integers OL of a CM algebra L ⊂ End0 (A0) with [L : Q] = 2 dim(A0), the representation of OL/pOL on Lie(A0) turns out to be quite useful, despite the fact that it is not an isogeny invariant. Its class in a suitable K-group will be called the Lie type of (A0, OL → End(A0)). 3A CM algebra is a finite product of CM fields; see Definition 1.3.3.1. 4surely also part of the attraction
- 18. INTRODUCTION 3 The above discrepancy between the theories in characteristic zero and charac- teristic p 0 is the basic phenomenon underlying this entire book. Before dis- cussing its content, we recall the following theorem of Honda and Tate ([50, §2, Thm. 1] and [121, Thm. 2]). For an abelian variety A0 over a finite field κ there is a finite extension κ of κ and an isogeny (A0)κ → B0 such that B0 admits a CM lifting over a local domain of characteristic zero with residue field κ . This result has been used in the study of Shimura varieties, for settings where the ground field is an algebraic closure of Fp and isogeny classes (of structured abelian varieties) are the objects of interest; see [135]. Our starting point comes from the following questions which focus on controlling ground field extensions and isogenies. For an abelian variety A0 over a finite field κ, to ensure the existence of a CM lifting over a local domain with characteristic zero and residue field κ of finite degree over κ, (a) may we choose κ = κ? (b) is an isogeny (A0)κ → B0 necessary? These questions are formulated in various precise forms in 1.8. An isogeny is necessary. Question (b) was answered in 1992 (see [93]) as follows. There exist (many) abelian varieties over Fp that do not admit any CM lifting to characteristic zero. The main point of [93] is that a CM liftable abelian variety over Fp can be defined over a small finite field. This idea is further pursued in Chapter 3, where the size, or more accurately the minima5 of the size, of all possible fields of definition of the p-divisible group of a given abelian variety over Fp is turned into an obstruction for the existence of a CM lifting to characteristic 0. This is used to show (in 3.8.3) that in “most” isogeny classes of non-ordinary abelian varieties of dimension ⩾ 2 over finite fields there is a member that has no CM lift to characteristic 0. (In dimension 1 a CM lift to characteristic 0 always exists, over the valuation ring of the minimal possible p-adic field, by Deuring Lifting Theorem; see 1.7.4.6.) We also provide effectively computable examples of abelian varieties over explicit finite fields such that there is no CM lift to characteristic 0. A field extension might be necessary—depending on what you ask. Bearing in mind the necessity to modify a given abelian variety over a finite field to guarantee the existence of a CM lifting, we rephrase question (a) in a more precise version (a) below. (a) Given an abelian variety A0 over a finite field κ of characteris- tic p, is it necessary to extend scalars to a strictly larger finite field κ ⊃ κ (depending on A0) to ensure the existence of a κ -rational isogeny (A0)κ → B0 such that B0 admits a CM lifting over a characteristic 0 local domain R with residue field κ ? It turns out there are two quite different answers to question (a) , depending on whether one requires the local domain R of characteristic 0 to be normal. The subtle distinction between using normal or general local domains for the lifting 5The size of a finite field κ1 is smaller than the size of a finite field κ2 if κ1 is isomorphic to a subfield of κ2, or equivalently if #κ1 | #κ2. Among the sizes of a family of finite fields there may not be a unique minimal element.
- 19. 4 INTRODUCTION went unnoticed for a long time. Once this distinction came in focus, answers to the resulting questions became available. If we ask for a CM lifting over a normal domain up to isogeny, in general a base field extension before modification by an isogeny is necessary. This is explained in 2.1.2, where we formulate the “residual reflex obstruction”, the idea for which goes as follows. Over an algebraically closed field K of characteristic zero, we know that a simple CM abelian variety B with K-valued CM type Φ (for the action of a CM field L) is defined over a number field in K containing the reflex field E(Φ) of Φ. Suppose that for every K-valued CM type Φ of L, the residue field of E(Φ) at any prime above p is not contained in the finite field κ with which we began in question (a). In such cases, for every CM structure L → End0 (A0) on A0 and any abelian variety B0 over κ which is κ-isogenous to A0, there is no L-linear CM lifting of B0 over a normal local domain R of characteristic zero with residue field κ.6 In 2.3.1–2.3.3 we give such an example, a supersingular abelian surface A0 over Fp2 with End(A0) = Z[ζ5] for any p ≡ ±2 (mod 5). A much broader class of examples is given in 2.3.5, consisting of absolutely simple abelian varieties (with arbitrarily large dimension) over Fp for infinitely many p. Note that passing to the normalization of a complete local noetherian domain generally enlarges the residue field. Hence, if we drop the condition that the mixed characteristic local domain R be normal then the obstruction in the preceding consideration dissolves. And in fact we were put on the right track by mathematics itself. The phenomenon is best illustrated in the example in 4.1.2, which is the same as the example in 2.3.1 already mentioned: an abelian surface C0 over Fp2 with CM order Z[ζ5] that, even up to isogeny, is not CM liftable to a normal local domain of characteristic zero. On the other hand, this abelian surface C0 is CM liftable to an abelian scheme C over a mixed characteristic non-normal local domain of characteristic zero, though the maximal subring of Z[ζ5] whose action lifts to C is non-Dedekind locally at p; see 4.1.2.7 This example is easy to construct, and the proof of the existence of a CM lifting, possibly after applying an Fp2 -rational isogeny, is not difficult either. In Chapter 4 we show that the general question of existence of a CM lifting after an appropriate isogeny can be reduced to the same question for (a mild gen- eralization of) the example in 4.1.2, enabling us to prove: every abelian variety A0 over a finite field κ admits an isogeny A0 → B0 over κ such that B0 admits a CM lifting to a mixed characteristic local domain with residue field κ. There are refined lifting problems, such as specifying at the beginning which CM structure on A0 is to be lifted, or even what its CM type should be on a geometric fiber in characteristic 0. These matters will also be addressed. 6The source of obstructions is that the base field κ might be too small to contain at least one characteristic p residue field of the reflex field E(Φ) for at least one CM type Φ on L. Thus, the field of definition of the generic fiber of the hypothetical lift may be too big. Likewise, an obstruction for question (b) is that the field of definition of the p-divisible group A0[p∞] may be too big (in a sense that is made precise in 3.8.3 and illustrated in 3.8.4–3.8.5). 7No modification by isogeny is necessary in this example, but the universal deformation for C0 with its Z[ζ5]-action is a non-algebraizable formal abelian scheme over W(Fp2 ).
- 20. INTRODUCTION 5 Our basic method is to “localize” various CM lifting problems to the corre- sponding problems for p-divisible groups. Although global properties of abelian va- rieties are often lost in this localization process, the non-rigid nature of p-divisible groups can be an advantage. In Chapter 3 the size of fields of definition of a p- divisible group in characteristic p appears as an obstruction to the existence of CM lifting. The reduction steps in Chapter 4 rely on a classification and descent of CM p-divisible groups in characteristic p with the help of their Lie types (see 4.2.2, 4.4.2). In addition, the “Serre tensor construction” is applied to p-divisible groups, both in characteristic p and in mixed characteristic (0, p); see 1.7.4 and 4.3.1 for this general construction. Survey of the contents. In Chapter 1 we start with a survey of general facts about CM abelian varieties and their endomorphism algebras. In particular, we discuss the deformation theory of abelian varieties and p-divisible groups, and we review results in Honda-Tate theory that describe isogeny classes and endomor- phism algebras of abelian varieties over a finite field in terms of Weil integers. We conclude by formulating various CM lifting questions in 1.8. These are studied in the following chapters. We will see that the questions can be answered with some precision. In Chapter 2 we formulate and study the “residual reflex condition”. Using this condition we construct several examples of abelian varieties over finite fields κ such that, even after applying a κ-isogeny, there is no CM lifting to a normal local domain with characteristic zero and residue field of finite degree over κ; see 2.3. It is remarkable that many such examples exist, but we do not know whether we have characterized all possible examples; see 2.3.7. We then study algebraic Hecke characters and review part of the theory of com- plex multiplication due to Shimura and Taniyama. Using the relationship between algebraic Hecke characters for a CM field L and CM abelian varieties with CM by L (the precise statement of which we review and prove), we use global methods to show that the residual reflex condition is the only obstruction to the existence of CM lifting up to isogeny over a normal local domain of characteristic zero. We also give another proof by local methods (such as p-adic Hodge theory). In Chapter 3 we take up methods described in [93]. In that paper classical CM theory in characteristic zero was used. Here we use p-divisible groups instead of abelian varieties and show that the size of fields of definition of a p-divisible group in characteristic p is a non-trivial obstruction to the existence of a CM lifting. In 3.3 we study the notion of isogeny for p-divisible groups over a base scheme (including its relation with duality). We show, in one case of the CM lifting problem left open in [93, Question C], that an isogeny is necessary. Our methods also provide effectively computed examples. Some facts about CM p-divisible groups explained in 3.7 are used in 3.8 to get an upper bound of a field of definition for the closed fiber of a CM p-divisible group. In Appendix 3.9, we use the construction (in 3.7) of a p-divisible group with any given p-adic CM type over the reflex field to produce a semisimple abelian crystalline p-adic representation of the local Galois group such that its restriction to the inertia group is “algebraic” with algebraic part that we may prescribe arbitrarily in accordance with some necessary conditions (see 3.9.4 and 3.9.8).
- 21. 6 INTRODUCTION In Chapter 4 we show CM liftability after an isogeny over the finite ground field (lifting over a characteristic zero local domain that need not be normal). That is, every CM structure (A0, L → End0 (A0)) over a finite field κ has an isogeny over κ to a CM structure (B0, L → End0 (B0)) that admits a CM lifting; (see 4.1.1). This statement is immediately reduced to the case when L is a CM field (not just a CM algebra) and the whole ring OL of integers of L operates on A0, which we assume. Our motivation comes from the proof in 4.1.2 (using an algebraization argument at the end of 4.1.3) that the counterexample in 2.3.1 to CM lifting over a normal local domain satisfies this property. In general, after an easy reduction to the isotypic case, we apply the Serre-Tate deformation theorem to localize the problem at p-adic places v of the maximal totally real subfield L+ of a CM field L ⊆ End0 (A0) of degree 2 dim(A0). This reduces the existence of a CM lifting for the abelian variety A0 to a corresponding problem for the CM p-divisible group A0[v∞ ] attached to v.8 We formulate several properties of v with respect to the CM field L; any one of them ensures the existence of a CM lifting of A0[v∞ ]κ after applying a κ-isogeny to A0[v∞ ] (see 4.1.6, 4.1.7, and 4.5.7). These properties involve the ramification and residue fields of L and L+ relative to v. If v violates all of these properties then we call it bad (with respect to L/L+ and κ). Let Lv := L ⊗L+ L+ v . After applying a preliminary κ-isogeny to arrange that OL ⊂ End(A0), for v that are not bad we apply an OL-linear κ-isogeny to arrange that the Lie type of the OL,v- factor of Lie(A0) (i.e., its class in a certain K-group of (OL,v/(p)) ⊗ κ-modules) is “self-dual”. Under the self-duality condition (defined in 4.4.3) we produce an OL,v- linear CM lifting of A0[v∞ ]κ by specializing a suitable OL,v-linear CM v-divisible group in mixed characteristic; see 4.4.6. We use an argument with deformation rings to eliminate the intervention of κ: if every p-adic place v of L+ is not bad then there exists a κ-isogeny A0 → B0 such that OL ⊂ End(B0) and the pair (B0, OL → End(B0)) admits a lift to characteristic 0 without increasing κ. If some p-adic place v of the totally real field L+ is bad then the above argument does not work because in that case no member of the OL,v-linear κ-isogeny class of the p-divisible group A0[v∞ ] has a self-dual Lie type. Instead we change A0[v∞ ] by a suitable OL,v-linear κ-isogeny so that its Lie type becomes as symmetric as possible, a condition whose precise formulation is called “striped”. Such a p-divisible group is shown to be isomorphic to the Serre tensor construction applied to a special class of 2-dimensional p-divisible groups of height 4 that are similar to the ones arising from the abelian surface counterexamples in 2.3.1; we call these toy models (see 4.1.3, especially 4.1.3.2). These “toy models” are sufficiently special that we can analyze their CM lift- ing properties directly; see 4.2.10 and 4.5.15(iii). After this key step we deduce the existence of a CM lifting of A0[v∞ ]κ from corresponding statements for (the p-divisible group version of) toy models. In the final step, once again we use de- formation theory to produce an abelian variety B0 isogenous to (the original) A0 over κ and a CM lifting of B0 over a possibly non-normal 1-dimensional complete local noetherian domain of characteristic 0 with residue field κ. Although OL acts 8See 1.4.5.3 for the statement of the Serre–Tate deformation theorem, and 2.2.3 and 4.6.3.1 for a precise statement of the algebraization criterion that is used in this localization step.
- 22. INTRODUCTION 7 on the closed fiber, we can only ensure that a subring of OL of finite index9 acts on the lifted abelian scheme (see 4.6.4). Appendix A. In Appendix A.1 we provide a self-contained development of the proof of the p-part of Tate’s isogeny theorem for abelian varieties over finite fields of characteristic p, as well as a proof of Tate’s formula for the local invariants at p-adic places for endomorphism algebras of simple abelian varieties over such fields. (An exposition of these results is also given in [79]; our treatment uses less input from non-commutative algebra.) Appendices A.2 and A.3 provide purely algebraic proofs of the Main Theorem of Complex Multiplication for abelian varieties, as well as a converse result, both of which are used in essential ways in Chapter 2. In Appendix A.4 we use Shimura’s method to show that an algebraic Hecke character with a given algebraic part can be constructed over the field of moduli of the algebraic part, with control over places of bad reduction. In the special case of the reflex norm of a CM type (L, Φ), combining this construction of algebraic Hecke characters with the converse to the Main Theorem of CM in A.3 proves that over the associated field of moduli M ⊂ Q (a subfield of the Hilbert class field of the reflex field E(L, Φ)) there exists a CM abelian variety A with CM type (L, Φ) such that A has good reduction at all p-adic places of M; see A.4.6.5. Since M is the smallest possible field of definition given (L, Φ), this existence result is optimal in terms of its field of definition. Typically M = E(L, Φ), and this is regarded as a “class group obstruction” to finding A with its CM structure by L over E(L, Φ), a well-known phenomenon in the classical CM theory of elliptic curves. (In the “local” setting of CM p-divisible groups over p-adic integer rings there are no class group problems and one gets a better result: in 3.7 we use the preceding global construction over the field of moduli to prove that for any p-adic CM type (F, Φ) and the associated p-adic reflex field E ⊂ Qp over Qp there exists a CM p-divisible group over OE with p-adic CM type (F, Φ).) Appendix B. In Appendices B.1 and B.2, we give two versions of a more di- rect (but more complicated) proof of the existence of CM liftings for a higher- dimensional generalization of the toy model.10 The first version uses Raynaud’s theory of group schemes of type (p, . . . , p). The second version uses recent devel- opments in p-adic Hodge theory. We hope that material described there will be useful in the future. In Appendix B.3 we compare several Dieudonné theories over a perfect base field of characteristic p 0. In Appendix B.4 we give a formula for the Dieudonné module of the closed fiber of a finite flat commutative group scheme, constructed using integral p-adic Hodge theory; this formula is used in B.2. 9This subring of finite index can be taken to be Z + pOL. 10In the original proof of our main CM lifting result in 4.1.1, the case of a bad place v|p of L+ was reduced through the Serre tensor construction to this existence result. Both B.1 and B.2 are logically independent of results in Chapter 4. Readers who cannot wait to see a proof of the existence of a CM lifting (without modification by any isogeny) for such a higher-dimensional toy model may proceed directly to B.1 or B.2, after consulting 4.2 for the definition of the Lie type of an O-linear p-divisible group and related notation.
- 23. 8 INTRODUCTION References (1) Abelian varieties. In Mumford’s book [82] the theory of abelian varieties is developed over an algebraically closed base field, and we need the theory over a general field; references addressing this extra generality are Milne’s article [76] (which rests on [82]) and the forthcoming book [45]. Since [45] is not yet in final form we do not refer to it in the main text, but the reader should keep in mind that many results for which we refer to [82] and [76] are also treated in [45]. We refer the reader to [83, Ch. 6, §1–§2] for a self- contained development of the elementary properties of abelian schemes, which we freely use. (For example, the group law is necessarily commutative and is determined by the identity section, as in the theory over a field.) (2) Semisimple algebras. We assume familiarity with the classical theory of finite-dimensional semisimple algebras over fields (including the theory of their splitting fields and maximal commutative subfields). A suitable reference for this material is [53, §4.1–4.6]; another reference is [11]. In 1.2.2–1.2.4 we review some of the facts that we need from that theory. (3) Descent theory and formal schemes. In many places, we need to use the techniques of descent theory and Grothendieck topologies (especially the fppf topology, though in some situations we use the fpqc topology to perform de- scent from a completion). This is required for arguments with group schemes, even over a field, such as in considerations with short exact sequences. For accounts of descent theory, we refer the reader to [10, §6.1–6.2], and to [39, Part 1] for a more exhaustive discussion. These techniques are discussed in a manner well-suited to group schemes in [98] and [30, Exp. IV–VIA]. Our arguments with deformation theory rest on the theory of formal schemes, especially Grothendieck’s formal GAGA and algebraization theo- rems. A succinct overview of these matters is given in [39, Part 4], and the original references [34, I, §10; III1, §4–§5] are also highly recommended. (4) Dieudonné theory and p-divisible groups. To handle p-torsion phenom- ena in characteristic p 0 we use Dieudonné theory and p-divisible groups. Brief surveys of some basic definitions and properties in this direction are given in 1.4, 3.1.2–3.1.6, and B.3.5.1–B.3.5.5. We refer the reader to [119], [71] and [110, §6] for more systematic discussions of basic facts concerning p-divisible groups, and to [29] and [41, Ch. II–III] for self-contained devel- opments of (contravariant) Dieudonné theory, with applications to p-divisible groups. Contravariant Dieudonné theory is used in Chapters 1–4. Covariant Dieudonné theory is used in Appendix B.1 because the alter- native proof there of the main result of Chapter 4 uses a covariant version of p-adic Hodge theory. A brief summary of covariant Dieudonné can be found in B.3.5.6–B.3.6.7. We recommend [136] for Cartier theory; an older standard reference is [69]. A very useful technique within the deformation theory of p-divisible groups is Grothendieck–Messing theory, which is developed from scratch in [75]. Al- though we do not provide an introduction to this topic, we hope that our applications of it may inspire an interested reader who is not familiar with this technique to learn more about it.
- 24. NOTATION AND TERMINOLOGY 9 Notation and terminology • Numerical labeling of text items and displayed expressions. – We use “x.y.z”, “x.y.z.w”, etc. for text items (sub-subsections, results, remarks, definitions, etc.), arranged lexicographically without repetition. – Any labeling of displayed expressions (equations, commutative diagrams, etc.) is indicated with parentheses, so “see (x.y.z)” means that one should look at the zth displayed expression in subsection x.y. This convention avoids confusion with the use of “x.y.z” to label a text item. – Any label for a text item is uniquely assigned, so even though “see x.y.z” does not indicate if it is a sub-subsection or theorem (or lemma, etc.), there is no ambiguity for finding it in this book. • Convention on notation. – p denotes a prime number. – CM fields are usually denoted by L. – K often stands for an arbitrary field, κ is usually used to denote either a residue field or a finite field of characteristic p. – V ∨ denotes the dual of a finite-dimensional vector space V over a field. – k denotes a perfect field, often of characteristic p 0. In 4.2–4.6, k is an algebraically closed field of characteristic p. – K0 is the fraction field of W(k), where k is a perfect field of characteristic p 0 and W(k) is the ring of p-adic Witt vectors with entries in k. – Abelian varieties are usually written as A, B, or C, and p-divisible groups are often denoted as G or as X or Y . – The p-divisible group attached to an abelian variety or an abelian scheme A is denoted by A[p∞ ]; its subgroup scheme of pn -torsion points is A[pn ]. • Fields and their extensions. – For a field K, we write K to denote an algebraic closure and Ks to denote a separable closure. – An extension of fields K /K is primary if K is separably algebraically closed in K (i.e., the algebraic closure of K in K is purely inseparable over K). – For a number field L we write OL to denote its ring of integers. Similar notation is used for non-archimedean local fields. – If q is a power of a prime p, Fq denotes a finite field with size q (sometimes understood to be the unique subfield of order q in a fixed algebraically closed field of characteristic p). If κ and κ are abstract finite fields with respective sizes q = pn and q = pn for integers n, n ⩾ 1 then κ ∩ κ denotes the unique subfield of either κ or κ with size pgcd(n,n ) ; the context will always make clear if this is being considered as a subfield of either κ or κ . Likewise, κκ denotes κ ⊗κ∩κ κ , a common extension of κ and κ with size plcm(n,n ) . • Base change. – If T → S is a map of schemes and S is an S-scheme, then TS denotes the S -scheme T ×S S if S is understood from context. – When S = Spec(R) and S = Spec(R ) are affine, we may write TR to denote T ⊗R R := T ×Spec(R) Spec(R ) when R is understood from context.
- 25. 10 INTRODUCTION • Abelian varieties and homomorphisms between them. – The dual of an abelian variety A is denoted At . – For an abelian variety A over a field K and a prime not divisible by char(K), upon choosing a separable closure Ks of K (often understood from context) the -adic Tate module T(A) denotes lim ← − A[n ](Ks) and V(A) denotes Q ⊗Z T(A). – For any abelian varieties A and B over a field K, Hom(A, B) denotes the group of homomorphisms A → B over K, and Hom0 (A, B) denotes Q ⊗Z Hom(A, B). (Since Hom(A, B) → Hom(AK, BK) is injective, Hom(A, B) is a finite free Z-module since the same holds over K by [82, §19, Thm. 3].) – When B = A we write End(A) and End0 (A) respectively, and call End0 (A) the endomorphism algebra of A (over K). The endomorphism algebra End0 (A) is an invariant which only depends on A up to isogeny over K, in contrast with the endomorphism ring End(A). – We write A ∼ B to denote that abelian varieties A and B over K are K-isogenous. – To avoid any possible confusion with notation found in the literature, we emphasize that what we call Hom(A, B) and Hom0 (A, B) are sometimes denoted by others as HomK(A, B) and Hom0 K(A, B).11 • Adeles and local fields. – We write AL to denote the adele ring of a number field L, AL,f to denote the factor ring of finite adeles, and A and Af in the case L = Q. – If v is a place of a number field L then Lv denotes the completion of L with respect to v; OL,v denotes the valuation ring OLv of Lv in case v is non-archimedean, with residue field κv whose size is denoted qv. – For a place w of Q we define Lw := Qw ⊗Q L = v|w Lv, and in case w is the -adic place for a prime we define OL, := Z ⊗Z OL = v| OL,v. • Class field theory and reciprocity laws. – The Artin maps of local and global class field theory are taken with the arithmetic normalization, which is to say that local uniformizers are carried to arithmetic Frobenius elements.12 – recL : A× L /L× → Gal(Lab /L) denotes the arithmetically normalized glob- al reciprocity map for a number field L. – The composition of A× L A× L /L× with recL is denoted rL. – For a non-archimedean local field F we write rF : F× → Gal(Fab /F) to denote the arithmetically normalized local reciprocity map. • Frobenius and Verschiebung. – For a commutative group scheme N over an Fp-scheme S, N(p) denotes the base change of N by the absolute Frobenius endomorphism of S. The relative Frobenius homomorphism is denoted FrN/S : N → N(p) , and the 11with the notation Hom(A, B) and Hom0 (A, B) then reserved to mean the analogues for AK and BK over K, or equivalently for AKs and BKs over Ks (see Lemma 1.2.1.2). 12Recall that for a non-archimedean local field F with residue field of size q, an element of Gal(Fs/F) is called an arithmetic (resp. geometric) Frobenius element if its effect on the residue field of Fs is the automorphism x → xq (resp. x → x1/q); this automorphism of the residue field is likewise called the arithmetic (resp. geometric) Frobenius automorphism. We choose the arithmetic normalization of class field theory so that uniformizers correspond to Frobenius endo- morphisms of abelian varieties in the Main Theorem of Complex Multiplication.
- 26. NOTATION AND TERMINOLOGY 11 Verschiebung homomorphism for S-flat N of finite presentation denoted VerN/S : N(p) → N see [30, VIIA, 4.2–4.3]). If S is understood from context then we may denote these as FrN and VerN respectively. For n ⩾ 1, the pn -fold relative Frobenius and Verschiebung homomor- phisms N → N(pn ) and N(pn ) → N are respectively denoted FrN/S,pn and VerN/S,pn . – For a perfect field k with char(k) = p 0 and the unique lift σ : W(k) → W(k) of the Frobenius automorphism y → yp of k, a Dieudonné module over k is a W(k)-module M equipped with additive endomorphisms F : M → M and V : M → M such that F ◦ V = [p]M = V ◦ F, F(c m) = σ(c) F(m), and c V(m) = V(σ(c) m) for all c ∈ W(k) and m ∈ M; these are the left modules over the Dieudonné ring Dk (see 1.4.3.1). – The semilinear operators F and V on a Dieudonné module M corre- spond to respective W(k)-linear maps M(p) → M and M → M(p) , where M(p) := W(k) ⊗σ,W (k) M.
- 28. CHAPTER 1 Algebraic theory of complex multiplication The theory of complex multiplication. . . is not only the most beautiful part of mathematics but also of all science. — David Hilbert 1.1. Introduction 1.1.1. Lifting questions. A natural question early in the theory of abelian vari- eties is whether every abelian variety in positive characteristic admits a lift to char- acteristic 0. That is, given an abelian variety A0 over a field κ with char(κ) 0, does there exist a local domain R of characteristic zero with residue field κ and an abelian scheme A over R whose special fiber Aκ is isomorphic to A0? We may also wish to demand that a specified polarization of A0 or subring of the endomor- phism algebra of A0 (or both) also lifts to A. (The functor A Aκ from abelian R-schemes to abelian varieties over κ is faithful, by consideration of finite étale torsion levels; see the beginning of 1.4.4.) Suppose there is an affirmative solution A to such a lifting problem over some local domain R as above. Let’s see that we can arrange for a solution to be found over a local noetherian domain (that is even complete). This rests on a direct limit technique (that is very useful throughout algebraic geometry), as follows. Observe that for the directed system of noetherian local subrings Ri with local inclusions Ri → R, we have R = lim − → Ri. In [34, IV3, §8–§12; IV4, §17] there is an exhaustive development of the technique of descent through direct limits. The principle is that if {Di} is a directed system of rings with limit D, and if we are given a “finitely presented” algebro-geometric situation over D (a diagram of finitely many D-schemes of finite presentation, equipped with with finitely many D-morphisms among them and perhaps some finitely presented quasi-coherent sheaves on them, some of which may be D-flat, etc.) then the entire structure descends to Di for sufficiently large i. Moreover, if we increase i enough then we can also descend “reasonable” properties (such as flatness for morphisms or sheaves, and properness, surjectivity, smoothness, and having geometrically connected fibers for morphisms), any two descents become isomorphic after increasing i some more, and so on. The results of this direct limit formalism are intuitively plausible, but their proofs can be rather non-obvious to the uninitiated (e.g., descending the properties of flatness and surjectivity). We will often use this limit formalism without much explanation, and we hope that the plausibility of such results is sufficient for a non- expert reader to follow the ideas. Everything we need is completed proved in the cited sections of [34]. As a basic example, since the condition of being an abelian 13
- 29. 14 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION scheme amounts to a group scheme diagram for a smooth proper R-scheme having geometrically connected fibers, the abelian scheme A over R descends to an abelian scheme over Ri0 for some sufficiently large i0. The residue field κi0 of Ri0 is merely a subfield of κ. By [34, 0III, 10.3.1], there is a faithfully flat local extension Ri0 → R with R noetherian and having residue field κ over κi0 . By faithful flatness, every minimal prime of R has residue characteristic 0, so we can replace R with its quotient by such a prime to obtain a solution over a complete local noetherian domain with residue field κ. Typically our liftings will be equipped with additional structure such as a po- larization, and so the existence of an affirmative solution for our lifting problem (for a given A0) often amounts to an appropriate deformation ring R for A0 (over a Cohen ring for κ) admitting a generic point in characteristic 0; the coordinate ring of the corresponding irreducible component of Spec(R) is such an R. If κ → κ is an extension of fields and W → W is the associated extension of Cohen rings then often there is a natural isomorphism R W ⊗W R relating the corresponding de- formation rings for A0 and (A0)κ (see 1.4.4.5, 1.4.4.13, and 1.4.4.14). Thus, if R has a generic point of characteristic 0 then so does R. Hence, to prove an affirmative answer to lifting questions as above it is usually enough to consider algebraically closed κ. For example, the general lifting problem for polarized abelian varieties (allowing polarizations for which the associated symmetric isogeny A0 → At 0 is not separable) was solved affirmatively by Norman-Oort [85, Cor. 3.2] when κ = κ, and the general case follows by deformation theory (via 1.4.4.14 with O = Z). 1.1.2. Refinements. When a lifting problem as above has an affirmative solution, it is natural to ask if the (complete local noetherian) base ring R for the lifting can be chosen to satisfy nice ring-theoretic properties, such as being normal or a discrete valuation ring. Slicing methods allow one to find an R with dim(R) = 1 (see 2.1.1 for this argument), but normalization generally increases the residue field. Hence, asking that the complete local noetherian domain R be normal or a discrete valuation ring with a specified residue field κ is a non-trivial condition unless κ is algebraically closed. We are interested in versions of the lifting problem for finite κ when we lift not only the abelian variety but also a large commutative subring of its endomorphism algebra. To avoid counterexamples it is sometimes necessary to weaken the lifting problem by permitting the initial abelian variety A0 to be replaced with another in the same isogeny class over κ. In 1.8 we will precisely formulate several such lifting problems involving complex multiplication, and the main result of our work is a rather satisfactory solution to these lifting problems. 1.1.3. Purpose of this chapter. Much of the literature on complex multiplica- tion involves either (i) working over an algebraically closed ground field, (ii) making unspecified finite extensions of the ground field, or (iii) restricting attention to sim- ple abelian varieties. To avoid any uncertainty about the degree of generality in which various foundational results in the theory are valid, as well as to provide a convenient reference for subsequent considerations, in this chapter we provide an extensive review of the algebraic theory of complex multiplication over a general base field. This includes special features of the theory over finite fields and over fields of characteristic 0, and for some important proofs we refer to the original literature (e.g., papers of Tate). Some arithmetic aspects (such as reflex fields and
- 30. 1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 15 the Main Theorem of Complex Multiplication) are discussed in Chapter 2, and Appendix A provides proofs of the Main Theorem of Complex Multiplication and some results of Tate over finite fields. 1.2. Simplicity, isotypicity, and endomorphism algebras 1.2.1. Simple abelian varieties. An abelian variety A over a field K is simple (over K) if it is non-zero and contains no non-zero proper abelian subvarieties. Simplicity is not generally preserved under extension of the base field; see Example 1.6.3 for some two-dimensional examples over finite fields and over Q. An abelian variety A over K is absolutely simple (over K) if AK is simple. 1.2.1.1. Lemma. If A is absolutely simple over a field K then for any field ex- tension K /K, the abelian variety AK over K is simple. Proof. This is an application of direct limit and specialization arguments, as we now explain. Assume for some K /K that there is a non-zero proper abelian sub- variety B ⊂ AK . By replacing K with an algebraic closure we may arrange that K and then especially K is algebraically closed. (The algebraically closed property for K is unimportant, but it is crucial that we have it for K.) By expressing K as a direct limit of finitely generated K-subalgebras, there is a finitely generated K-subalgebra R ⊂ K such that B = BK for an abelian scheme B → Spec(R) that is a closed R-subgroup of AR. The constant positive dimension of the fibers of B → Spec(R) is strictly less than dim(A), as we may check using the K -fiber B ⊂ AK . Since K is algebraically closed we can choose a K-point x of Spec(R). The fiber Bx is a non-zero proper abelian subvariety of A, contrary to the simplicity of A over K. For a pair of abelian varieties A and B over a field K, Hom0 (AK , BK ) can be strictly larger than Hom0 (A, B) for some separable algebraic extension K /K. For example, if E is an elliptic curve over Q then considerations with the tangent line over Q force End0 (E) = Q, but it can happen that End0 (EL) = L for an imaginary quadratic field L (e.g., E : y2 = x3 − x and L = Q( √ −1)). Scalar extension from number fields to C or from an imperfect field to its perfect closure are useful techniques in the study of abelian varieties, so there is natural interest in considering ground field extensions that are not separable algebraic (e.g., non-algebraic or purely inseparable). It is an important fact that allowing such general extensions of the base field does not lead to more homomorphisms: 1.2.1.2. Lemma (Chow). Let K /K be an extension of fields that is primary (i.e., K is separably algebraically closed in K ). For abelian varieties A and B over K, the natural map Hom(A, B) → Hom(AK , BK ) is bijective. Proof. See [23, Thm. 3.19] for a proof using faithfully flat descent (which is reviewed at the beginning of [23, §3]). An alternative proof is to show that the locally finite type Hom-scheme Hom(A, B) over K is étale. We shall be interested in certain commutative rings acting faithfully on abelian varieties, so we need non-trivial information about the structure of endomorphism algebras of abelian varieties. The study of such rings rests on the following funda- mental result.
- 31. 16 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION 1.2.1.3. Theorem (Poincaré reducibility). Let A be an abelian variety over a field K. For any abelian subvariety B ⊂ A, there is is abelian subvariety B ⊂ A such that the multiplication map B × B → A is an isogeny. In particular, if A = 0 then there exist pairwise non-isogenous simple abelian varieties C1, . . . , Cs over K such that A is isogenous to Cei i for some ei ⩾ 1. Proof. When K is algebraically closed this result is proved in [82, §19, Thm. 1]. The same method works for perfect K, as explained in [76, Prop. 12.1]. (Perfectness is implicit in the property that the underlying reduced scheme of a finite type K- group is a K-subgroup. For a counterexample over any imperfect field, see [25, Ex. A.3.8].) The general case can be pulled down from the perfect closure via Lemma 1.2.1.2; see the proof of [23, Cor. 3.20] for the argument. 1.2.1.4. Corollary. For a non-zero abelian variety A over a field K and a primary extension of fields K /K, every abelian subvariety B of AK has the form BK for a unique abelian subvariety B ⊂ A. Proof. By the Poincaré reducibility theorem, abelian subvarieties of A are pre- cisely the images of maps A → A, and similarly for AK . Since scalar extension commutes with the formation of images, the assertion is reduced to the bijectivity of End(A) → End(AK ), which follows from Lemma 1.2.1.2. Since any map between simple abelian varieties over K is either 0 or an isogeny, by general categorical arguments the collection of Ci’s (up to isogeny) in the Poincaré reducibility theorem is unique up to rearrangement, and the multiplic- ities ei are also uniquely determined. 1.2.1.5. Definition. The Ci’s in the Poincaré reducibility theorem (considered up to isogeny) are the simple factors of A. By the uniqueness of the simple factors up to isogeny, we deduce: 1.2.1.6. Corollary. Let A be a non-zero abelian variety over a field, with simple factors C1, . . . , Cs. The non-zero abelian subvarieties of A are generated by the images of maps Ci → A from the simple factors. 1.2.2. Central simple algebras. Using notation from the Poincaré reducibility theorem, for a non-zero abelian variety A we have End0 (A) Matei (End0 (Ci)) where {Ci} is the set of simple factors of A and the ei’s are the corresponding multiplicities. Each End0 (Ci) is a division algebra, by simplicity of the Ci’s. Thus, to understand the structure of endomorphism algebras of abelian varieties we need to understand matrix algebras over division algebras, especially those of finite di- mension over Q. We therefore next review some general facts about such rings. Although we have used K to denote the ground field for abelian varieties above, in what follows we will use K to denote the ground field for central simple algebras; the two are certainly not to be confused, since for abelian varieties in positive characteristic the endomorphism algebras are over fields of characteristic 0.
- 32. 1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 17 1.2.2.1. Definition. A central simple algebra over a field K is a non-zero asso- ciative K-algebra of finite dimension such that K is the center and the underlying ring is simple (i.e., has no non-trivial two-sided ideals). A central division algebra over K is a central simple algebra over K whose underlying ring is a division algebra. Among the most basic examples of central simple algebras over a field K are the matrix algebras Matn(K) for n ⩾ 1. The most general case is given by: 1.2.2.2. Proposition (Wedderburn’s Theorem). Every central simple algebra D over a field K is isomorphic to Matn(Δ) = EndΔ(Δ⊕n ) for some n ⩾ 1 and some central division algebra Δ over K (where Δ⊕n is a left Δ-module). Moreover, n is uniquely determined by D, and Δ is uniquely determined up to K-isomorphism. Proof. This is a special case of a general structure theorem for simple rings; see [53, Thm. 4.2] and [53, §4.4, Lemma 2]. In addition to matrix algebras, another way to make new central simple algebras from old ones is to use tensor products: 1.2.2.3. Lemma. If D and D are central simple algebras over a field K, then so is D ⊗K D . For any extension field K /K, DK := K ⊗K D is a central simple K -algebra. Proof. The first part is [53, §4.6, Cor. 3]; the second is [53, §4.6, Cor. 1, 2]. 1.2.3. Splitting fields. It is a general fact that for any central division algebra Δ over a field K, ΔKs is a matrix algebra over Ks (so [Δ : K] is a square). In other words, Δ is split by a finite separable extension of K. There is a refined structure theory concerning splitting fields and maximal commutative subfields of central simple algebras over fields; [53, §4.1–4.6] gives a self-contained development of this material. An important result in this direction is: 1.2.3.1. Proposition. Let D be a central simple algebra over a field F, with [D : F] = n2 . An extension field F /F with degree n embeds as an F-subalgebra of D if and only if F splits D (i.e., DF Matn(F )). Moreover, if D is a division algebra then every maximal commutative subfield of D has degree n over F. Proof. The first assertion is a special case of [53, Thm. 4.12]. Now assume that D is a division algebra and consider a maximal commutative subfield F . In such cases F splits D (by [53, §4.6, Cor. to Thm. 4.8]), so n|[F : F] by [53, Thm. 4.12]. To establish the reverse divisibility it suffices to show that for any central simple algebra D of dimension n2 over F, every commutative subfield of D has F-degree at most n. If A is any simple F-subalgebra of D and its centralizer in D is denoted ZD(A) then n2 = [A : F][ZD(A) : F] by [53, §4.6, Thm. 4.11]. Thus, if A is also commutative (so A is contained in ZD(A)) then [A : F] ⩽ n. The second assertion in Proposition 1.2.3.1 does not generalize to central simple algebras; e.g., perhaps D = Matn(F) with F having no degree-n extension fields. In general, for a splitting field F /F of a central simple F-algebra D, the choice of isomorphism DF Matn(F ) is ambiguous up to composition with the
- 33. 18 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION action of AutF (Matn(F )), so it is useful to determine this automorphism group. The subgroup of inner automorphisms is GLn(F )/F× , arising from conjugation against elements of Matn(F )× = GLn(F ). In general, the inner automorphisms are the only ones: 1.2.3.2. Theorem (Skolem–Noether). For a central simple algebra D over a field F, the inclusion D× /F× → AutF (D) carrying u ∈ D× to (d → udu−1 ) is an equality. That is, all automorphisms are inner. Proof. This is [53, §4.6, Cor. to Thm. 4.9]. We finish our discussion of central simple algebras by using the Skolem–Noether theorem to build the K-linear reduced trace map TrdD/K : D → K for a central simple algebra D over a field K. 1.2.3.3. Construction. Let D be a central simple algebra over an arbitrary field K. It splits over a separable closure Ks, which is to say that there is a Ks-algebra isomorphism f : DKs Matn(Ks) onto the n × n matrix algebra for some n ⩾ 1. By the Skolem-Noether theorem, all automorphisms of a matrix algebra are given by conjugation by an invertible matrix. Hence, f is well-defined up to composition with an inner automorphism. The matrix trace map Tr : Matn(Ks) → Ks is invariant under inner automor- phisms and is equivariant for the natural action of Gal(Ks/K), so the composition of the matrix trace with f is a Ks-linear map DKs → Ks that is independent of f and Gal(Ks/K)-equivariant. Thus, this descends to a K-linear map TrdD/K : D → K that is defined to be the reduced trace. In other words, the reduced trace map is a twisted form of the usual matrix trace, just as D is a twisted form of a matrix algebra. (For d ∈ D, the K-linear left multiplication map x → d · x on D has trace [D : K] TrdD/K(x), as we can see by scalar extension to Ks and a direct computation for matrix algebras. The elimination of the coefficient [D : K] is the reason for the word “reduced”.) 1.2.4. Brauer groups. For applications to abelian varieties it is important to classify division algebras of finite dimension over Q (such as the endomorphism algebra of a simple abelian variety over a field). If Δ is such a ring then its center Z is a number field and Δ is a central division algebra over Z. More generally, the set of isomorphism classes of central division algebras over an arbitrary field has an interesting abelian group structure. This comes out of the following definition. 1.2.4.1. Definition. Central simple algebras D and D over a field K are similar if there exist n, n ⩾ 1 such that the central simple K-algebras D ⊗K Matn(K) = Matn(D) and D ⊗K Matn (K) = Matn (D ) are K-isomorphic. The Brauer group Br(K) is the set of similarity classes of central simple algebras over K, and [D] denotes the similarity class of D. For classes [D] and [D ], define [D][D ] := [D ⊗K D ]. This composition law on Br(K) is well-defined and makes it into an abelian group with inversion given by [D]−1 = [Dopp ], where Dopp is the “opposite algebra”. By Proposition 1.2.2.2, each element in Br(K) is represented (up to isomorphism)
- 34. 1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 19 by a unique central division algebra over K. In this sense, Br(K) is an abelian group structure on the set of isomorphism classes of such division algebras. 1.2.4.2. Example. The computation of the Brauer group of a number field in- volves computing the Brauer groups of local fields, so we now clear up any possible confusion concerning sign conventions in the description of Brauer groups for non- archimedean local fields. Upon choosing a separable closure Ks of an arbitrary field K, there are two natural procedures to define a functorial group isomorphism Br(K) H2 (Ks/K, K× s ): a conceptual method via non-abelian cohomology as in [107, Ch. X, §5] and an explicit method via crossed-product algebras. By [107, Ch. X, §5, Exer. 2], these procedures are negatives of each other. We use the conceptual method of non-abelian cohomology, but we do not need to make that method explicit here and so we refer the interested reader to [107] for the details. Let K be a non-archimedean local field with residue field κ and let Kun denote its maximal unramified subextension in Ks (with κ the residue field of Kun ). It is known from local class field theory that the natural map H2 (Kun /K, Kun× ) → H2 (Ks/K, K× s ) is an isomorphism, and the normalized valuation mapping Kun× → Z induces an isomorphism H2 (Kun /K, Kun× ) H2 (Kun /K, Z) δ H1 (Gal(Kun /K), Q/Z) = H1 (Gal(κ/κ), Q/Z). There now arises the question of choice of topological generator for Gal(κ/κ): arith- metic or geometric Frobenius? We choose to work with arithmetic Frobenius. (In [103, §1.1] and [107, Ch. XIII, §3] the arithmetic Frobenius generator is also used.) Via evaluation on the chosen topological generator, our conventions lead to a composite isomorphism invK : Br(K) Q/Z for non-archimedean local fields K. If one uses the geometric Frobenius convention, then by also adopting the crossed-product algebra method to define the isomor- phism Br(K) H2 (Ks/K, K× s ) one would get the same composite isomorphism invK since the two sign differences cancel out in the composite. (Beware that in [103] and [107] the Brauer group of a general field K is defined to be H2 (Ks/K, K× s ), and so the issue of choosing between non-abelian cohomology or crossed-product algebras does not arise in the founda- tional aspects of the theory. However, this issue implicitly arises in the relationship of Brauer groups and central simple algebras, such as in [103, Appendix to §1] where the details are omitted.) Since Br(R) is cyclic of order 2 and Br(C) is trivial, for archimedean local fields K there is a unique injective homomorphism invK : Br(K) → Q/Z. By [103, §1.1, Thm. 3], for a finite extension K /K of non-archimedean local fields, composition with the natural map rK K : Br(K) → Br(K ) satisfies (1.2.4.1) invK ◦ rK K = [K : K] · invK. By [107, Ch. XIII, §3, Cor. 3], invK(Δ) has order [Δ : K] for any central division algebra Δ over K. These assertions are trivially verified to hold for archimedean local fields K as well.
- 35. 20 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION 1.2.4.3. Theorem. Let L be a global field. There is an exact sequence 0 // Br(L) // v Br(Lv) invLv // Q/Z // 0 where the direct sum is taken over all places of L and the first map is defined via extension of scalars. Proof. This is [120, §9.7, §11.2]. For a global field L and central division algebra Δ over L, invv(Δ) denotes invLv (ΔLv ). Theorem 1.2.4.3 says that a central division algebra Δ over a global field L is uniquely determined up to isomorphism by its invariants invv(Δ), and that these may be arbitrarily assigned subject to the conditions invv(Δ) = 0 for all but finitely many v and invv(Δ) = 0. Moreover, the order of [Δ] in Br(L) is the least common “denominator” of the local invariants invv(Δ) ∈ Q/Z. If K is any field then for a class c ∈ Br(K) its period is its order and its index is [Δ : K] with Δ the unique central division algebra over K representing the class c. It is a classical fact that the period divides that index and that these integers have the same prime factors (see [107, X.5], especially Lemma 1 and Exercise 3), but in general equality does not hold. For example, there are function fields of complex 3-folds for which some order-2 elements in the Brauer group cannot be represented by a quaternion algebra; examples are given in [61, §4], and there are examples with less interesting fields as first discovered by Brauer. We have noted above that over local fields there is equality of period and index (the archimedean case being trivial). The following deep result is an analogue over global fields. 1.2.4.4. Theorem. For a central division algebra Δ over a global field L, the order of [Δ] in Br(L) is [Δ : L]. As a special (and very important) case, elements of order 2 in Br(L) are pre- cisely the Brauer classes of quaternion division algebras for a global field L; as noted above, this fails for more general fields. Since Theorem 1.2.4.4 does not seem to be explicitly stated in any of the standard modern references on class field theory (though there is an allusion to it at the end of [4, Ch. X, §2]), and the structure theory of endomorphism algebras of abelian varieties rests on it, here is a proof. Proof. Let Δ have degree n2 over L and let d be the order of [Δ] in Br(L), so d|n. Note that d is the least common multiple of the local orders dv of [ΔLv ] ∈ Br(Lv) for each place v of L, with dv = 1 for complex v, dv|2 for real v, and dv = 1 for all but finitely many v. Using these formal properties of the dv’s, we may call upon the full power of global class field theory via Theorem 6 in [4, Ch. X] to infer the existence of a cyclic extension L /L of degree d such that [L v : Lv] is a multiple of dv for every place v of L (here, v is any place on L over v, and the constraint on the local degree is only non-trivial when dv 1). In the special case d = 2 (the only case we will require) one only needs weak approximation and Krasner’s Lemma rather than class field theory: take L to split a separable quadratic polynomial over L that closely approximates ones that define quadratic separable extensions of Lv for each v such that dv = 2. By (1.2.4.1), restriction maps on local Brauer groups induce multiplication by the local degree on the local invariants, so ΔL is locally split at all places of L . Thus, by the injectivity of the map from the global Brauer group into the direct
- 36. 1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 21 sum of the local ones (for L ) we conclude that the Galois extension L /L of degree d splits Δ. (The existence of cyclic splitting fields for all Brauer classes is proved for number fields in [120] and is proved for all global fields in [128], but neither reference seems to control the degree of the global cyclic extension.) It is a general fact for Brauer groups of arbitrary fields [107, Ch. X, §5, Lemma 1] that every Brauer class split by a Galois extension of degree r is represented by a central simple algebra with degree r2 . Applying this fact from algebra in our situation, [Δ] = [D] for a central simple algebra D of degree d2 over L. But each Brauer class is represented by a unique central division algebra, and so D must be L-isomorphic to a matrix algebra over Δ. Since [D : L] = d2 and [Δ : L] = n2 with d|n, this forces d = n as desired. 1.2.5. Homomorphisms and isotypicity. The study of maps between abelian varieties over a field rests on the following useful injectivity result. 1.2.5.1. Proposition. Let A and B be abelian varieties over a field K. For any prime (allowing = char(K)), the natural map Z ⊗Z Hom(A, B) → Hom(A[∞ ], B[∞ ]) is injective, where the target is the Z-module of maps of -divisible groups over K (i.e., compatible systems of K-group maps A[n ] → B[n ] for all n ⩾ 1). Proof. Without loss of generality, K is algebraically closed (and hence perfect). When = char(K) the assertion is a reformulation of the well-known analogous injectivity with -adic Tate modules (and such injectivity in turn underlies the proof of Z-module finiteness of Hom(A, B)). The proof in terms of Tate modules is given in [82, §19, Thm. 3] for = char(K), and when phrased in terms of - divisible groups it works even when = p = char(K) 0. For the convenience of the reader, we now provide the argument for = p in such terms. We will use that the torsion-free Z-module Hom(A, B) is finitely generated, and our argument works for any (especially = char(K)). Choose a Z-basis {f1, . . . , fn} of Hom(A, B). For c1, . . . , cn ∈ Z it suffices to show that if cifi kills A[] then |ci for all i. Indeed, if we can prove this then consider the case when cifi kills A[∞ ]. Certainly ci = c i for some c i ∈ Z, and ( c ifi) · kills A[n ] for all n 0. But the map A[n ] → A[n−1 ] induced by -multiplication is faithfully flat since it is the pullback along A[n−1 ] → A of the faithfully flat map : A → A, so c ifi kills A[n−1 ] for all n 0. In other words, the kernel of the map in the Proposition would be -divisible, yet this kernel is a finitely generated Z-module, so it would vanish as desired. Now consider c1, . . . , cn ∈ Z such that cifi kills A[]. For the purpose of proving ci ∈ Z for all i, it is harmless to add to each ci any element of Z. Hence, we may and do assume ci ∈ Z for all i, so cifi : A → B makes sense and kills A[]. Since : A → A is a faithfully flat homomorphism with kernel A[], by fppf descent theory any K-group scheme homomorphism A → G that kills A[] factors through : A → A (see [30, IV, 5.1.7.1] and [98]). Thus, cifi = · h for some h ∈ Hom(A, B). Writing h = mifi with mi ∈ Z, we get ci ⊗fi = · 1⊗mifi in Z ⊗Z Hom(A, B). This implies ci = mi for all i, so we are done. A weakening of simplicity that is sometimes convenient is:
- 37. 22 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION 1.2.5.2. Definition. An abelian variety A over a field K is isotypic if it is isogenous to Ce for a simple abelian variety C over K with e ⩾ 1; that is, up to isogeny, A has a unique simple factor. For a simple factor C of an abelian variety A over K, the C-isotypic part of A is the isotypic subvariety of A generated by the images of all maps C → A. An isotypic part of A is a C-isotypic part for some such C. Clearly End0 (A) is a semisimple Q-algebra. It is simple if and only if A is isotypic, and it is a division algebra if and only if A is simple. By the Poincaré reducibility theorem, every non-zero abelian variety A over a field K is naturally isogenous to the product of its distinct isotypic parts, and these distinct parts admit no non-zero maps between them. Hence, if {Bi} is the set of isotypic parts of A then End0 (A) = End0 (Bi) with each End0 (Bi) a simple algebra of finite dimension over Q. Explicitly, if Ci is the unique simple factor of Bi then a choice of isogeny Bi → Cei i defines an isomorphism from End0 (Bi) onto the matrix algebra Matei (End0 (Ci)) over the division algebra End0 (Ci). Beware that the composite “diagonal” ring map End0 (Ci) → Matei (End0 (Ci)) End0 (Bi) is canonical only when End0 (Ci) is commutative. In general isotypicity is not preserved by extension of the ground field. To make examples illustrating this possibility, as well as other examples in the theory of abelian varieties, we need the operation of Weil restriction of scalars. For a field K and finite K-algebra K , the Weil restriction functor ResK/K from quasi- projective K -schemes to separated (even quasi-projective) K-schemes of finite type is characterized by the functorial identity ResK/K(X )(A) = X (K ⊗K A) for K-algebras A. Informally, Weil restriction is an algebraic analogue of viewing a complex manifold as a real manifold with twice the dimension. In particular, if K /K is an extension of fields then ResK/K(X ) is K -smooth and equidimensional when X is K-smooth and equidimensional, with dim(ResK/K(X )) = [K : K] · dim(X ). We refer the reader to [10, §7.6] for a self-contained development of the con- struction and properties of Weil restriction (replacing K with more general rings), and to [25, A.5] for a discussion of further properties (especially of interest for group schemes). In general the formation of Weil restriction naturally commutes with any extension of the base field, and for K equal to the product ring Kn we have that ResK/K carries a disjoint union n i=1 Si of quasi-projective K-schemes (viewed as a K -scheme) to the product Si. Thus, the natural isomorphism ResK/K(X )Ks Res(K⊗K Ks)/Ks (X K⊗K Ks ) implies that if K is a field separable over K then ResK/K(A ) is an abelian variety over K of dimension [K : K]dim(A ) for any abelian variety A over K (since K ⊗K Ks K [K :K] s ). If K /K is a field extension of finite degree that is not separable then ResK/K(X ) is never proper when X is smooth and proper of positive dimension [25, Ex. A.5.6]. 1.2.6. Example. Consider a separable quadratic extension of fields K /K and a simple abelian variety A over K . Let σ ∈ Gal(K /K) be the non-trivial element, so K ⊗K K K × K via x ⊗ y → (xy, σ(x)y). Thus, the Weil restriction A := ResK/K(A ) satisfies AK A × σ∗ (A ), so AK is not isotypic if and only if A is not isogenous to its σ-twist. Hence, for K = R examples of non-isotypic
- 38. 1.3. COMPLEX MULTIPLICATION 23 AK are obtained by taking A to be an elliptic curve over C with analytic model C/(Z ⊕ Zτ) for τ ∈ C − R such that 1, τ, τ, ττ are Q-linearly independent. (In Example 1.6.4 we give examples with K = Q.) In cases when AK is non-isotypic, A is necessarily simple. Indeed, if A is not simple then a simple factor of A would be a K-descent of a member of the isogeny class of A , contradicting that A and σ∗ (A ) are not isogenous. Thus, we have exhibited examples in characteristic 0 for which isotypicity is lost after a ground field extension. The failure of isotypicity to be preserved after a ground field extension does not occur over finite fields: 1.2.6.1. Proposition. If A is an isotypic abelian variety over a finite field K then AK is isotypic for any extension field K /K. Proof. By the Poincaré reducibility theorem, it is equivalent to show that End0 (AK ) is a simple Q-algebra, so by Lemma 1.2.1.2 we may replace K with the algebraic closure of K in K . That is, we can assume that K /K is algebraic. Writing K = lim − → K i with {K i} denoting the directed system of subfields of finite degree over K, we have End(AK ) = lim − → End(AK i ). But End(AK ) is finitely gen- erated as a Z-module, so for large enough i we have End0 (AK ) = End0 (AK i ). We may therefore replace K with K i for sufficiently large i to reduce to the case when K /K is of finite degree. Let q = #K. The key point is to show that for any abelian variety B over K and any g ∈ Gal(K /K), B and g∗ (B ) are isogenous. Since Gal(K /K) is generated by the q-Frobenius σq, it suffices to show that B and B(q) := σ∗ q (B ) are isogenous. The purely inseparable relative q-Frobenius morphism B → B(q) (arising from the absolute q-Frobenius map B → B over the q-Frobenius of Spec(K )) is such an isogeny. Hence, the Weil restriction ResK/K(B ) satisfies ResK/K(B )K g g∗ (B ) ∼ B[K :K] . Take B to be a simple factor of AK (up to isogeny), so ResK/K(B ) is an isogeny factor of ResK/K (AK ) ∼ A[K :K] . By the simplicity of A and the Poincaré reducibility theorem, it follows that ResK/K(B ) is isogenous to a power of A. Extending scalars, ResK/K(B )K is therefore isogenous to a power of AK . But ResK/K(B )K ∼ B[K :K] , so non-trivial powers of AK and B are isogenous. By the simplicity of B and Poincaré reducibility, this forces B to be the only simple factor of AK (up to isogeny), so AK is isotypic. 1.3. Complex multiplication 1.3.1. Commutative subrings of endomorphism algebras. The following fact motivates the study of complex multiplication in the sense that we shall con- sider. 1.3.1.1. Theorem. Let A be an abelian variety over a field K with g := dim(A) 0, and let P ⊂ End0 (A) be a commutative semisimple Q-subalgebra. Then [P : Q] ⩽ 2g, and if equality holds then P is its own centralizer in End0 (A). If equality holds
- 39. 24 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION and moreover P is a field of degree 2g over Q then A is isotypic and P is a maximal commutative subfield of End0 (A). Proof. Consider the decomposition P = Li into a product of fields. Using the primitive idempotents of P, we get a corresponding decomposition Ai of A in the isogeny category of abelian varieties over K, with each Ai = 0 and each Li a commutative subfield of End0 (Ai) compatibly with the inclusion End0 (Ai) ⊂ End0 (A) and the equality Li = P. Since dim(A) = dim(Ai), to prove that [P : Q] ⩽ 2g it suffices to treat the Ai’s separately, which is to say that we may and do assume that P = L is a field. Since D = End0 (A) is of finite rank over Q, clearly [L : Q] is finite. Choose a prime different from char(K). Recall that V(A) denotes Q ⊗Z T(A) for T(A) := lim ← − A[n ](Ks). The injectivity of the natural map L := Q ⊗Q L → EndQ (V(A)) (see Proposition 1.2.5.1) implies that L acts faithfully on the Q-vector space V(A) of rank 2g. But L = w| Lw, where w runs over all -adic places of L, so each corresponding factor module V(A)w over Lw is non-zero as a vector space over Lw. Hence, 2g = dimQ V(A) = w| dimQ V(A)w ⩾ w| [Lw : Q] = [L : Q] with equality if and only if V(A) is free of rank 1 over L. Assume that equality holds, so V(A) is free of rank 1 over L. If A is not isotypic then by passing to an isogenous abelian variety we may arrange that A = B × B with B and B non-zero abelian varieties such that Hom(B, B ) = 0 = Hom(B , B). Hence, End0 (A) = End0 (B) × End0 (B ) and so L embeds into End0 (B). But 2 dim(B) 2 dim(A) = [L : Q], so we have a contradiction (since B = 0). It remains to prove, without assuming P is a field, that if [P : Q] = 2g then P is its own centralizer in End0 (A). (In case P is a field, so A is isotypic and hence End0 (A) is simple, such a centralizer property would imply that P is a maximal commutative subfield of End0 (A), as desired.) Consider once again the ring decomposition P = Li and the corresponding isogeny decomposition Ai of A as at the beginning of this proof. We have [Li : Q] ⩽ 2 dim(Ai) for all i, and these inequalities add up to an equality when summed over all i, so in fact [Li : Q] = 2 dim(Ai) for all i. The preceding analysis shows that each V(Ai) is free of rank 1 over Li, := Q ⊗Q Li, and so likewise V(A) is free of rank 1 over P. Hence, EndP (V(A)) = P, so if Z(P) denotes the centralizer of P in End0 (A) then the P-algebra map Z(P) = Q ⊗Q Z(P) → EndQ (V(A)) is injective (Proposition 1.2.5.1) and lands inside EndP (V(A)) = P. In other words, the inclusion P ⊂ Z(P) of Q-algebras becomes an equality after scalar extension to Q, so P = Z(P) as desired. The preceding theorem justifies the interest in the following concept. 1.3.1.2. Definition. An abelian variety A of dimension g 0 over a field K admits sufficiently many complex multiplications (over K) if there exists a commu- tative semisimple Q-subalgebra P in End0 (A) with rank 2g over Q.
- 40. 1.3. COMPLEX MULTIPLICATION 25 The reason for the terminology in Definition 1.3.1.2 is due to certain examples with K = C and P a number field such that the analytic uniformization of A(C) expresses the P-action in terms of multiplication of complex numbers; see Example 1.5.3. The classical theory of complex multiplication focused on the case of Defini- tion 1.3.1.2 in which P is a field, but it is useful to allow P to be a product of several fields (i.e., a commutative semisimple Q-algebra). For example, by Theorem 1.3.1.1 this is necessary if we wish to consider the theory of complex multiplication with A that is not isotypic, or more generally if we want Definition 1.3.1.2 to be preserved under the formation of products. The theory of Shimura varieties provides further reasons not to require P to be a field. Note that we do not consider A to admit sufficiently many complex multipli- cations merely if it does so after an extension of the base field K. 1.3.2. Example. The elliptic curve y2 = x3 −x admits sufficiently many complex multiplications over Q( √ −1) but not over Q. More generally, End0 (E) = Q for every elliptic curve E over Q (since the tangent line at the origin is too small to support a Q-linear action by an imaginary quadratic field), so in our terminology an elliptic curve over Q does not admit sufficiently many complex multiplications. 1.3.2.1. Proposition. Let A be a non-zero abelian variety over a field K. The following are equivalent. (1) The abelian variety A admits sufficiently many complex multiplications. (2) Each isotypic part of A admits sufficiently many complex multiplications. (3) Each simple factor of A admits sufficiently many complex multiplications. See Definition 1.2.5.2 for the terminology used in (2). Proof. Let {Bi} be the set of isotypic parts of A, so End0 (Bi) Matei (End0 (Ci)) where Ci is the unique simple factor of Bi and ei ⩾ 1 is its multiplicity as such. Since End0 (A) = End0 (Bi), (2) implies (1). It is clear that (3) implies (2). Conversely, assume that End0 (A) contains a Q-algebra P satisfying [P : Q] = 2 dim(A). There is a unique decomposition P = Li with fields L1, . . . , Ls, and [Li : Q] = 2 dim(A). We saw in the proof of Theorem 1.3.1.1 that by replacing A with an isogenous abelian variety we may arrange that A = Ai with each Ai a non-zero abelian variety having Li ⊂ End0 (Ai) compatibly with the embedding End0 (Ai) ⊂ End0 (A) and the equality Li = P. Thus, [Li : Q] ⩽ 2 dim(Ai) for all i (by Theorem 1.3.1.1), and adding this up over all i yields an equality, so each Ai admits sufficiently many complex multiplications using Li. Since each simple factor of A is a simple factor of some Ai, to prove (3) we are therefore reduced to the case when P = L is a field. Applying Theorem 1.3.1.1 once again, L is its own centralizer in End0 (A) and A is isotypic, say with unique simple factor C appearing with multiplicity e. In particular, End0 (A) = Mate(D) for the division algebra D = End0 (C) of finite rank over Q. If Z denotes the center of D then D is a central division algebra over Z, and L contains Z since L is its own centralizer in End0 (A) = Mate(D). Letting d = dim(C), Mate(D) contains the maximal commutative subfield L of degree 2g/[Z : Q] = (2d/[Z : Q])e over Z. As we noted in the proof of Proposition 1.2.3.1 (parts of which are carried out for central simple algebras that may not be division algebras), the Z-degree of
- 41. 26 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION Mate(D) is the product of the Z-degrees of L and the centralizer of L in Mate(D). But L is its own centralizer, so e2 [D : Z] = dimZMate(D) = [L : Z]2 = e2 (2d/[Z : Q])2 . We conclude that 2d/[Z : Q] = [D : Z], so (by Proposition 1.2.3.1) 2d/[Z : Q] is the common Z-degree of all maximal commutative subfields of the central division algebra D = End0 (C) over Z, or equivalently 2d is the Q-degree of all such fields. But 2d = 2 dim(C), so choosing any maximal commutative subfield of D shows that C admits sufficiently many complex multiplications. 1.3.3. CM algebras and CM abelian varieties. The following three conditions on a number field L are equivalent: (1) L has no real embeddings but is quadratic over a totally real subfield, (2) for every embedding j : L → C, the subfield j(L) ⊂ C is stable under complex conjugation and the involution x → j−1 (j(x)) in Aut(L) is non-trivial and independent of j, (3) there is a non-trivial involution τ ∈ Aut(L) such that for every embedding j : L → C we have j(τ(x)) = j(x) for all x ∈ L. The proof of the equivalence is easy. When these conditions hold, τ in (3) is unique and its fixed field is the maximal totally real subfield L+ ⊂ L (over which L is quadratic). The case L+ = Q corresponds to the case when L is an imaginary quadratic field. 1.3.3.1. Definition. A CM field is a number field L satisfying the equivalent conditions (1), (2), and (3) above. A CM algebra is a product L1 × · · · × Ls of finitely many CM fields (with s ⩾ 1). The reason for this terminology is due to the following important result (along with Example 1.5.3). 1.3.4. Theorem (Tate). Let A be an abelian variety of dimension g 0 over a field K. Suppose A admits sufficiently many complex multiplications. Then there exists a CM algebra P ⊂ End0 (A) with [P : Q] = 2 dim(A). In case A is isotypic we can take P to be a CM field. The proof of this theorem (which ends with the proof of Lemma 1.3.7.1) will require some effort, especially since we consider an arbitrary base field K. Before we start the proof, it is instructive to consider an example. 1.3.4.1. Example. Consider A = E2 with an elliptic curve E over K = C such that L := End0 (E) is an imaginary quadratic field. The endomorphism algebra End0 (A) = Mat2(L) is simple and contains as its maximal commutative subfields all quadratic extensions of L. Those extensions which are biquadratic over Q are CM fields, and the rest are not CM fields. Hence, in the setup of Theorem 1.3.4, even when A is isotypic and char(K) = 0 there can be maximal commutative semisimple subalgebras of End0 (A) that are not CM algebras. However, if char(K) = 0 and A is simple (over K) then End0 (A) is a CM field; see Proposition 1.3.6.4.
- 42. 1.3. COMPLEX MULTIPLICATION 27 1.3.5. We will begin the proof of Theorem 1.3.4 now, but at a certain point we will need to use deeper input concerning the fine structure of endomorphism algebras of simple abelian varieties over general fields. At that point we will digress to review the required structure theory, and then we will complete the argument. By Proposition 1.3.2.1, every simple factor of A admits sufficiently many com- plex multiplications. Thus, to prove the existence of the CM subalgebra P in Theo- rem 1.3.4 it suffices to treat the case when A is simple. Note that in the simple case such a CM subalgebra is automatically a field, since the endomorphism algebra is a division algebra. Let us first show that the result in the simple case implies that in the general isotypic case we can find P as a CM field. For isotypic A, by passing to an isogenous abelian variety we can arrange that A = Am for a simple abelian variety A over K and some m ⩾ 1. Thus, if g = dimA then g = mg and End0 (A ) contains a CM field P of degree 2g over Q. But End0 (A) Matm(End0 (A )) and this contains Matm(P ). To find a CM field P ⊂ End0 (A) of degree 2g = 2g m over Q it therefore suffices to construct a degree-m extension field P of P such that P is a CM field. Let P+ be the maximal totally real subfield of P , so for any totally real field P+ of finite degree over P+ the ring P = P+ ⊗P + P is a field quadratic over P+ and it is totally complex, so it is a CM field and clearly [P : Q] = [P : P ][P : Q] = 2g [P+ : P+ ]. Hence, to find the required CM field P in the isotypic case it suffices to construct a degree-m totally real extension of P+ . To do this, first recall the following basic fact from number theory [15, §6]: 1.3.5.1. Theorem (weak approximation). For any number field L and finite set S of places of L, the map L → v∈S Lv has dense image. Proof. This is [15, §6]. Applying this to P+ , we can construct a monic polynomial f of degree m in P+ [u] that is very close to a totally split monic polynomial of degree m at each real place and is very close to an irreducible (e.g., Eisenstein) polynomial at a single non-archimedean place. It follows that f is totally split at each real place of P+ and is irreducible over P+ , so the ring P+ = P+ [u]/(f) is a totally real field of degree m over P+ as required. 1.3.5.2. We may and do assume for the remainder of the argument that A is simple. In this case D = End0 (A) is a central division algebra over a number field Z, so the commutative semisimple Q-subalgebra P ⊂ D is a field, and the proof of Proposition 1.3.2.1 shows that the common Q-degree of all maximal commutative subfields of D is 2g. Hence, our problem is to construct a maximal commutative subfield of D that is a CM field. Let TrdD/Q = TrZ/Q ◦ TrdD/Z, where TrdD/Z is the reduced trace. An abelian variety over any field admits a polarization, so choose a polarization of A over K. Let x → x∗ denote the associated Rosati involution on D (so (xy)∗ = y∗ x∗ and x∗∗ = x). 1.3.5.3. Lemma. The quadratic form x → TrdD/Q(xx∗ ) on D is positive-definite. Proof. For any central simple algebra D over any field K whatsoever, let n = [D : K] and define the variant TrmD/K : D → K of the reduced trace to be the
- 43. 28 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION map that sends each y ∈ D to the trace of the K-linear map my : D → D defined by d → yd. We have TrmD/K = n · TrdD/K, as may be checked by extending scalars to Ks and directly computing with elementary matrices (see 1.2.3.3). Hence, in the setting of interest with D = End0 (A) and K = Z we see that it is equivalent to prove positive-definiteness for the quadratic form x → TrmD/Q(xx∗ ), where TrmD/Q = TrZ/Q ◦ TrmD/Z. The positive-definiteness for TrmD/Q can be verified by replacing D with End0 K (AK), to which [82, §21, Thm. 1] applies. Lemma 1.3.5.3 says that x → x∗ is a positive involution of D (relative to the linear form TrdD/Q). The existence of such an involution severely constrains the possibilities for D. First we record the consequences for the center Z. 1.3.5.4. Lemma. The center Z of D = End0 (A) is either totally real or a CM field, and in the latter case its canonical complex conjugation is induced by the Rosati involution defined by any polarization of A over K. Proof. Fix a polarization and consider the associated Rosati involution x → x∗ on the center Z of D. Clearly Z is stable under this involution. The positive- definite TrdD/Q(xx∗ ) on D restricts to [D : Z] · TrZ/Q(xx∗ ) on Z, so TrZ/Q(xx∗ ) is positive-definite on Z. If x∗ = x for all x ∈ Z then the rational quadratic form TrZ/Q(x2 ) is positive-definite on Z, so by extending scalars to R we see that Tr(R⊗QZ)/R(x2 ) is positive-definite. This forces the finite étale R-algebra R ⊗Q Z to have no complex factors. Hence, Z is a totally real field in such cases. It remains to show that if the involution x → x∗ is non-trivial on Z for some choice of polarization then Z is a CM field (so the preceding argument would imply that the Rosati involution arising from any polarization of A is non-trivial on Z) and its intrinsic complex conjugation is equal to this involution on Z. Let Z+ be the subfield of fixed points in Z for this involution, so [Z : Z+ ] = 2 and 2 TrZ+/Q is the restriction to Z+ of TrZ/Q. Hence, TrZ+/Q(x2 ) is positive-definite on Z+ , so Z+ is totally real. We aim to prove that Z has no real places, so we assume otherwise and seek a contradiction. Let v be a real place of Z. Since the involution x → x∗ is non-trivial on Z and the field Zv R has no non-trivial field automorphisms, the real place v on Z is not fixed by the involution x → x∗ . Thus, the real place v∗ obtained from v under the involution is a real place of Z distinct from v, and so the positive-definiteness of TrZ/Q(xx∗ ) implies (after scalar extension to R) the positive-definiteness of Tr(Zv×Zv∗ )/R(xx∗ ), where x → x∗ on Zv × Zv∗ = R × R is the involution that swaps the factors. In other words, this is the quadratic form (c, c ) → 2cc , which by inspection is not positive-definite. 1.3.6. Albert’s classification. To go further with the proof of Theorem 1.3.4, we need to review properties of endomorphism algebras of simple abelian varieties over arbitrary fields. 1.3.6.1. Definition. An Albert algebra is a pair consisting of a division algebra D of finite dimension over Q and a positive involution x → x∗ on D. For any Albert algebra D and any algebraically closed field K, there exists a simple abelian variety A over K such that End0 (A) is Q-isomorphic to D (with the
- 44. 1.3. COMPLEX MULTIPLICATION 29 given involution on D arising from a polarization on A); see [1], [2], [3], [112, §4.1, Thm. 5], and [46, Thm. 13]. For a survey and further references on this topic, see [92]. We will not need this result. Instead, we are interested in the non-trivial constraints on the Albert algebras that arise from polarized simple abelian varieties A over an arbitrary field K when char(K) and dim A are fixed. Before listing these constraints, it is convenient to record Albert’s classification of general Albert algebras (omitting a description of the possibilities for the involution). 1.3.6.2. Theorem (Albert). Let (D, (·)∗ ) be an Albert algebra. For any place v of the center Z, let v∗ denote the pullback of v along x → x∗ . Exactly one of the following occurs: Type I: D = Z is a totally real field. Type II: D is a central quaternion division algebra over a totally real field Z such that D splits at each real place of Z. Type III: D is a central quaternion division algebra over a totally real field Z such that D is non-split at each real place of Z. Type IV: D is a central division algebra over a CM field Z such that for all finite places v of Z, invv(D) + invv∗ (D) = 0 in Q/Z and moreover D splits at such a v if v = v∗ . Proof. See [82, §21, Thm. 2] (which also records the possibilities for the involu- tion). 1.3.6.3. Let A be a simple abelian variety over a field K, D = End0 (A), and Z the center of D. Let Z+ be the maximal totally real subfield of Z, so either Z = Z+ or Z is a totally complex quadratic extension of Z+ . The invariants e = [Z : Q], e0 = [Z+ : Q], d = [D : Z], and g = dim(A) satisfy some divisibility restrictions: • whenever char(K) = 0, the integer ed2 = [D : Q] divides 2g (proof: there is a subfield K0 ⊆ K finitely generated over Q such that A descends to an abelian variety A0 over K0 and the D-action on A in the isogeny category over K descends to an action on A0 in the isogeny category over K0, so upon choosing an embedding K0 → C we get a Q-linear action by the division algebra D on the 2g-dimensional homology H1(A0(C), Q)), • the action of D on V(A) with = char(K) implies (via Cor. to Thm. 4 of [82, §19], whose proof is valid over any base field) that ed|2g, • the structure of symmetric elements in Q ⊗Z Hom(A, At ) Q ⊗Z Pic(A)/Pic0 (A) (via [82, §20, Cor. to Thm. 3], whose proof is valid over any base field) yields that [L : Q]|g for every commutative subfield L ⊂ D whose elements are invariant under the involution. • for Type II in any characteristic we have 2e|g (which coincides with the general divisibility ed2 |2g when char(K) = 0 since d = 2 for Type II). To prove it uniformly across all characteristics, first note that for Type II we have R ⊗Q D = (R ⊗Q Z) ⊗Z D = v|∞ Zv ⊗Z D Mat2(Zv)e .
- 45. 30 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION Moreover, by [82, §21, Thm. 2] it can be arranged that under this composite isomorphism the positive involution on D goes over to transpose on each factor Mat2(Zv) = Mat2(R). Thus, for D of Type II the fixed part of the involution on D has Q-dimension 2e and hence Z-degree 2. By centrality of Z in the division algebra D, the condition x∗ = x for x in D of Type II therefore defines a necessarily commutative quadratic extension Z of Z inside D, so g is divisible by [Z : Q] = 2e as desired. The preceding results are summarized in the following table, taken from the end of [82, §21]. (As we have just seen, the hypothesis there that K is algebraically closed is not necessary.) The invariants of D = End0 (A) are given in the first three columns. In the last two columns we give some necessary divisibility restrictions on these invariants. Type e d char(K) = 0 char(K) 0 I e = e0 1 e | g e | g II e = e0 2 2e | g 2e | g III e = e0 2 2e | g e | g IV e = 2e0 d e0d2 | g e0d | g We refer the reader to [82, §21], and to [92] for further information on these invariants. Using the above table, we can prove the following additional facts when the simple A admits sufficiently many complex multiplications. 1.3.6.4. Proposition. Let A be a simple abelian variety of dimension g 0 over a field K, and assume that A admits sufficiently many complex multiplications. Let D = End0 (A). (1) If char(K) = 0 then D is of Type IV with d = 1 and e = 2g (so D is a CM field, by Theorem 1.3.6.2). (2) If char(K) 0 then D is of Type III or Type IV. Proof. By simplicity, D is a division algebra. Its center Z is a commutative field. First suppose char(K) = 0. Let P ⊂ D be a commutative semisimple Q- subalgebra with [P : Q] = 2g. Since D is a division algebra, P is a field. The above table (or the discussion preceding it) says that the degree [D : Q] = ed2 divides [P : Q] = 2g, so the inclusion P ⊂ D is an equality. Thus, D is commutative (i.e., d = 1), so D = Z is a commutative field and hence e := [Z : Q] = 2g by the complex multiplication hypothesis. The table shows that in characteristic 0 we have e|g for Types I, II, and III, so D is of Type IV. Suppose char(K) 0. In view of the divisibility relations in the table in positive characteristic, D is not of Type I since in such cases D is a commutative field whose Q-degree divides dim(A), contradicting the existence of sufficiently many complex multiplications. For Type II we have 2e|g, yet the integer 2e = 2[Z : Q] is the Q- degree of a maximal commutative subfield of the central quaternion division algebra D over Z, so there are no such subfields with Q-degree 2g. Since a commutative semisimple Q-subalgebra of D is a field (as D is a division algebra), Type II is not possible if the simple A has sufficiently many complex multiplications.
- 46. 1.3. COMPLEX MULTIPLICATION 31 1.3.7. Returning to the proof of Theorem 1.3.4, recall that we reduced the proof to the case of simple A. Proposition 1.3.6.4(1) settles the case of characteristic 0, and Proposition 1.3.6.4(2) gives that D = End0 (A) is an Albert algebra of Type III or IV when char(K) 0. If D is of Type III then the center Z is totally real and d is even, whereas if D is of Type IV then Z is CM. Thus, we can apply the following general lemma to conclude the proof. 1.3.7.1. Lemma (Tate). Let D be a central division algebra of degree d2 over a number field Z that is totally real or CM. If Z is totally real then assume that d is even. There exists a maximal commutative subfield L ⊂ D that is a CM field. The parity condition on d is necessary when Z is totally real, since d = [L : Z] by maximality of L in D. Proof. By Proposition 1.2.3.1, any degree-d extension of Z that splits D is a maximal commutative subfield of D. Hence, we just need to find a degree-d extension L of Z that is a CM field and splits D. Let Σ be a finite non-empty set of finite places of Z containing the finite places at which D is non-split. By the structure of Brauer groups of local fields, for any v ∈ Σ the central simple Zv-algebra Dv := Zv ⊗Z D of rank d2 over Zv is split by any extension of Zv of degree d. First assume that Z is totally real, so d is even. By weak approximation (Theorem 1.3.5.1), there is a monic polynomial f over Z of degree d/2 that is close to a monic irreducible polynomial of degree d/2 over Zv for all v ∈ Σ (and in particular f is irreducible over all such Zv, and hence over Z since Σ is non-empty). We can also arrange that for each real place v of Z the polynomial f viewed over Zv R is close to a totally split monic polynomial of degree d/2 and hence is totally split over Zv. Thus, Z := Z[u]/(f) is a totally real extension field of Z with degree d/2. By the same method, we can construct a quadratic extension L/Z that is unramified quadratic over each place v over a place in Σ and is also totally complex (by using approximations to irreducible quadratics over R at the real places of Z ). This L is a CM field and it is designed so that Zv ⊗Z L is a degree-d field extension of Zv for all v ∈ Σ. Hence, DL is split at all places of L (the archimedean ones being obvious), so DL is split. Assume next that Z is a CM field. Let Z+ ⊂ Z be the maximal totally real subfield. By the same weak approximation procedure as above (replacing d/2 with d), we can construct a degree d totally real extension Z+ /Z+ such that for each place v0 of Z+ beneath a place v ∈ Σ, the extension Z+ /Z+ has a unique place v 0 over v0 and is totally ramified (resp. unramified) at v 0 when Z/Z+ is unramified (resp. ramified) at v. Hence, (Z+ )v 0 and Zv are linearly disjoint over (Z+ )v0 . We conclude that Z+ and Z are linearly disjoint over Z , so L := Z+ ⊗Z+ Z is a field and each v ∈ Σ has a unique place w over it in L. Clearly [Lw : Zv] = d for all such w, so L splits D. By construction, L is visibly CM. We have proved Lemma 1.3.7.1. This also finishes the proof of Theorem 1.3.4. 1.3.7.2. Corollary. An isotypic abelian variety A with sufficiently many complex multiplications remains isotypic after any extension of the base field.
- 47. 32 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION Proof. By Theorem 1.3.4, the endomorphism algebra End0 (A) contains a com- mutative field with Q-degree 2 dim(A). This property is preserved after any ground field extension (even though the endomorphism algebra may get larger), so by the final part of Theorem 1.3.1.1 isotypicity is preserved as well. 1.3.8. CM abelian varieties. It turns out to be convenient to view the CM algebra P in Theorem 1.3.4 as an abstract ring in its own right, and to thereby regard the embedding P → End0 (A) as additional structure on A. This is encoded in the following concept. 1.3.8.1. Definition. Let A be an abelian variety over a field K, and assume that A has sufficiently many complex multiplications. Let j : P → End0 (A) be an embedding of a CM algebra P with [P : Q] = 2 dim(A). Such a pair (A, j) is called a CM abelian variety (with complex multiplication by P). Note that in this definition we are requiring P to be embedded in the endo- morphism algebra of A over K (and not merely in the endomorphism algebra after an extension of K). For example, according to this definition, no elliptic curve over Q admits a CM structure (even if such a structure exists after an extension of the base field). As an application of Theorem 1.3.4, we establish the following result concerning the possibilities for Z of Type III in Proposition 1.3.6.4(2). This will not be used later. 1.3.8.2. Proposition. Let A, K, and D be as in Proposition 1.3.6.4(2) with p = char(K) 0, and let Z be the center of D, g = dim(A), d = [D : Z], and e = [Z : Q]. We have ed = 2g, and if D is of Type III (so d = 2) then either Z = Q or Z = Q( √ p). Note that in this proposition, K is an arbitrary field with char(K) 0; K is not assumed to be finite. Proof. We always have ed|2g, but ed = [D : Q] and D contains a field P of Q-degree 2g, so 2g|ed. Thus, ed = 2g. Now we can assume A is of Type III, so the field Z is totally real. Since A is of finite type over K and D is finite-dimensional over Q, by direct limit considerations we can descend to the case when K is finitely generated over Fp. Let S be a separated integral Fp-scheme of finite type whose function field is K. Since A is an abelian variety over the direct limit K of the coordinate rings of the non-empty affine open subschemes of S, by replacing S with a sufficiently small non-empty affine open subscheme we can arrange that A is the generic fiber of an abelian scheme A → S. Since S is connected, the fibers of the map A → S all have the same dimension, and this common dimension is g (as we may compute using the generic fiber A). The Z-module End(A) is finitely generated, and each endomorphism of A ex- tends uniquely to a U-endomorphism of AU for some non-empty open U in S (with U perhaps depending on the chosen endomorphism). Using a finite set of endomorphisms that spans End(A) allows us to shrink S so that all elements of End(A) extend to S-endomorphisms of A , or in other words End(A) = End(A ). We therefore have a specialization map D = End0 (A) → End0 (As) for every s ∈ S.
- 48. 1.4. DIEUDONNÉ THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 33 Fix a prime = p. Since S is connected and A [n ] is finite étale over S, an S-endomorphism of A [n ] is uniquely determined by its effect on a single geometric fiber over S. But maps between abelian varieties are uniquely determined by their effect on -adic Tate modules when is a unit in the base field, so we conclude (via consideration of -power torsion) that the specialization map D → End0 (As) is injective for all s ∈ S. We can therefore speak of an element of End0 (As) “lifting” over K in the sense that it is the image of a unique element of D = End0 (A) under the specialization mapping at s. This will be of interest when s is a closed point and we consider the qs-Frobenius endomorphism of As over the finite residue field κ(s) at s (with qs = #κ(s)). By Theorem 1.3.4, we can choose a CM field L ⊂ D with [L : Q] = 2g. In particular, for each s ∈ S the field L embeds into End0 (As) with [L : Q] = 2g = 2 dim(As), so each As is isotypic. By Theorem 1.3.1.1, L is its own centralizer in End0 (As). Take s to be a closed point of S, and let qs denote the size of the finite residue field κ(s) at s. The qs-Frobenius endomorphism ϕs ∈ End0 (As) is central, so it centralizes L and hence must lie in the image of L. In particular, ϕs lifts to an element of End0 (A) = D that is necessarily central (as we may compute after applying the injective specialization map D → End0 (As)). That is, ϕs ∈ Z ⊂ D for all closed points s ∈ S. Let Z be the subfield of Z generated over Q by the lifts of the endomorphisms ϕs as s varies through all closed points of S. Each Q[ϕs] is a totally real field since Z is totally real. By Weil’s Riemann Hypothesis for abelian varieties over finite fields (see the discussion following Definition 1.6.1.2), under any embedding ι : Q[ϕs] → C we have each ι(ϕs)ι(ϕs) = qs for qs = #κ(s) ∈ pZ , so the real number ι(ϕs) is a power of √ p. Hence, the subfield Q[ϕs] ⊂ Z is either Q or Q( √ p), so the subfield Z ⊂ Z is either Q or Q( √ p). Let η be a geometric generic point of S, and let Γ be the associated absolute Galois group for the function field of S. Because each A [n ] is finite étale over S, the representation of Γ on V(A) factors through the quotient π1(S, η). The Chebotarev Density Theorem for π1(S, η) [97, App. B.9] says that the Frobenius elements at the closed points of S generate a dense subgroup of the quotient π1(S, η) of Γ. Thus, the image of Q[Γ] in EndQ (V(A)) is equal to the subalgebra Z := Q ⊗Q Z generated by the endomorphisms ϕs. We therefore have an injective map Q ⊗Q D → EndQ[Γ](V(A)) = EndZ (V(A)). By Zarhin’s theorem [134] (see [80, XII, §2] for a proof valid for all p, especially allowing p = 2) this injection is an isomorphism, so we conclude that Z is central in EndZ (V(A)). But the center of this latter matrix algebra is Z , so the inclusion Z ⊂ Z is an equality. Hence, the inclusion Z ⊂ Z is an equality as well. Since Z is either Q or Q( √ p), we are done. 1.4. Dieudonné theory, p-divisible groups, and deformations To solve problems involving lifts from characteristic p to characteristic 0, we need a technique for handling p-torsion phenomena in characteristic p 0. The two main tools for this purpose in what we shall do are Dieudonné theory and p-divisible groups. For the convenience of the reader we review the basic facts in this direction, and for additional details we refer to [119], [110, §6], and [75] for
- 49. 34 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION p-divisible groups, and [41, Ch. II–III] for (contravariant) Dieudonné theory with applications to p-divisible groups. 1.4.1. Exactness. We shall frequently use exact sequence arguments with abelian varieties and finite group schemes over fields, as well as with their relative analogues over more general base schemes. It is assumed that the reader has some familiarity with these notions, but we now provide a review of this material. 1.4.1.1. Definition. Let S be a scheme, and let T be a Grothendieck topology on the category of S-schemes. (For our purposes, only the étale, fppf, and fpqc topologies will arise.) A diagram 1 → G → G f → G → 1 of S-group schemes is short exact for the topology T if G → G is an isomorphism onto ker(f) and the map f is a T -covering. By [30, Exp. IV, 5.1.7.1], in such cases G represents the quotient sheaf G/G for the chosen Grothendieck topology. By [31, Exp. V, Thm. 4.1(iv), Rem. 5.1], if G is a quasi-projective group scheme over a noetherian ring R and if G is a finite flat R-subgroup of G then the fppf quotient sheaf G/G is represented by a quasi- projective R-group (also denoted G/G ), and the resulting map of group schemes G → G/G is an fppf G -torsor (so G/G is R-flat if G is). 1.4.1.2. The Cartier dual ND of a commutative finite locally free group scheme N over a base scheme S is the commutative finite locally free group scheme which represents the fppf sheaf functor Hom(N, Gm) : S HomS-gp(NS , Gm) on the category of S-schemes. The structure sheaf OND of ND is canonically isomorphic to the OS-linear dual of the structure sheaf ON of N, and the co-multiplication (respectively multiplication) map for OND is the OS-linear dual of the multiplication (respectively co-multiplication) map for ON . The functor N ND on the category of commutative finite locally free group schemes over S swaps closed immersions and quotient maps, preserves exactness, and is an involution in the sense that there is a natural isomorphism fN : N (ND )D satisfying (fN )D = f−1 ND . See [87, Prop. 2.9] for further details. As an application, if the S-homomorphism j : G → G is a closed immersion between finite locally free commutative group schemes then we can use Cartier duality to give a direct proof that the the fppf quotient sheaf G/G is represented by a finite locally free S-group (without needing to appeal to general existence results for quotients by G -actions on quasi-projective S-schemes). The key point is that the Cartier dual map jD : GD → GD between finite flat S-schemes is faithfully flat, as this holds on fibers over S (since injective maps between Hopf algebras over a field are always faithfully flat [126, 14.1]). Such flatness implies that H := ker(jD ) is a finite locally free commutative S-group, so HD makes sense and the dual map q : G (GD )D → HD is faithfully flat. It is clear that G ⊂ ker(q), and this inclusion between finite locally free S-schemes is an isomorphism by comparison of fibral degrees, so HD represents G/G . The following result is useful for constructing commutative group schemes G → S that are finite and fppf (equivalently, finite and locally free over S).
- 50. 1.4. DIEUDONNÉ THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 35 1.4.1.3. Proposition. Let S be a scheme, and let G and G be finitely presented separated S-group schemes with G affine and flat over S. For any exact sequence 1 → G → G → G → 1 of group sheaves for the fppf topology on the category of S-schemes, G is represented by a finitely presented S-group that is flat and affine over G . Moreover, G and G are S-finite if and only if G is S-finite. See [87, 17.4] for a generalization (using the fpqc topology). Proof. For any G -scheme T viewed as an S-scheme, let g ∈ G (T) correspond to the given S-morphism T → G . Consider the set Eg (T) that is the preimage under G(T) → G (T) of g . This is a sheaf of sets on the category of G -schemes equipped with the fppf topology, and as such it is a left G -torsor (strictly speaking, a left torsor for the G -group G G ) due to the given exact sequence. In particular, the fppf sheaves of sets Eg and G G over G are isomorphic fppf-locally over G . Since G is fppf affine over S and fppf descent is effective for affine morphisms, it follows that Eg as an fppf sheaf of sets over G is represented by an affine fppf G - scheme (which is therefore affine fppf over S when G is). It is elementary to check that this affine G -scheme viewed as an S-scheme has its functor of points naturally identified with G (since for any S-scheme T and g ∈ G(T), visibly g ∈ Eg (T) for the point g ∈ G (T) arising from g), so G is represented by an S-group. Separatedness of G over S and exactness imply that G is closed in G. More- over, G → G is a left G G -torsor for the fppf topology over G , so it is finite when G is S-finite. Thus, if G and G are S-finite then G is S-finite. Conversely, if G is S-finite then its closed subscheme G is S-finite, so the quotient G/G exists as an S-finite scheme. But G represents this quotient, so G is S-finite too. 1.4.1.4. Remark. If 1 → G → G → G → 1 is an exact sequence of separated fppf S-groups with G and G abelian schemes then G is an abelian scheme. Indeed, since G → G is an fppf torsor for the G -group G G that is smooth and proper with geometrically connected fibers, G → G is smooth and proper with geometrically connected fibers. The map G → S is also smooth and proper with geometrically connected fibers, so G → S is as well. Hence, G is an abelian scheme. It is also true that if G is an abelian scheme and G is a closed S-subgroup of G that is also an abelian scheme then the fppf quotient sheaf G/G is represented by an abelian scheme. We will give an elementary proof of this over fields in Lemma 1.7.4.4 using the Poincaré reducibility theorem (which is only available over fields). In general the proof requires a detour through algebraic spaces. 1.4.2. Duality for abelian schemes. In [83, §6.1], duality is developed for pro- jective abelian schemes, building on the case of an algebraically closed ground field. Projectivity is imposed primarily due to the projectivity hypotheses in Grothendieck’s work on Hilbert schemes. The projective case is sufficient for our needs because any abelian scheme over a discrete valuation ring is projective (this follows from Lemma 2.1.1.1, to which the interested reader may now turn). For both technical and aesthetic reasons, it is convenient to avoid the projectivity hy- pothesis. We now sketch Grothendieck’s results on duality in the projective case, as well as Artin’s improvements that eliminated the projectivity assumption.
- 51. 36 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION 1.4.2.1. Let A → S be an abelian scheme, and let PicA/S be the functor assigning to any S-scheme T the group of isomorphism classes of pairs (L , i) consisting of an invertible sheaf L on AT and a trivialization i : e∗ T (L ) OT along the identity section eT of AT → T. This is an fppf group sheaf on the category of S-schemes, and its restriction to the category of S -schemes (for an S-scheme S ) is PicAS /S . Let Pic0 A/S ⊂ PicA/S be the subfunctor classifying pairs (L , i) that lie in the identity component of the Picard scheme on geometric fibers. By [7, Exp. XIII, Thm. 4.7(i)] (see [39, §9.6] for the projective case), the inclusion j : Pic0 A/S → PicA/S is an open subfunctor; i.e., it is relatively representable by open immersions. This means that for any S-scheme T and (L , i) ∈ PicA/S(T), Pic0 A/S ×PicA/S T as a functor on T-schemes is represented by an open subscheme U ⊂ T; explicitly, there is an open subscheme U ⊂ T such that a T-scheme T lies over U if and only if the T -pullback of (L , i) lies in Pic0 on geometric fibers over T . By Grothendieck’s work on Picard schemes (see [39, Part 5]), if A → S is projective Zariski-locally on S then PicA/S is represented by a locally finitely pre- sented and separated S-scheme and the open subscheme At representing Pic0 A/S is quasi-projective Zariski-locally on S and finitely presented. For noetherian S, functorial criteria show that At is proper and smooth (see [83, §6.1]), hence an abelian scheme; the case of general S (with A projective Zariski-locally on S) then follows by descent to the noetherian case. To drop the projectivity hypothesis, one has to use algebraic spaces. Infor- mally, an algebraic space over S is an fppf sheaf on the category of S-schemes that is “well-approximated” by a representable functor (relative to the étale topology), so concepts from algebraic geometry such as smoothness, properness, and connect- edness can be defined and behave as expected; see [60]. By Artin’s work on relative Picard functors as algebraic spaces (see [5, Thm. 7.3]), PicA/S is always a separated algebraic space locally of finite presentation, and by [7, Exp. XIII, Thm. 4.7(iii)] the open algebraic subspace Pic0 A/S is finitely presented over S. The functorial arguments that prove smoothness and properness for Pic0 A/S when A is projective work without projectivity because the same criteria are avail- able for algebraic spaces. Thus, Pic0 A/S is smooth and proper over S in the sense of algebraic spaces. Consequently, by a theorem of Raynaud (see [38, Thm. 1.9]), Pic0 A/S is represented by an S-scheme At ; this must be an abelian scheme, called the dual abelian scheme. Its formation commutes with any base change on S, and it is contravariant in A in an evident manner. 1.4.2.2. Over A × At there is a Poincaré bundle PA/S provided by the universal property of At , exactly as in the theory of duality for abelian varieties over a field. In particular, PA/S is canonically trivialized along e × idAt . Let e ∈ At (S) be the identity, so for any S-scheme T the point e T ∈ At (T) corresponds to OAT equipped with the canonical trivialization of e∗ T (OAT ). Thus, setting T = A gives that PA/S is also canonically trivialized along idA × e . Hence, the pullback of PA/S along the flip At × A A × At defines a canonical S-morphism ιA/S : A → Att . This morphism carries the identity to the identity, so it is a homomorphism. By applying the duality theory over fields to the fibers of A over S, it follows that ιA/S is an isomorphism; in other words, the pullback of PA/S along the flip At ×A A×At is uniquely isomorphic to PAt/S respecting trivializations along the identity sections of both factors. Such uniqueness implies that ιt A/S is inverse to ιAt/S.
- 52. 1.4. DIEUDONNÉ THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 37 A homomorphism f : A → At is symmetric when the map ft ◦ ιA/S : A Att → At is equal to f. Writing f† := ft ◦ιA/S, the equality ιt A/S = ι−1 At/S and the functoriality of ιA/S in A (applied with respect to f) implies f†† = f, so if we abuse notation by writing ft rather than f† then (ft )t = f. We say f is symmetric when ft = f (or more accurately, f† = f). This property holds if it does so on fibers over S, because homomorphisms f, f : A ⇒ B between abelian schemes coincide if fs = f s for all s ∈ S. Indeed, for noetherian S such rigidity is [83, Cor. 6.2], and the general case reduces to this because equality on all fibers descends through direct limits (since it says that the finitely presented ideal of (f, f )−1 (ΔA/S) in OA is nilpotent). A polarization of A is a homomorphism f : A → At that is a polarization on geometric fibers. Any such f is necessarily symmetric. The properties of polar- izations are developed in [83, §6.2] for projective abelian schemes, but the only purpose of imposing projectivity at the outset (even though it is a consequence of the definition, due to [34, IV3, 9.6.4]) is to ensure the existence of the dual abelian scheme, so such an assumption may be eliminated. 1.4.2.3. Definition. A homomorphism ϕ : A → B between abelian schemes over a scheme S is an isogeny when it is surjective with finite fibers. (Equivalently, the homomorphims ϕs are isogenies in the sense of abelian varieties for each s ∈ S.) Since quasi-finite proper morphisms are finite by [34, IV4, 18.12.4] (or by [34, IV3, 8.11.1] with finite presentation hypothesis, which suffices for us), any isogeny between abelian schemes is a finite morphism. Moreover, by the fibral flatness criterion [34, IV3, 11.3.11], such maps are flat. Hence, if ϕ as above is an isogeny then it is finite locally free (and surjective), so the closed subgroup ker(ϕ) is a finite locally free commutative S-group scheme. Thus, B represents the fppf quotient sheaf A/ker(ϕ). For example, setting ϕ = [n]A for n ⩾ 1 gives A/A[n] A. Turning this around, suppose we are given the abelian scheme A and a closed S-subgroup N ⊂ A that is finite locally free over S. Consider the fppf quotient sheaf A/N. We claim that this quotient is (represented by) an abelian scheme, so the map A → A/N with kernel N is an isogeny. It suffices to work Zariski-locally on S, so we may assume that N → S has all fibers with the same order n ⩾ 1. We then have N ⊂ A[n], due to the following result (proved in [123, §1]): 1.4.2.4. Theorem (Deligne). Let S be a scheme and let H be a commutative S- group scheme for which the structural morphism H → S is finite and locally free. If the fibers Hs have rank n for all s ∈ S then H is killed by n. The quotient sheaf A/N is an fppf torsor over A/A[n] A with fppf covering group A[n]/N that is finite (and hence affine) over S. It then follows from effective fppf descent for affine morphisms that the quotient A/N is represented by a scheme finite over A/A[n] = A, and the map A → A/N is an fppf A[n]/N-torsor, so the S-proper S-smooth A is finite locally free over A/N (as A[n]/N is finite locally free over S). Hence, A/N is proper and smooth since A is, and likewise its fibers over S are geometrically connected. Thus, A/N is an abelian scheme as desired.
- 53. 38 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION 1.4.2.5. Theorem. Let ϕ : A → B be an isogeny between abelian schemes over a scheme S, and let N = ker(ϕ). Duality applied to the exact sequence 0 → N −→ A ϕ −→ B → 0 functorially yields an exact sequence 0 → ND −→ Bt ϕt −→ At → 0. That is, the map ϕt is an isogeny whose kernel is canonically isomorphic to ND . Moreover, double duality for abelian schemes and for finite locally free commu- tative group schemes are compatible up to a sign: if we identify ϕ and ϕtt via ιA/S and ιB/S then the natural isomorphism (ND )D ker((ϕt )t ) = ker(ϕtt ) ker(ϕ) = N is the negative of the canonical isomorphism provided by Cartier duality. We refer the reader to [86, Thm. 1.1, Cor. 1.3] for a proof based on arguments that relativize the ones over an algebraically closed field in [82]. (An alternative approach, at least for the first part, is [87, Thm. 19.1], resting on the link between dual abelian schemes and Ext-sheaves given in [87, Thm. 18.1].) The special case ϕ = [n]A : A → A implies that naturally A[n]D = At [n] for every n ⩾ 1 because [n]t A = [n]At (by [87, 18.3]); this identification respects multiplicative change in n. 1.4.3. Constructions and definitions. Let us now focus on constructions spe- cific to the theory of finite commutative group schemes over a perfect field k of characteristic p 0. Let W = W(k) be the ring of Witt vectors of k; e.g., if k is finite of size q = pr then W is the ring of integers in an unramified extension of Qp of degree r. Let σ be the unique automorphism of W that reduces to the map x → xp on the residue field k. 1.4.3.1. Definition. The Dieudonné ring Dk over k is W[F, V], where F and V are indeterminates subject to the relations (1) FV = VF = p, (2) Fc = σ(c)F and cV = Vσ(c) for all c ∈ W. Explicitly, elements of Dk have unique expressions as finite sums a0 + j0 ajFj + j0 bjVj with coefficients in W (so the center of Dk is clearly Zp[Fr , Vr ] if k has finite size pr and it is Zp otherwise; i.e., if k is infinite). Some of the main conclusions in classical Dieudonné theory, as developed from scratch in [41, Ch. I–III], are summarized in the following theorem. 1.4.3.2. Theorem. There is an additive anti-equivalence of categories G M∗ (G) from the category of finite commutative k-group schemes of p-power order to the category of left Dk-modules of finite W-length. Moreover, the following hold. (1) A group scheme G has order pW (M∗ (G)) , where W (·) denotes W-length. (2) If k → k is an extension of perfect fields with associated extension W → W of Witt rings (e.g., the absolute Frobenius automorphism of k) then the functor W ⊗W (·) on Dieudonné modules is naturally identified with the base-change
- 54. 1.4. DIEUDONNÉ THEORY, p-DIVISIBLE GROUPS, AND DEFORMATIONS 39 functor on finite commutative group schemes. In particular, M∗ (G(p) ) σ∗ (M∗ (G)) as W-modules. (3) Let FrG/k : G → G(p) be the relative Frobenius morphism. The σ-semilinear action on M∗ (G) induced by M∗ (FrG/k) through the isomorphism M∗ (G(p) ) σ∗ (M∗ (G)) equals the action of F, and G is connected if and only if F is nilpotent on M∗ (G). (4) There is a natural k-linear isomorphism M∗ (G)/FM∗ (G) Lie(G)∨ respect- ing extension of the perfect base field. (5) For the Cartier dual GD , naturally M∗ (GD ) HomW (M∗ (G), K/W) with K = W[1/p], using the operators F() : m → σ((V(m))) and V() : m → σ−1 ((F(m))) on K/W-valued linear forms . For an abelian scheme A → S with fibers of constant dimension g ⩾ 1 and its finite commutative pn -torsion subgroup scheme A[pn ] with order (pn )2g , the directed system A[p∞ ] := (A[pn ])n⩾1 satisfies the following definition (with h = 2g). 1.4.3.3. Definition. A p-divisible group of height h ⩾ 0 over a scheme S is a directed system G = (Gn)n⩾1 of commutative S-groups Gn such that: Gn is killed by pn , each Gn → S is finite and locally free, [p]: Gn+1 → Gn is faithfully flat for every n ⩾ 1, G1 → S has constant degree ph , and Gn is identified with Gn+1[pn ] for all n ⩾ 1. The (Serre) dual p-divisible group Gt is the directed system (GD n ) of Cartier dual group schemes GD n with the transition maps GD n → GD n+1 that are Cartier dual to the quotient maps [p] : Gn+1 → Gn. As an illustration, if A → S is an abelian scheme with fibers of dimension g ⩾ 1 then the isomorphisms A[n]D At [n] respecting multiplicative change in n (as noted immediately below Theorem 1.4.2.5) yield a canonical isomorphism between the Serre dual A[p∞ ]t and the p-divisible group At [p∞ ] of the dual abelian scheme At (see [86, Prop. 1.8] or [87, Thm. 19.1]). 1.4.3.4. Remark. In view of the sign discrepancy for comparisons of double du- ality in Theorem 1.4.2.5, if ϕ : A → At is an S-homomorphism and f : A[p∞ ] → At [p∞ ] A[p∞ ]t is the associated homomorphism between p-divisible groups then the dual homo- morphism ϕt : A → At (strictly speaking, ϕt ◦ ιA/S via double duality for abelian schemes) has as its associated homomorphism A[p∞ ] → A[p∞ ]t the negative1 of ft (using double duality for p-divisible groups). It follows that if ϕ is symmetric with respect to double duality for abelian schemes then f is skew-symmetric with respect to double duality for p-divisible groups. The converse is also true: we can see immediately via skew-symmetry of f that ϕ and ϕt induce the same homomorphism between p-divisible groups, and to conclude that ϕ = ϕt it suffices to check on fibers due to the rigidity of abelian schemes (as in 1.4.2.2). On fibers we can apply the faithfulness of passage to p-divisible groups over fields via 1.2.5.1 with = p. 1A related sign issue in the double duality for commutative finite group schemes over perfect fields is discussed in a footnote at the end of B.3.5.5.
- 55. Random documents with unrelated content Scribd suggests to you:
- 56. melancholious dremeth of sorrow, the Cholarike, of firy things, and the Flematike, of Raine, Snow,' c.; id. lib. vi. c. 27. 4123. the humour of malencolye. 'The name (melancholy) is imposed from the matter, and disease denominated from the material cause, as Bruel observes, μελαγχολία quasi μελαιναχόλη, from black choler.' Fracastorius, in his second book of Intellect, calls those melancholy 'whom abundance of that same depraved humour of black choler hath so misaffected, that they become mad thence, and dote in most things or in all, belonging to election, will, or other manifest operations of the understanding.'—Burton's Anat. of Melancholy, p. 108, ed. 1805. 4128. 'That cause many a man in sleep to be very distressed.' 4130. Catoun. Dionysius Cato, de Moribus, l. ii. dist. 32: somnia ne cures. 'I observe by the way, that this distich is quoted by John of Salisbury, Polycrat. l. ii. c. 16, as a precept viri sapientis. In another place, l. vii. c. 9, he introduces his quotation of the first verse of dist. 20 (l. iii.) in this manner:—Ait vel Cato vel alius, nam autor incertus est.'—Tyrwhitt. Cf. note to G. 688. 4131. do no fors of = take no notice of, pay no heed to. Skelton, i. 118, has 'makyth so lytyll fors,' i. e. cares so little for. 4153. 'Wormwood, centaury, pennyroyal, are likewise magnified and much prescribed, especially in hypochondrian melancholy, daily to be used, sod in whey. And because the spleen and blood are often misaffected in melancholy, I may not omit endive, succory, dandelion, fumitory, c., which cleanse the blood.'—Burton's Anat. of Mel. pp. 432, 433. See also p. 438, ed. 1845. 'Centauria abateth wombe-ache, and cleereth sight, and vnstoppeth the splene and the reines'; Batman upon Bartholomè, lib. xvii. c. 47. 'Fumus terre [fumitory] cleanseth and purgeth Melancholia, fleme, and cholera'; id. lib. xvii. c. 69. 'Medicinal herbs were grown in every garden, and were dried or made into decoctions, and kept for use'; Wright, Domestic Manners, p. 279.
- 57. 4154. ellebor. Two kinds of hellebore are mentioned by old writers; 'white hellebore, called sneezing powder, a strong purger upward' (Burton's Anat. of Mel. pt. 2. § 4. m. 2. subsec. 1.), and 'black hellebore, that most renowned plant, and famous purger of melancholy.'—Ibid. subsec. 2. 4155. catapuce, caper-spurge, Euphorbia Lathyris. gaytres (or gaytrys) beryis, probably the berries of the buck-thorn, Rhamnus catharticus; which (according to Rietz) is still called, in Swedish dialects, the getbärs-trä (goat-berries tree) or getappel (goat-apple). I take gaytre to stand for gayt-tre, i. e. goat-tree; a Northern form, from Icel. geit (gen. geitar), a goat. The A. S. gāte-trēow, goat-tree, is probably the same tree, though the prov. Eng. gaiter-tree, gatten- tree, or gatteridge-tree is usually applied to the Cornus sanguinea or cornel-tree, the fruits of which 'are sometimes mistaken for those of the buck-thorn, but do not possess the active properties of that plant'; Eng. Cyclop., s. v. Cornus. The context shews that the buck- thorn is meant. Langham says of the buck-thorn, that 'the beries do purge downwards mightily flegme and choller'; Garden of Health, 1633, p. 99 (New E. Dict., s. v. Buckthorn). This is why Chanticleer was recommended to eat them. 4156. erbe yve, herb ive or herb ivy, usually identified with the ground-pine, Ajuga chamæpitys. mery, pleasant, used ironically; as the leaves are extremely nauseous. 4160. graunt mercy, great thanks; this in later authors is corrupted into grammercy or gramercy. 4166. so mote I thee, as I may thrive (or prosper). Mote = A. S. mōt-e, first p. s. pr. subj. 4174. Oon of the gretteste auctours. 'Cicero, De Divin. l. i. c. 27, relates this and the following story, but in a different order, and with so many other differences, that one might be led to suspect that he was here quoted at second-hand, if it were not usual with Chaucer, in these stories of familiar life, to throw in a number of natural
- 58. circumstances, not to be found in his original authors.'—Tyrwhitt. Warton thinks that Chaucer took it rather from Valerius Maximus, who has the same story; i. 7. He has, however, overlooked the statement in l. 4254, which decides for Cicero. I here quote the whole of the former story, as given by Valerius. 'Duo familiares Arcades iter una facientes, Megaram venerunt; quorum alter ad hospitem se contulit, alter in tabernam meritoriam devertit. Is, qui in hospitio venit, vidit in somnis comitem suam orantem, ut sibi cauponis insidiis circumvento subveniret: posse enim celeri ejus accursu se imminenti periculo subtrahi. Quo viso excitatus, prosiluit, tabernamque, in qua is diversabatur, petere conatus est. Pestifero deinde fato ejus humanissimum propositum tanquam supervacuum damnavit, et lectum ac somnum repetiit. Tunc idem ei saucius oblatus obsecravit, ut qui auxilium vitae suae ferre neglexisset, neci saltem ultionem non negaret. Corpus enim suum à caupone trucidatum, tum maxime plaustro ad portam ferri stercore coöpertum. Tam constantibus familiaris precibus compulsus, protinus ad portam cucurrit, et plaustrum, quod in quiete demonstratum erat, comprehendit, cauponemque ad capitale supplicium perduxit.' Valerii Maximi, lib. i. c. 7 (De Somniis). Cf. Cicero, De Divinatione, i. 27. 4194. oxes; written oxe in Hl. Cp. Ln; where oxe corresponds to the older English gen. oxan, of an ox—oxe standing for oxen (as in Oxenford, see note on l. 285 of Prologue). Thus oxes and oxe are equivalent. 4200. took of this no keep, took no heed to this, paid no attention to it. 4211. sooth to sayn, to say (tell) the truth. 4232. gapinge. The phrase gaping upright occurs elsewhere (see Knightes Tale, A. 2008), and signifies lying flat on the back with the mouth open. Cf. 'Dede he sate uprighte,' i. e. he lay on his back dead. The Sowdone of Babyloyne, l. 530.
- 59. 4235. Harrow, a cry of distress; a cry for help. 'Harrow! alas! I swelt here as I go.'—The Ordinary; see vol. iii. p. 150, of the Ancient Drama. See F. haro in Godefroy and Littré; and note to A. 3286. 4237. outsterte (Elles., c.); upsterte (Hn., Harl.) 4242. A common proverb. Skelton, ed. Dyce, i. 50, has 'I drede mordre wolde come oute.' 4274. And preyde him his viáge for to lette, And prayed him to abandon his journey. 4275. to abyde, to stay where he was. 4279. my thinges, my business-matters. 4300. 'Kenelm succeeded his father Kenulph on the throne of the Mercians in 821 [Haydn, Book of Dates, says 819] at the age of seven years, and was murdered by order of his aunt, Quenedreda. He was subsequently made a saint, and his legend will be found in Capgrave, or in the Golden Legend.'—Wright. St. Kenelm's day is Dec. 13. Alban Butler, in his Lives of the Saints, says:—[Kenulph] 'dying in 819, left his son Kenelm, a child only seven years old [see l. 4307] heir to his crown, under the tutelage of his sister Quindride. This ambitious woman committed his person to the care of one Ascobert, whom she had hired to make away with him. The wicked minister decoyed the innocent child into an unfrequented wood, cut off his head, and buried him under a thorn- tree. His corpse is said to have been discovered by a heavenly ray of light which shone over the place, and by the following inscription:— In Clent cow-pasture, under a thorn, Of head bereft, lies Kenelm, king born.' Milton tells the story in his History of Britain, bk. iv. ed. 1695, p. 218, and refers us to Matthew of Westminster. He adds that the
- 60. 'inscription' was inside a note, which was miraculously dropped by a dove on the altar at Rome. Our great poet's verson of it is:— 'Low in a Mead of Kine, under a thorn, Of Head bereft, li'th poor Kenelm King-born.' Clent is near the boundary between Staffordshire and Worcestershire. Neither of these accounts mentions Kenelm's dream, but it is given in his Life, as printed in Early Eng. Poems, ed. Furnivall (Phil. Soc. 1862), p. 51, and in Caxton's Golden Legend. St. Kenelm dreamt that he saw a noble tree with waxlights upon it, and that he climbed to the top of it; whereupon one of his best friends cut it down, and he was turned into a little bird, and flew up to heaven. The little bird denoted his soul, and the flight to heaven his death. 4307. For traisoun, i. e. for fear of treason. 4314. Cipioun. The Somnium Scipionis of Cicero, as annotated by Macrobius, was a favourite work during the middle ages. See note to l. 31 of the Parl. of Foules. 4328. See the Monkes Tale, B. 3917, and the note, p. 246. 4331. Lo heer Andromacha. Andromache's dream is not to be found in Homer. It is mentioned in chapter xxiv. of Dares Phrygius, the authority for the history of the Trojan war most popular in the middle ages. See the Troy-book, ed. Panton and Donaldson (E.E.T.S.), l. 8425; or Lydgate's Siege of Troye, c. 27. 4341. as for conclusioun, in conclusion. 4344. telle ... no store, set no store by them; reckon them of no value; count them as useless. 4346. never a del, never a whit, not in the slightest degree.
- 61. 4350. This line is repeated from the Compleynt of Mars, l. 61. 4353-6. 'By way of quiet retaliation for Partlet's sarcasm, he cites a Latin proverbial saying, in l. 344, 'Mulier est hominis confusio,' which he turns into a pretended compliment by the false translation in ll. 345, 346.'—Marsh. Tyrwhitt quotes it from Vincent of Beauvais, Spec. Hist. x. 71. Chaucer has already referred to this saying above; see p. 207, l. 2296. 'A woman, as saith the philosofre [i. e. Vincent], is the confusion of man, insaciable, c.'; Dialogue of Creatures, cap. cxxi. 'Est damnum dulce mulier, confusio sponsi'; Adolphi Fabulae, x. 567; pr. in Leyser, Hist. Poet. Med. Aevi, p. 2031. Cf. note to D. 1195. 4365. lay, for that lay. Chaucer omits the relative, as is frequently done in Middle English poetry; see note to l. 4090. 4377. According to Beda, the creation took place at the vernal equinox; see Morley, Eng. Writers, 1888, ii. 146. Cf. note to l. 4045. 4384. See note on l. 4045 above. 4395. Cf. Man of Lawes Tale, B. 421, and note. See Prov. xiv. 13. 4398. In the margin of MSS. E. and Hn. is written 'Petrus Comestor,' who is probably here referred to. 4402. See the Squieres Tale, F. 287, and the note. 4405. col-fox; explained by Bailey as a 'coal-black fox'; and he seems to have caught the right idea. Col- here represents M. E. col, coal; and the reference is to the brant-fox, which is explained in the New E. Dict. as borrowed from the G. brand-fuchs, 'the German name of a variety of the fox, chiefly distinguished by a greater admixture of black in its fur; according to Grimm, it has black feet, ears, and tail.' Chaucer expressly refers to the black-tipped tail and ears in l. 4094 above. Mr. Bradley cites the G. kohlfuchs and Du. koolvos, similarly formed; but the ordinary dictionaries do not give these names. The
- 62. old explanation of col-fox as meaning 'deceitful fox' is difficult to establish, and is now unnecessary. 4412. undern; see note to E. 260. 4417. Scariot, i. e. Judas Iscariot. Genilon; the traitor who caused the defeat of Charlemagne, and the death of Roland; see Book of the Duchesse, 1121, and the note in vol. i. p. 491. 4418. See Vergil, Æn. ii. 259. 4430. bulte it to the bren, sift the matter; cf. the phrase to boult the bran. See the argument in Troilus, iv. 967; cf. Milton, P. L. ii. 560. 4432. Boece, i. e. Boethius. See note to Kn. Tale, A. 1163. Bradwardyn. Thomas Bradwardine was Proctor in the University of Oxford in the year 1325, and afterwards became Divinity Professor and Chancellor of the University. His chief work is 'On the Cause of God' (De Causâ Dei). See Morley's English Writers, iv. 61. 4446. colde, baneful, fatal. The proverb is Icelandic; 'köld eru opt kvenna-ráð,' cold (fatal) are oft women's counsels; Icel. Dict. s. v. kaldr. It occurs early, in The Proverbs of Alfred, ed. Morris, Text 1, l. 336:—'Cold red is quene red.' Cf. B. 2286, and the note. 4450-6. Imitated from Le Roman de la Rose, 15397-437. 4461. Phisiologus. 'He alludes to a book in Latin metre, entitled Physiologus de Naturis xii. Animalium, by one Theobaldus, whose age is not known. The chapter De Sirenis begins thus:— Sirenae sunt monstra maris resonantia magnis Vocibus, et modulis cantus formantia multis, Ad quas incaute veniunt saepissime nautae, Quae faciunt sompnum nimia dulcedine vocum.'—Tyrwhitt.
- 63. See The Bestiary, in Dr. Morris's Old English Miscellany, pp. 18, 207; Philip de Thaun, Le Bestiaire, l. 664; Babees Book, pp. 233, 237; Mätzner's Sprachproben, i. 55; Gower, C.A. i. 58; and cf. Rom. Rose, Eng. Version, 680 (in vol. i. p. 122). 4467. In Douglas's Virgil, prol. to Book xi. st. 15, we have— 'Becum thow cowart, craudoun recryand, And by consent cry cok, thi deid is dycht'; i. e. if thou turn coward, (and) a recreant craven, and consent to cry cok, thy death is imminent. In a note on this passage, Ruddiman says—'Cok is the sound which cocks utter when they are beaten.' But it is probable that this is only a guess, and that Douglas is merely quoting Chaucer. To cry cok! cok! refers rather to the utterance of rapid cries of alarm, as fowls cry when scared. Brand (Pop. Antiq., ed. Ellis, ii. 58) copies Ruddiman's explanation of the above passage. 4484. Boethius wrote a treatise De Musica, quoted by Chaucer in the Hous of Fame; see my note to l. 788 of that poem (vol. iii. p. 260). 4490. 'As I hope to retain the use of my two eyes.' So Havelok, l. 2545:— 'So mote ich brouke mi Rith eie!' And l. 1743:—'So mote ich brouke finger or to.' And l. 311:—'So brouke i euere mi blake swire!' swire = neck. See also Brouke in the Glossary to Gamelyn. 4502. daun Burnel the Asse. 'The story alluded to is in a poem of Nigellus Wireker, entitled Burnellus seu Speculum Stultorum, written in the time of Richard I. In the Chester Whitsun Playes, Burnell is used as a nickname for an ass. The original word was probably brunell, from its brown colour; as the fox below is called Russel, from
- 64. his red colour.'—Tyrwhitt. The Latin story is printed in The Anglo- Latin Satirists of the Twelfth Century, ed. T. Wright, i. 55; see also Wright's Biographia Britannica Literaria, Anglo-Norman Period, p. 356. There is an amusing translation of it in Lowland Scotch, printed as 'The Unicornis Tale' in Small's edition of Laing's Select Remains of Scotch Poetry, ed. 1885, p. 285. It tells how a certain young Gundulfus broke a cock's leg by throwing a stone at him. On the morning of the day when Gundulfus was to be ordained and to receive a benefice, the cock took his revenge by not crowing till much later than usual; and so Gundulfus was too late for the ceremony, and lost his benefice. Cf. Warton, Hist. E. P., ed. 1871, ii. 352; Lounsbury, Studies in Chaucer, ii. 338. As to the name Russel, see note to l. 4039. 4516. See Rom. of the Rose (E. version), 1050. MS. E. alone reads courtes; Hn. Cm. Cp. Pt. have court; Ln. courte; Hl. hous. 4519. Ecclesiaste; not Ecclesiastes, but Ecclesiasticus, xii. 10, 11, 16. Cf. Tale of Melibeus, B. 2368. 4525. Tyrwhitt cites the O. F. form gargate, i. e. (throat), from the Roman de Rou. Several examples of it are given by Godefroy. 4537. O Gaufred. 'He alludes to a passage in the Nova Poetria of Geoffrey de Vinsauf, published not long after the death of Richard I. In this work the author has not only given instructions for composing in the different styles of poetry, but also examples. His specimen of the plaintive style begins thus:— 'Neustria, sub clypeo regis defensa Ricardi, Indefensa modo, gestu testare dolorem; Exundent oculi lacrimas; exterminet ora Pallor; connodet digitos tortura; cruentet Interiora dolor, et verberet aethera clamor; Tota peris ex morte sua. Mors non fuit eius, Sed tua, non una, sed publica mortis origo. O Veneris lacrimosa dies! O sydus amarum!
- 65. Illa dies tua nox fuit, et Venus illa venenum. Illa dedit vulnus,' c. These lines are sufficient to show the object and the propriety of Chaucer's ridicule. The whole poem is printed in Leyser's Hist. Poet. Med. Ævi, pp. 862-978.'—Tyrwhitt. See a description of the poem, with numerous quotations, in Wright's Biographia Britannica Literaria, Anglo-Norman Period, p. 400; cf. Lounsbury, Studies, ii. 341. 4538. Richard I. died on April 6, 1199, on Tuesday; but he received his wound on Friday, March 26. 4540. Why ne hadde I = O that I had. 4547. streite swerd = drawn (naked) sword. Cf. Aeneid, ii. 333, 334: — 'Stat ferri acies mucrone corusco Stricta, parata neci.' 4548. See Aeneid, ii. 550-553. 4553. Hasdrubal; not Hannibal's brother, but the King of Carthage when the Romans burnt it, B.C. 146. Hasdrubal slew himself; and his wife and her two sons burnt themselves in despair; see Orosius, iv. 13. 3, or Ælfred's translation, ed. Sweet, p. 212. Lydgate has the story in his Fall of Princes, bk. v. capp. 12 and 27. 4573. See note to Ho. Fame, 1277 (in vol. iii. p. 273). 'Colle furit'; Morley, Eng. Writers, 1889, iv. 179. 4584. Walsingham relates how, in 1381, Jakke Straw and his men killed many Flemings 'cum clamore consueto.' He also speaks of the noise made by the rebels as 'clamor horrendissimus.' See Jakke in Tyrwhitt's Glossary. So also, in Riley's Memorials of London, p. 450, it is said, with respect to the same event—'In the Vintry was a very great massacre of Flemings.'
- 66. 4590. houped. See Piers Plowman, B. vi. 174; 'houped after Hunger, that herde hym,' c. 4616. Repeated in D. 1062. 4633. 'Mes retiengnent le grain et jettent hors la paille'; Test. de Jean de Meun, 2168. 4635. my Lord. A side-note in MS. E. explains this to refer to the Archbishop of Canterbury; doubtless William Courtenay, archbishop from 1381 to 1396. Cf. note to l. 4584, which shews that this Tale is later than 1381; and it was probably earlier than 1396. Note that good men is practically a compound, as in l. 4630. Hence read good, not gōd-e. Epilogue to the Nonne Preestes Tale. 4641. Repeated from B. 3135. 4643. Thee wer-e nede, there would be need for thee. 4649. brasil, a wood used for dyeing of a bright red colour; hence the allusion. It is mentioned as being used for dyeing leather in Riley's Memorials of London, p. 364. 'Brazil-wood; this name is now applied in trade to the dye-wood imported from Pernambuco, which is derived from certain species of Cæsalpinia indigenous there. But it originally applied to a dye-wood of the same genus which was imported from India, and which is now known in trade as Sappan. The history of the word is very curious. For when the name was applied to the newly discovered region in S. America, probably, as Barros alleges, because it produced a dye-wood similar in character to the brazil of the East, the trade-name gradually became appropriated to the S. American product, and was taken away from that of the E. Indies. See some further remarks in Marco Polo, ed. Yule, 2nd ed. ii. 368-370.
- 67. 'This is alluded to also by Camoẽs (Lusiad, x. 140). Burton's translation has:— But here, where earth spreads wider, ye shall claim Realms by the ruddy dye-wood made renowned; These of the 'Sacred Cross' shall win the name, By your first navy shall that world be found. 'The medieval forms of brazil were many; in Italian, it is generally verzi, verzino, or the like.'—Yule, Hobson-Jobson, p. 86. Again—'Sappan, the wood of Cæsalpinia sappan; the baqqam of the Arabs, and the Brazil-wood of medieval commerce. The tree appears to be indigenous in Malabar, the Deccan, and the Malay peninsula.'— id. p. 600. And in Yule's edition of Marco Polo, ii. 315, he tells us that 'it is extensively used by native dyers, chiefly for common and cheap cloths, and for fine mats. The dye is precipitated dark-brown with iron, and red with alum.' Cf. Way's note on the word in the Prompt. Parv. p. 47. Florio explains Ital. verzino as 'brazell woode, or fernanbucke [Pernambuco] to dye red withall.' The etymology is disputed, but I think brasil and Ital. verzino are alike due to the Pers. wars, saffron; cf. Arab. warīs, dyed with saffron or wars. greyn of Portingale. Greyn, mod. E. grain, is the term applied to the dye produced by the coccus insect, often termed, in commerce and the arts, kermes; see Marsh, Lectures on the E. Language, Lect. III. The colour thus produced was 'fast,' i. e. would not wash out; hence the phrase to engrain, or to dye in grain, meaning to dye of a fast colour. Various tones of red were thus produced, one of which was crimson, and another carmine, both forms being derivatives of kermes. Of Portingale means 'imported from Portugal.' In the Libell of English Policy, cap. ii. (l. 132), it is said that, among 'the
- 68. commoditees of Portingale' are:—'oyl, wyn, osey [Alsace wine], wex, and graine.' 4652. to another, to another of the pilgrims. This is so absurdly indefinite that it can hardly be genuine. Ll. 4637-4649 are in Chaucer's most characteristic manner, and are obviously genuine; but there, I suspect, we must stop, viz. at the word Portingale. The next three lines form a mere stop-gap, and are either spurious, or were jotted down temporarily, to await the time of revision. The former is more probable. This Epilogue is only found in three MSS.; (see footnote, p. 289). In Dd., Group G follows, beginning with the Second Nun's Tale. In the other two MSS., Group H follows, i. e. the Manciple's Tale; nevertheless, MS. Addit. absurdly puts the Nunne, in place of another. The net result is, that, at this place, the gap is complete; with no hint as to what Tale should follow. It is worthy of note that this Epilogue is preserved in Thynne and the old black-letter editions, in which it is followed immediately by the Manciple's Prologue. This arrangement is obviously wrong, because that Prologue is not introduced by the Host (as said in l. 4652). In l. 4650, Thynne has But for Now; and his last line runs—'Sayd to a nother man, as ye shal here.' I adopt his reading of to for unto (as in the MSS.). NOTES TO GROUP C. The Phisiciens Tale. For remarks on the spurious Prologues to this Tale, see vol. iii. p. 434. For further remarks on the Tale, see the same, p. 435, where its original is printed in full.
- 69. 1. The story is told by Livy, lib. iii.; and, of course, his narrative is the source of all the rest. But Tyrwhitt well remarks, in a note to l. 12074 (i. e. C. 140):—'In the Discourse, c., I forgot to mention the Roman de la Rose as one of the sources of this tale; though, upon examination, I find that our author has drawn more from thence, than from either Gower or Livy.' It is absurd to argue, as in Bell's Chaucer, that our poet must necessarily have known Livy 'in the original,' and then to draw the conclusion that we must look to Livy only as the true source of the Tale. For it is perfectly obvious that Tyrwhitt is right as regards the Roman de la Rose; and the belief that Chaucer may have read the tale 'in the original' does not alter the fact that he trusted much more to the French text. In this very first line, he is merely quoting Le Roman, ll. 5617, 8:— 'Qui fu fille Virginius, Si cum dist Titus Livius.' The story in the French text occupies 70 lines (5613-5682, ed. Méon); the chief points of resemblance are noted below. Gower has the same story, Conf. Amant. iii. 264-270; but I see no reason why Chaucer should be considered as indebted to him. It is, however, clear that, if Chaucer and Gower be here compared, the latter suffers considerably by the comparison. Gower gives the names of Icilius, to whom Virginia was betrothed, and of Marcus Claudius. But Chaucer omits the name Marcus, and ignores the existence of Icilius. The French text does the same. 11. This is the 'noble goddesse Nature' mentioned in the Parl. of Foules, ll. 368, 379. Cf. note to l. 16. 14. Pigmalion, Pygmalion; alluding to Ovid, Met. x. 247, where it is said of him:— 'Interea niueum mira feliciter arte Sculpit ebur, formamque dedit, qua femina nasci
- 70. Nulla potest; operisque sui concepit amorem.' In the margin of E. Hn. is the note—'Quere in Methamorphosios'; which supplies the reference; but cf. note to l. 16 below, shewing that Chaucer also had in his mind Le Roman de la Rose, l. 16379. So also the author of the Pearl, l. 750; see Morris, Allit. Poems. 16. In the margin of E. Hn. we find the note:—'Apelles fecit mirabile opus in tumulo Darii; vide in Alexandri libro .1.º [Hn. has .6.º]; de Zanze in libro Tullii.' This note is doubtless the poet's own; see further, as to Apelles, in the note to D. 498. Zanzis, Zeuxis. The corruption of the name was easy, owing to the confusion in MSS. between n and u.[26] In the note above, we are referred to Tullius, i. e. Cicero. Dr. Reid kindly tells me that Zeuxis is mentioned, with Apelles, in Cicero's De Oratore, iii. § 26, and Brutus, § 70; also, with other artists, in Academia, ii. § 146; De Finibus, ii. § 115; and alone, in De Inventione, ii. § 52, where a long story is told of him. Cf. note to Troil. iv. 414. However, the fact is that Chaucer really derived his knowledge of Zeuxis from Le Roman de la Rose (ed. Méon, l. 16387); for comparison with the context of that line shews numerous points of resemblance to the present passage in our author. Jean de Meun is there speaking of Nature, and of the inability of artists to vie with her, which is precisely Chaucer's argument here. The passage is too long for quotation, but I may cite such lines as these:— 'Ne Pymalion entaillier' (l. 16379), 'voire Apelles Que ge moult bon paintre appelles, Biautés de li james descrive Ne porroit,' c. (l. 16381). 'Zeuxis neis par son biau paindre Ne porroit a tel forme ataindre,' c. (l. 16387).
- 71. Si cum Tules le nous remembre Ou livre de sa retorique'; (l. 16398). Here the reference is to the passage in De Oratore, iii. § 26. 'Mes ci ne péust-il riens faire Zeuxis, tant séust bien portraire, Ne colorer sa portraiture, Tant est de grant biauté Nature.' (l. 16401). A little further on, Nature is made to say (l. 16970):— 'Cis Diex méismes, par sa grace,... Tant m'ennora, tant me tint chere, Qu'il m'establi sa chamberiere ... Por chamberiere! certes vaire, Por connestable, et por vicaire.' 20. See just above; and cf. Parl. of Foules, 379—'Nature, the vicaire of thalmighty lord.' 32-4. Cf. Le Rom. de la Rose, 16443-6. 35. From this line to l. 120, Chaucer has it all his own way. This fine passage is not in Le Roman, nor in Gower. 37. I. e. she had golden hair; cf. Troil. iv. 736, v. 8. 49. Perhaps Chaucer found the wisdom of Pallas in Vergil, Aen. v. 704.— 'Tum senior Nautes, unum Tritonia Pallas Quem docuit, multaque insignem reddidit arte.' 50. fácound, eloquence; cf. facóunde in Parl. Foules, 558. 54. Souninge in, conducing to; see A. 307, B. 3157, and notes.
- 72. 58. Bacus, Bacchus, i. e. wine; see next note. 59. youthe, youth; such is the reading in MSS. E. Hn., and edd. 1532 and 1561. MS. Cm. has lost a leaf; the rest have thought, which gives no sense. It is clear that the reading thought arose from misreading the y of youthe as þ (th). How easily this may be done appears from Wright's remark, that the Lansdowne MS. has youthe, whilst, in fact, it has þouht. Tyrwhitt objects to the reading youthe, and proposes slouthe, wholly without authority. But youthe, meaning 'youthful vigour,' is right enough; I see no objection to it at all. Rather, it is simply taken from Ovid, Ars Amat. i. 243:— 'Illic saepe animos iuuenum rapuere puellae; Et Venus in uinis, ignis in igne fuit.' Only a few lines above (l. 232), Bacchus occurs, and there is a reference to wine, throughout the context. Cf. the Romaunt of the Rose, l. 4925:— 'For Youthe set man in al folye ... In leccherye and in outrage.' Cf. note to l. 65. 60. Alluding to a proverbial phrase, occurring in Horace, Sat. ii. 3. 321, viz. 'oleum adde camino'; and elsewhere. 65. This probably refers to the same passage in Ovid as is mentioned in the note to l. 59. For we there find (l. 229):— 'Dant etiam positis aditum conuiuia mensis; Est aliquid, praeter uina, quod inde petas ... Vina parant animos, faciuntque caloribus aptos'; c.
- 73. 79. See A. 476, and the note. Chaucer is here thinking of the same passage in Le Roman de la Rose. I quote a few lines (3930-46):— 'Une vielle, que Diex honnisse! Avoit o li por li guetier, Qui ne fesoit autre mestier Fors espier tant solement Qu'il ne se maine folement.... Bel-Acueil se taist et escoute Por la vielle que il redoute, Et n'est si hardis qu'il se moeve, Que la vielle en li n'aperçoeve Aucune fole contenance, Qu'el scet toute la vielle dance.' See the English version in vol. i. p. 205, ll. 4285-4300. 82. See the footnote for another reading. The line there given may also be genuine. It is deficient in the first foot. 85. This is like our proverb:—'Set a thief to catch [or take] a thief.' An old poacher makes a good gamekeeper. 98. Cf. Prov. xiii. 24; P. Plowman, B. v. 41. 101. See a similar proverb in P. Plowman, C. x. 265, and my note on the line. The Latin lines quoted in P. Plowman are from Alanus de Insulis, Liber Parabolarum, cap. i. 31; they are printed in Leyser, Hist. Poet. Med. Aevi, 1721, p. 1066, in the following form:— 'Sub molli pastore capit lanam lupus, et grex Incustoditus dilaceratur eo.' 117. The doctour, i. e. the teacher; viz. St. Augustine. (There is here no reference whatever to the 'Doctor' or 'Phisicien' who is supposed to tell the tale.) In the margin of MSS. E. Hn. is written 'Augustinus'; and the matter is put beyond doubt by a passage in the Persones
- 74. Tale, l. 484:—'and, after the word of seint Augustin, it [Envye] is sorwe of other mannes wele, and Ioye of othere mennes harm.' See note to l. 484. The same idea is exactly reproduced in P. Plowman, B. v. 112, 113. Cf. 'Inuidus alterius macrescit rebus opimis'; Horace, Epist. i. 2. 57. 135. From Le Roman, l. 5620-3; see vol. iii. p. 436. 140. cherl, dependant. It is remarkable that, throughout the story, MSS. E. Hn. and Cm. have cherl, but the rest have clerk. In ll. 140, 142, 153, 164, the Camb. MS. is deficient; but it at once gives the reading cherl in l. 191, and subsequently. Either reading might serve; in Le Roman, l. 5614, the dependant is called 'son serjant'; and in l. 5623, he is called 'Li ribaus,' i. e. the ribald, which Chaucer Englishes by cherl. But when we come to C. 289, the MSS. gives us the choice of 'fals cherl' and 'cursed theef'; very few have clerk (like MS. Sloane 1685). Cf. vol. iii. p. 437. 153, 154. The 'churl's' name was Marcus Claudius, and the 'judge' was 'Appius Claudius.' Chaucer simply follows Jean de Meun, who calls the judge Apius; and speaks of the churl as 'Claudius li chalangieres' in l. 5675. 165. Cf. Le Roman, l. 5623-7; see vol. iii. p. 436. 168-9. From Le Roman, 5636-8, as above. 174. The first foot is defective; read—Thou | shalt have | al, c. al right, complete justice. MS. Cm. has alle. 184. Cf. Le Roman, l. 5628-33. 203. From Le Roman, 5648-54. 207-253. The whole of this fine passage appears to be original. There is no hint of it in Le Roman de la Rose, except as regards l.
- 75. 225, where Le Roman (l. 5659) has:—'Car il par amors, sans haïne.' We may compare the farewell speech of Virginius to his daughter in Webster's play of Appius and Virginia, Act iv. sc. 1. 240. Iepte, Jephtha; in the Vulgate, Jephte. See Judges, xi. 37, 38. MSS. E. Hn. have in the margin—'fuit illo tempore Jephte Galaandes' [error for Galaadites]. This reference by Virginia to the book of Judges is rather startling; but such things are common enough in old authors, especially in our dramatists. 255. Here Chaucer returns to Le Roman, 5660-82. The rendering is pretty close down to l. 276. 280. Agryse of, shudder at; 'nor in what kind of way the worm of conscience may shudder because of (the man's) wicked life'; cf. 'of pitee gan agryse,' B. 614. When agryse is used with of, it is commonly passive, not intransitive; see examples in Mätzner and in the New E. Dictionary. Cf. been afered, i. e. be scared, in l. 284. 'Vermis conscientiae tripliciter lacerabit'; Innocent III., De Contemptu Mundi, l. iii. c. 2. 286. Cf. Pers. Tale, I. 93:—'repentant folk, that stinte for to sinne, and forlete [give up] sinne er that sinne forlete hem.' Words of the Host. In the Six-text Edition, pref. col. 58, Dr. Furnivall calls attention to the curious variations in this passage, in the MSS., especially in ll. 289-292, and in 297-300; as well as in ll. 487, 488 in the Pardoneres Tale. I note these variations below, in their due places. 287. wood, mad, frantic, furious; esp. applied to the transient madness of anger. See Kn. Tale, A. 1301, 1329, 1578; also Mids. Nt. Dr. ii. 1. 192. Cf. G. wüthend, raging. 288. Harrow! also spelt haro; a cry of astonishment; see A. 3286, 3825, B. 4235, c. 'Haro, the ancient Norman hue and cry; the
- 76. exclamation of a person to procure assistance when his person or property was in danger. To cry out haro on any one, to denounce his evil doings'; Halliwell. Spenser has it, F. Q. ii. 6. 43; see Harrow in Nares, and the note above, to A. 3286. On the oaths used by the Host, see note to l. 651 below. 289. fals cherl is the reading in E. Hn., and is evidently right; see note to l. 140 above. It is supported by several MSS., among which are Harl. 7335, Addit. 25718, Addit. 5140, Sloane 1686, Barlow 20, Hatton 1, Camb. Univ. Lib. Dd. 4. 24 and Mm. 2. 5, and Trin. Coll. Cam. R. 3. 3. A few have fals clerk, viz. Sloane 1685, Arch. Seld. B. 14, Rawl. Poet. 149, Bodley 414. Harl. 7333 has a fals thef, Acursid Iustise; out of which numerous MSS. have developed the reading a cursed theef, a fals Iustice, which rolls the two Claudii into one. It is clearly wrong, but appears in good MSS., viz. in Cp. Pt. Ln. Hl. See vol. iii. pp. 437-8, and the note to l. 291 below. 290. shamful. MSS. Ln. Hl. turn this into schendful, i. e. ignominious, which does not at all alter the sense. It is a matter of small moment, but I may note that of the twenty-five MSS. examined by Dr. Furnivall, only the two above-named MSS. adopt this variation. 291, 292. Here MSS. Cp. Ln. Hl., as noted in the footnote, have two totally different lines; and this curious variation divides the MSS. (at least in the present passage) into two sets. In the first of these we find E. Hn. Harl. 7335, Addit. 25718, Addit. 5140, Sloane 1685 and 1686, Barlow 20, Arch. Seld. B. 14, Rawl. Poet. 149, Hatton 1, Bodley 414, Camb. Dd. 4. 24, and Mm. 2. 5, Trin. Coll. Cam. R. 3. 3. In the second set we find Cp. Ln. Hl., Harl. 1758, Royal 18. C. 2, Laud 739, Camb. Ii. 3. 26, Royal 17. D. 15, and Harl. 7333. There is no doubt as to the correct reading; for the 'false cherl' and 'false justice' were two different persons, and it was only because they had been inadvertently rolled into one (see note to l. 289) that it became possible to speak of 'his body,' 'his bones,' and 'him.' Hence the lines are rightly given in the text which I have adopted.
- 77. There is a slight difficulty, however, in the rime, which should be noted. We see that the t in advocats was silent, and that the word was pronounced (ad·vokaa·s), riming with allas (alaa·s), where the raised dot denotes the accent. That this was so, is indicated by the following spellings:—Pt. aduocas, and so also in Harl. 7335, Addit. 5140, Bodl. 414; Rawl. Poet. 149 has advocas; whilst Sloane 1685, Sloane 1686, and Camb. Mm. 2. 5 have aduocase, and Barlow 20, advocase. MS. Trin. Coll. R. 3. 3 has aduocasse. The testimony of ten MSS. may suffice; but it is worth noting that the F. pl. aduocas occurs in Le Roman de la Rose, 5107. 293. 'Alas! she (Virginia) bought her beauty too dear'; she paid too high a price; it cost her her life. 297-300. These four lines are genuine; but several MSS., including E. Hn. Pt., omit the former pair (297-8), whilst several others omit the latter pair. Ed. 1532 contains both pairs, but alters l. 299. 299. bothe yiftes, both (kinds of) gifts; i. e. gifts of fortune, such as wealth, and of nature, such as beauty. Compare Dr. Johnson's poem on the Vanity of Human Wishes, imitated from the tenth satire of Juvenal. 303. is no fors, it is no matter. It must be supplied, for the sense. Sometimes Chaucer omits it is, and simply writes no fors, as in E. 1092, 2430. We also find I do no fors, I care not, D. 1234; and They yeve no fors, they care not, Romaunt of the Rose, 4826. Palsgrave has—'I gyue no force, I care nat for a thing, Il ne men chault.' 306. Ypocras is the usual spelling, in English MSS., of Hippocrates; see Prologue A. 431. So also in the Book of the Duchess, 571, 572:— 'Ne hele me may physicien, Noght Ypocras, ne Galien.' In the present passage it does not signify the physician himself, but a beverage named after him. 'It was composed of wine, with spices
- 78. and sugar, strained through a cloth. It is said to have taken its name from Hippocrates' sleeve, the term apothecaries gave to a strainer'; Halliwell's Dict. s. v. Hippocras. In the same work, s. v. Ipocras, are several receipts for making it, the simplest being one copied from Arnold's Chronicle:—'Take a quart of red wyne, an ounce of synamon, and half an unce of gynger; a quarter of an ounce of greynes, and long peper, and halfe a pounde of sugar; and brose all this, and than put them in a bage of wullen clothe, made therefore, with the wyne; and lete it hange over a vessel, tyll the wyne be rune thorowe.' Halliwell adds that—'Ipocras seems to have been a great favourite with our ancestors, being served up at every entertainment, public or private. It generally made a part of the last course, and was taken immediately after dinner, with wafers or some other light biscuits'; c. See Pegge's Form of Cury, p. 161; Babees Book, ed. Furnivall, pp. 125-128, 267, 378; Skelton, ed. Dyce, ii. 285; and Nares's Glossary, s. v. Hippocras. Galianes. In like manner this word (hitherto unexplained as far as I am aware) must signify drinks named after Galen, whose name is spelt Galien (in Latin, Galienus) not only in Chaucer, but in other authors. See the quotation above from the Book of the Duchess. Speght guessed the word to mean 'Galen's works.' 310. lyk a prelat, like a dignitary of the church, like a bishop or abbot. Mr. Jephson, in Bell's edition, suggests that the Doctor was in holy orders, and that this is why we are told in the Prologue, l. 438, that 'his studie was but litel on the bible.' I see no reason for this guess, which is quite unsupported. Chaucer does not say he is a prelate, but that he is like one; because he had been highly educated, as a member of a 'learned profession' should be. Ronyan is here of three syllables and rimes with man; in l. 320 it is of two syllables, and rimes with anon. It looks as if the Host and Pardoner were not very clear about the saint's name, only knowing him to swear by. In Pilkington's Works (Parker Society), we find a mention of 'St. Tronian's fast,' p. 80; and again, of 'St. Rinian's fast,'
- 79. p. 551, in a passage which is a repetition of the former. The forms Ronyan and Rinian are evidently corruptions of Ronan, a saint whose name is well known to readers of 'St. Ronan's Well.' Of St. Ronan scarcely anything is known. The fullest account that can easily be found is the following:— 'Ronan, B. and C. Feb. 7.—Beyond the mere mention of his commemoration as S. Ronan, bishop at Kilmaronen, in Levenax, in the body of the Breviary of Aberdeen, there is nothing said about this saint.... Camerarius (p. 86) makes this Ronanus the same as he who is mentioned by Beda (Hist. Ecc. lib. iii. c. 25). This Ronan died in A. D. 778. The Ulster annals give at [A. D.] 737 (736)—Mors Ronain Abbatis Cinngaraid. Ængus places this saint at the 9th of February,' c.; Kalendars of Scottish Saints, by Bp. A. P. Forbes, 1872, p. 441. Kilmaronen is Kilmaronock, in the county and parish of Dumbarton. There are traces of St. Ronan in about seven place-names in Scotland, according to the same authority. Under the date of Feb. 7 (February vol. ii. 3 B), the Acta Sanctorum has a few lines about St. Ronan, who, according to some, flourished under King Malduin, A. D. 664-684; or, according to others, about 603. The notice concludes with the remark—'Maiorem lucem desideramus.' Beda says that 'Ronan, a Scot by nation, but instructed in ecclesiastical truth either in France or Italy,' was mixed up in the controversy which arose about the keeping of Easter, and was 'a most zealous defender of the true Easter.' This controversy took place about A. D. 652, which does not agree with the date above. 311. Tyrwhitt thinks that Shakespeare remembered this expression of Chaucer, when he describes the Host of the Garter as frequently repeating the phrase 'said I well': Merry Wives of Windsor, i. 3. 11; ii. 1. 226; ii. 3. 93, 99. in terme, in learned terms; cf. Prol. A. 323. 312. erme, to grieve. For the explanation of unusual words, the Glossary should, in general, be consulted; the Notes are intended,
- 80. for the most part, to explain only phrases and allusions, and to give illustrations of the use of words. Such illustrations are, moreover, often omitted when they can easily be found by consulting such a work as Stratmann's Old English Dictionary. In the present case, for example, Stratmann gives twelve instances of the use of earm or arm as an adjective, meaning wretched; four examples of ermlic, miserable; seven of earming, a miserable creature; and five of earmthe, misery. These twenty-eight additional examples shew that the word was formerly well understood. We may further note that a later instance of ermen or erme, to grieve, occurs in Caxton's translation of Reynard the Fox, A. D. 1481; see Arber's reprint, p. 48, l. 5: 'Thenne departed he fro the kynge so heuyly that many of them ermed,' i. e. then departed he from the king so sorrowfully that many of them mourned, or were greatly grieved. 313. cardiacle, pain about the heart, spasm of the heart; more correctly, cardiake, as the l is excrescent. See Cardiacle and Cardiac in the New E. Dictionary. In Batman upon Bartholomè, lib. vii. c. 32, we have a description of 'Heart-quaking and the disease Cardiacle.' We thus learn that 'there is a double manner of Cardiacle,' called 'Diaforetica' and 'Tremens.' Of the latter, 'sometime melancholy is the cause'; and the remedies are various 'confortatives.' This is why the host wanted some 'triacle' or some ale, or something to cheer him up. 314. The Host's form of oath is amusingly ignorant; he is confusing the two oaths 'by corpus Domini' and 'by Christes bones,' and evidently regards corpus as a genitive case. Tyrwhitt alters the phrase to 'By corpus domini,' which wholly spoils the humour of it. triacle, a restorative remedy; see Man of Lawes Tale, B. 479. 315. moyste, new. The word retains the sense of the Lat. musteus and mustus. In Group H. 60, we find moysty ale spoken of as differing from old ale. But the most peculiar use of the word is in the
- 81. Prologue, A. 457, where the Wyf of Bath's shoes are described as being moyste and newe. corny, strong of the corn or malt; cf. l. 456. Skelton calls it 'newe ale in cornys'; Magnificence, 782; or 'in cornes,' Elynour Rummyng, 378. Baret's Alvearie, s. v. Ale, has: 'new ale in cornes, ceruisia cum recrementis.' It would seem that ale was thought the better for having dregs of malt in it. 318. bel amy, good friend; a common form of address in old French. We also find biaus douz amis, sweet good friend; as in— 'Charlot, Charlot, biaus doux amis'; Rutebuef; La Disputoison de Charlot et du Barbier, l. 57. Belamy occurs in an Early Eng. Life of St. Cecilia, MS. Ashmole 43, l. 161; and six other examples are given in the New Eng. Dictionary. Similar forms are beau filtz, dear son, Piers Plowman, B. vii. 162; beau pere, good father; beau sire, good sir. Cf. beldame. 321. ale-stake, inn-sign. Speght interprets this by 'may-pole.' He was probably thinking of the ale-pole, such as was sometimes set up before an inn as a sign; see the picture of one in Larwood and Hotten's History of Signboards, Plate II. But the ale-stakes of the fourteenth century were differently placed; instead of being perpendicular, they projected horizontally from the inn, just like the bar which supports a painted sign at the present day. At the end of the ale-stake a large garland was commonly suspended, as mentioned by Chaucer himself (Prol. 667), or sometimes a bunch of ivy, box, or evergreen, called a 'bush'; whence the proverb 'good wine needs no bush,' i. e. nothing to indicate where it is sold; see Hist. Signboards, pp. 2, 4, 6, 233. The clearest information about ale-stakes is obtained from a notice of them in the Liber Albus, ed. Riley, where an ordinance of the time of Richard II. is printed, the translation of which runs as follows: 'Also, it was ordained that whereas the ale-stakes, projecting in front of the taverns in Chepe and elsewhere in the said city, extend too far over the king's
- 82. highways, to the impeding of riders and others, and, by reason of their excessive weight, to the great deterioration of the houses to which they are fixed,... it was ordained,... that no one in future should have a stake bearing either his sign or leaves [i. e. a bush] extending or lying over the king's highway, of greater length than 7 feet at most,' c. And, at p. 292 of the same work, note 2, Mr. Riley rightly defines an ale-stake to be 'the pole projecting from the house, and supporting a bunch of leaves.' The word ale-stake occurs in Chatterton's poem of Ælla, stanza 30, where it is used in a manner which shews that the supposed 'Rowley' did not know what it was like. See my note on this; Essay on the Rowley Poems, p. xix; and cf. note to A. 667. 322. of a cake; we should now say, a bit of bread; the modern sense of 'cake' is a little misleading. The old cakes were mostly made of dough, whence the proverb 'my cake is dough,' i. e. is not properly baked; Taming of the Shrew, v. 1. 145. Shakespeare also speaks of 'cakes and ale,' Tw. Nt. ii. 3. 124. The picture of the 'Simnel Cakes' in Chambers' Book of Days, i. 336, illustrates Chaucer's use of the word in the Prologue, l. 668. 324. The Pardoner was so ready to tell some 'mirth or japes' that the more decent folks in the company try to repress him. It is a curious comment on the popular estimate of his character. He has, moreover, to refresh himself, and to think awhile before he can recollect 'some honest (i. e. decent) thing.' 327, 328. The Harleian MS. has— 'But in the cuppe wil I me bethinke Upon some honest tale, whil I drinke.' The Pardoneres Prologue. Title. The Latin text is copied from l. 334 below; it appears in the Ellesmere and Hengwrt MSS. The A. V. has—'the love of money is
- 83. the root of all evil'; 1 Tim. vi. 10. It is well worth notice that the novel by Morlinus, quoted in vol. iii. p. 442, as a source of the Pardoner's Tale, contains the expression—'radice malorum cupiditate affecti.' 336. bulles, bulls from the pope, whom he here calls his 'liege lord'; see Prol. A. 687, and Piers the Plowman, B. Prol. 69. See also Wyclif's Works, ed. Arnold, iii. 308. alle and somme, one and all. Cf. Clerkes Tale, E. 941, and the note. 337. patente; defined by Webster as 'an official document, conferring a right or privilege on some person or party'; c. It was so called because 'patent' or open to public inspection. 'When indulgences came to be sold, the pope made them part of his ordinary revenue; and, according to the usual way in those, and even in much later times, of farming the revenue, he let them out usually to the Dominican friars'; Massingberd, Hist. Eng. Reformation, p. 126. 345. 'To colour my devotion with.' For saffron, MS. Harl. reads savore. Tyrwhitt rightly prefers the reading saffron, as 'more expressive, and less likely to have been a gloss.' And he adds —'Saffron was used to give colour as well as flavour.' For example, in the Babees Book, ed. Furnivall, p. 275, we read of 'capons that ben coloured with saffron.' And in Winter's Tale, iv. 3. 48, the Clown says —'I must have saffron to colour the warden-pies.' Cf. Sir Thopas, B. 1920. As to the position of with, cf. Sq. Ta., F. 471, 641. 346. According to Tyrwhitt, this line is, in some MSS. (including Camb. Dd. 4. 24. and Addit. 5140), replaced by three, viz.— 'In euery village and in euery toun, This is my terme, and shal, and euer was, Radix malorum est cupiditas.' Here terme is an error for teme, a variant of theme; so that the last two lines merely repeat ll. 333-4.
- 84. 347. cristal stones, evidently hollow pieces of crystal in which relics were kept; so in the Prologue, A. 700, we have— 'And in a glas he hadde pigges bones.' 348. cloutes, rags, bits of cloth. 'The origin of the veneration for relics may be traced to Acts, xix. 12. Hence clouts, or cloths, are among the Pardoner's stock'; note in Bell's edition. 349. Reliks. In the Prologue, we read that he had the Virgin Mary's veil and a piece of the sail of St. Peter's ship. Below, we have mention of the shoulder-bone of a holy Jew's sheep, and of a miraculous mitten. See Heywood's impudent plagiarism from this passage in his description of a Pardoner, as printed in the note to l. 701 of Dr. Morris's edition of Chaucer's Prologue. See also a curious list of relics in Chambers' Book of Days, i. 587; and compare the humorous descriptions of the pardoner and his wares in Sir David Lyndesay's Satyre of the Three Estates, ll. 2037-2121. Chaucer probably here took several hints from Boccaccio's Decamerone, Day 6, Nov. 10, wherein Frate Cipolla produces many very remarkable relics to the public gaze. See also the list of relics in Political, Religious, and Love Poems, ed. Furnivall (E. E. T. S.), pp. xxxii, 126- 9. 350. latoun. The word latten is still in use in Devon and the North of England for plate tin, but as Halliwell remarks, that is not the sense of latoun in our older writers. It was a kind of mixed metal, somewhat resembling brass both in its nature and colour, but still more like pinchbeck. It was used for helmets (Rime of Sir Thopas, B. 2067), lavers (P. Pl. Crede, 196), spoons (Nares), sepulchral memorials (Way in Prompt. Parv.), and other articles. Todd, in his Illustrations of Chaucer, p. 350, remarks that the escutcheons on the tomb of the Black Prince are of laton over-gilt, in accordance with the Prince's instructions; see Nichols's Royal Wills, p. 67. He adds —'In our old Church Inventories a cross of laton frequently occurs.' See Prol. A. 699, and the note. I here copy the description of this
- 85. metal given in Batman upon Bartholomè; lib. xvi. c. 5. 'Of Laton. Laton is called Auricalcum, and hath that name, for, though it be brasse or copper, yet it shineth as gold without, as Isidore saith; for brasse is calco in Greeke. Also laton is hard as brasse or copper; for by medling of copper, of tinne, and of auripigment [orpiment] and with other mettal, it is brought in the fire to the colour of gold, as Isidore saith. Also it hath colour and likenesse of gold, but not the value.' 351. The expression 'holy Jew' is remarkable, as the usual feeling in the middle ages was to regard all Jews with abhorrence. It is suggested, in a note to Bell's edition, that it 'must be understood of some Jew before the Incarnation.' Perhaps the Pardoner wished it to be understood that the sheep was once the property of Jacob; this would help to give force to l. 365. Cp. Gen. xxx. The best comment on the virtues of a sheep's shoulder-bone is afforded by a passage in the Persones Tale (De Ira), I. 602, where we find—'Sweringe sodeynly withoute avysement is eek a sinne. But lat us go now to thilke horrible swering of adiuracioun and coniuracioun, as doon thise false enchauntours or nigromanciens in bacins ful of water, or in a bright swerd, in a cercle, or in a fyr, or in a shulder-boon of a sheep'; c. Cf. also a curious passage in Trevisa's tr. of Higden's Polychronicon, lib. i. cap. 60, which shews that it was known among the Flemings who had settled in the west of Wales. He tells us that, by help of a bone of a wether's right shoulder, from which the flesh had been boiled (not roasted) away, they could tell what was being done in far countries, 'tokens of pees and of werre, the staat of the reeme, sleynge of men, and spousebreche.' Selden, in his notes to song 5 of Drayton's Polyolbion, gives a curious instance of such divination, taken from Giraldus, Itin. i. cap. 11; and a writer in the Retrospective Review, Feb. 1854, p. 109, says it is 'similar to one described by Wm. de Rubruquis as practised among the Tartars.' And see spade-bone in Nares. Cf. Notes and Queries, 1 S. ii. 20.
- 86. In Part I. of the Records of the Folk-lore Society is an article by Mr. Thoms on the subject of divination by means of the shoulder-bone of a sheep. He shews that it was still practised in the Scottish Highlands down to the beginning of the present century, and that it is known in Greece. He further cites some passages concerning it from some scarce books; and ends by saying—'let me refer any reader desirous of knowing more of this wide-spread form of divination to Sir H. Ellis's edition of Brand's Popular Antiquities, iii. 179, ed. 1842, and to much curious information respecting Spatulamancia, as it is called by Hartlieb, and an analogous species of divination ex anserino sterno, to Grimm's Deutsche Mythologie, 2nd ed. p. 1067.' 355. The sense is—'which any snake has bitten or stung.' The reference is to the poisonous effects of the bite of an adder or venomous snake. The word worm is used by Shakespeare to describe the asp whose bite was fatal to Cleopatra; and it is sometimes used to describe a dragon of the largest size. In Icelandic, the term 'miðgarðsormr,' lit. worm of the middle-earth, signifies a great sea-serpent encompassing the entire world. 363. Fastinge. This word is spelt with a final e in all seven MSS.; and as it is emphatic and followed by a slight pause, perhaps the final e should be pronounced. Cp. A. S. fæstende, the older form of the present participle. Otherwise, the first foot consists of but one syllable. 366. For heleth, MS. Hl. has kelith, i. e. cooleth. 379. The final e in sinne must not be elided; it is preserved by the caesura. Besides, e is only elided before h in the case of certain words. 387. assoile, absolve. In Michelet's Life of Luther, tr. by W. Hazlitt, chap. ii, there is a very similar passage concerning Tetzel, the Dominican friar, whose shameless sale of indulgences roused Luther to his famous denunciations of the practice. Tetzel 'went about from town to town, with great display, pomp, and expense, hawking the
- 87. commodity [i. e. the indulgences] in the churches, in the public streets, in taverns and ale-houses. He paid over to his employers as little as possible, pocketing the balance, as was subsequently proved against him. The faith of the buyers diminishing, it became necessary to exaggerate to the fullest extent the merit of the specific.... The intrepid Tetzel stretched his rhetoric to the very uttermost bounds of amplification. Daringly piling one lie upon another, he set forth, in reckless display, the long list of evils which this panacea could cure. He did not content himself with enumerating known sins; he set his foul imagination to work, and invented crimes, infamous atrocities, strange, unheard of, unthought of; and when he saw his auditors stand aghast at each horrible suggestion, he would calmly repeat the burden of his song:—Well, all this is expiated the moment your money chinks in the pope's chest.' This was in the year 1517. 390. An hundred mark. A mark was worth about 13s. 4d., and 100 marks about £66 13s. 4d. In order to make allowance for the difference in the value of money in that age, we must at least multiply by ten; or we may say in round numbers, that the Pardoner made at least £700 a year. We may contrast this with Chaucer's own pension of 20 marks, granted him in 1367, and afterwards increased till, in the very last year of his life, he received in all, according to Sir Harris Nicolas, as much as £61 13s. 4d. Even then his income did not quite attain to the 100 marks which the Pardoner gained so easily. 397. dowve, a pigeon; lit. a dove. See a similar line in the Milleres Tale, A. 3258. 402. namely, especially, in particular; cf. Kn. Ta. 410 (A. 1068). 406. blakeberied. The line means—'Though their souls go a- blackberrying'; i. e. wander wherever they like. This is a well-known crux, which all the editors have given up as unintelligible. I have been so fortunate as to obtain the complete solution of it, which was printed in Notes and Queries, 4 S. x. 222, xii. 45, and again in my
- 88. preface to the C-text of Piers the Plowman, p. lxxxvii. The simple explanation is that, by a grammatical construction which was probably due (as will be shewn) to an error, the verb go could be combined with what was apparently a past participle, in such a manner as to give the participle the force of a verbal substantive. In other words, instead of saying 'he goes a-hunting,' our forefathers sometimes said 'he goes a-hunted.' The examples of this use are at least seven. The clearest is in Piers Plowman, C. ix. 138, where we read of 'folk that gon a-begged,' i. e. folk that go a-begging. In Chaucer, we not only have 'goon a-begged,' Frank. Tale, F. 1580, and the instance in the present passage, but yet a third example in the Wyf of Bath's Tale, Group D. 354, where we have 'goon a- caterwawed,' with the sense of 'to go a-caterwauling'; and it is a fortunate circumstance that in two of these cases the idiomatic forms occur at the end of a line, so that the rime has preserved them from being tampered with. Gower (Conf. Amant. bk. i. ed. Chalmers, pp. 32, 33, or ed. Pauli, i. 110) speaks of a king of Hungary riding out 'in the month of May,' adding— 'This king with noble purueiance Hath for him-selfe his chare [car] arayed, Wherein he wolde ryde amayed,' c. that is, wherein he wished to ride a-Maying. Again (in bk. v, ed. Chalmers, p. 124, col. 2, or ed. Pauli, ii. 132) we read of a drunken priest losing his way:— 'This prest was dronke, and goth a-strayed'; i. e. he goes a-straying, or goes astray. The explanation of this construction I take to be this; the -ed was not really a sign of the past participle, but a corruption of the ending -eth (A. S.-að) which is sometimes found at the end of a verbal substantive. Hence it is that, in the passage from Piers Plowman above quoted, one of the best and earliest MSS. actually reads 'folk
- 89. Welcome to our website – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com