Rational points on curves -- BIMR 2025 --

Rational points on curves
BIMR 2025
Marc Masdeu
Universitat Autònoma de Barcelona - Centre de Recerca Matemàtica
July 1st, 2025

Points on a conic
Problem
Given a homogeneous quadratic equation in 3 variables
C : aX2
+ bY 2
+ cZ2
+ dXY + eXZ + fY Z = 0, a, b, c, d, e, f ∈ Z,
find all solutions (X, Y, Z) with X, Y, Z ∈ Z ( ⇐⇒ (X : Y : Z) ∈ P2(Q)).
Remark
By dividing out by Z2, equivalent to finding all solutions (x, y) to
ax2
+ by2
+ c + dxy + ex + fy = 0, x, y ∈ Q.
May be trivial: for xy = 1, all solutions are (t, 1/t), with t ∈ Q ∖ {0}.
Sometimes there are no solutions:
▶ x2
+ y2
= −1 has no solutions in R, let alone in Q.
▶ x2
+ y2
= 3 has no solutions in Q either. (Why?)
Marc Masdeu Rational points on curves July 1st
, 2025 1 / 35

Points on a conic: x2
+ y2
= 1
Goal
Find all rational solutions to the equation x2 + y2 = 1.
slope = t
y = t(x + 1)
x
y
(−1, 0)
P =

1−t2
1+t2 , 2t
1+t2

x2
+ y2
= 1
x2
+ t2
(x + 1)2
= 1
x2
+
2t2
1 + t2
x +
t2 − 1
1 + t2
= 0
(x − x0)(x − x1) = 0 =⇒ x0x1 =
t2 − 1
1 + t2
x0 = −1 =⇒ x1 =
1 − t2
1 + t2
, 2025 2 / 35

Parametrizing cubics
Technique in previous slide works for general conics.
Upshot: If a conic has one rational point, then it has infinitely many
and they can be easily parametrized.
Consider a cubic equation:
aX3
+bX2
Y +cXY 2
+dY 3
+eX2
Z+fXY Z+gY 2
Z+mXZ2
+nY Z2
+rZ3
= 0
Sometimes it has no solutions:
X3
+ 14Y 3
= 12Z3
(work modulo 7)
▶ Start with a solution (X, Y, Z) such that gcd(X, Y, Z) = 1.
▶ LHS can take values {0, ±1} modulo 7.
▶ RHS can take values {0, ±2} modulo 7.
▶ So any solution will satisfy 7 | X and 7 | Z.
▶ But then 7 | Y , which is a contradiction.
, 2025 3 / 35

Elliptic curves
Definition
A cubic E : y2 = x3 + ax + b is an elliptic curve if x3 + ax + b has no
repeated roots.
We write E(Q) for the set of the rational points of E.
▶ If K ⊇ Q is another field, write E(K) for the set of solutions where we
allow the coordinates to be in K.
E(Q) always has a point, namely O = (0 : 1 : 0) (“point at infinity”).
▶ The curve y2
= x3
− 108 has O as its only point: E(Q) = {O}.
▶ The curve y2
= x3
− 27 has two rational points: E(Q) = {O, (3, 0)}.
▶ The curve y2
= x3
+ 4 has three: E(Q) = {O, (0, 2), (0, −2)}.
▶ The curve y2
= x3
− x + 1 has infinitely many points:
E(Q) = {O, (1, −1), (−1, −1), (0, 1), (3, 5), (5, −11), (1
4 , −7
8 ), (−11
9 , 17
27 ), . . .}.
Is there any pattern??
, 2025 4 / 35

The cord construction
Given two points P, Q on E(Q), we can produce a third one.
P
Q
R
, 2025 5 / 35

The tangent construction
Given only one point P, we can also produce another one.
, 2025 6 / 35

Mordell’s Theorem
Louis Mordell
Theorem (Mordell 1922)
There are finitely many points
P1, . . . , Pt ∈ E(Q) such that every other
point in E(Q) can be obtained from those
by repeatedly applying the cord and
tangent constructions.
, 2025 7 / 35

The set E(Q) has a group structure
The cord construction gives an addition for points P, Q on E(Q):
P
Q
P + Q
R
Obviously commutative. . .
▶ . . . but proving associativity is much harder!
The point O is the neutral element.
To add a point to itself (P = Q), use the tangent construction.
, 2025 8 / 35

Mordell–Weil Theorem
Louis Mordell André Weil
Theorem (Mordell 1922, Weil 1928)
Let K be an algebraic number field (i.e. [K : Q] ∞).
Then E(K) is a finitely generated abelian group. So it is of the form
E(K) = (torsion) ⊕ Zr
, 2025 9 / 35

The torsion subgroup
Barry Mazur
Theorem (Mazur 1977)
Let E be an elliptic curve over Q. Then
E(Q)tors is isomorphic to:
Z/NZ with 1 ≤ N ≤ 10 or N = 12, or
Z/2Z ⊕ Z/2NZ with 1 ≤ N ≤ 4.
Furthermore, all of these occur.
Generalized to quadratic fields (Kamienny–Kenku–Momose 1990’s).
For number fields of larger degree, we have bounds on the largest
prime that can divide the order.
Merel (1996) proved that the size of the torsion subgroup is uniformly
bounded by a constant depending only on the degree of the field.
About the rank: know E/Q of rank up to 29 (Elkies–Klagsbrun
2024), but don’t know whether there is one of rank 30.
, 2025 10 / 35

Counting points in finite fields
Consider the curve E : y2 = x3 − x + 1.
Can think of it as an equation modulo 3. In that case it has 7 points:
{O, (0, 1), (0, 2), (1, 1), (1, 2), (2, 1), (2, 2)}
Or we can think modulo 5, where it has 8 points:
{O, (0, 1), (0, 4), (1, 1), (1, 4), (3, 0), (4, 1), (4, 4)}
Note that we should expect #E(Fp) to be roughly about p + 1.
▶ Let ap(E) = p + 1 − #E(Fp).
p 2 3 5 7 11 13 17 19 23 29 31 37
#E(Fp) 3 7 8 12 10 19 14 22 23 37 35 36
ap(E) 0 -3 -2 -4 2 -5 4 -2 1 -7 -3 2
, 2025 11 / 35

Bound for ap = p + 1 − #E(Fp)
2e4 4e4 6e4 8e4 1e5
p
-600
-400
-200
200
400
600
p+1 #E( p)
y=2 x
Theorem (1933)
p + 1 − #E(Fp) ≤ 2
√
p.
, 2025 12 / 35

The Sato–Tate conjecture
How the points distribute inside the parabola is the statement of the
Sato–Tate conjecture (1964).
Mikio Sato John Tate -1 -0.5 0 0.5 1
ap/2 p
500
1000
1500
2000
2500
3000
n
Theorem (Clozel–Harris-Shepherd-Barron–Taylor 2006 +
Barnet-Lamb–Gee-Geraghty 2009)
The Sato–Tate conjecture holds for all E over totally real fields (e.g. Q).
, 2025 13 / 35

The Birch and Swinnerton-Dyer conjecture
Bryan Birch Peter Swinnerton-Dyer
In the late 1950’s, Birch and Swinnerton-Dyer studied the asymptotic
behavior of the quantity
CE(x) =
Y
p≤x
#E(Fp)
p
as x → ∞
, 2025 14 / 35

Experimental rank data
Plots of CE(x) =
Q
p≤x
#E(Fp)
p against log(x) (doubly-logarithmic axes).
-0.4 -0.2 0 0.2 0.4 0.6
loglogx
0.5
1
1.5
2
2.5
3
3.5
logCE(x)
slope0.104
slope1.147
slope2.254
slope3.015
(5077.a1) y2 + y = x3 − 7x + 6
(389.a1) y2 + y = x3 + x2 − 2x
(37.a1) y2 + y = x3 − x
(11.a1) y2 + y = x3 − x2 − 7820x − 263580
Conjecture BSD (Birch–Swinnerton-Dyer)
CE(x) ∝ log(x)r
as x → ∞.
, 2025 15 / 35

The L-function of E
One can rephrase the BSD conjecture using L-functions.
Recall the “error” term
ap(E) = p + 1 − #E(Fp)
Define a function of a complex variable s:
L(E, s)“ = ”
Y
p prime
1 − app−s
+ p1−2s
−1
, ℜ(s) 3/2.
▶ Note that L(E, 1)“ = ”
Q p
#E(Fp) “ = ”CE(∞)−1
. . .
Conjecture BSD, second version
1 The L-function L(E, s) can be analytically continued to all C.
2 L(E, s) has a functional equation relating L(E, s) to L(E, 2 − s)
3
ords=1 L(E, s) = r.
, 2025 16 / 35

Plots of L(E, s) restricted to ℜ(s) = 1 (www.lmfdb.org)
y2 + y = x3 − x2 − 7820x − 263580
y2 + y = x3 + x2 − 2x
y2 + y = x3 − x
y2 + y = x3 − 7x + 6
, 2025 17 / 35

Remarks on BSD conjecture
The BSD conjecture is one of the CMI Problems of the Millenium.
Correct definition of L(E, s) involves also the behaviour of E at “bad
primes” (where the reduction of E modulo p is “singular”).
▶ This is encoded in the conductor N = cond(E).
Birch and Swinnerton-Dyer predicted also a formula for the leading
term of the Taylor expansion of L(E, s) at s = 1.
The first two statements of the refined conjecture are a
consequence of extremely deep theorems, known as “modularity”.
Conjecture extends to elliptic curves over other number fields.
▶ In this generality, one doesn’t even know whether L(E, s) can be
extended to all C.
, 2025 18 / 35

The theorems of Gross–Zagier and Kolyvagin
Robert Gross
Don Zagier Victor Kolyvagin
Theorem (Gross–Zagier 1986 + Kolyvagin 1989)
Suppose ords=1 L(E, s) ∈ {0, 1}. Then:
ords=1 L(E, s) = r.
, 2025 19 / 35

The main tool for BSD: Heegner points (1952)
Kurt Heegner
Heegner points are defined over (extensions of) quadratic fields K.
Only available when K = Q(
√
D) is imaginary: D 0.
We will further require the additional condition:
▶ Heegner hypothesis: p | N =⇒ p split in K.
This ensures that ords=1 L(E/K, s) is odd (so ≥ 1).
, 2025 20 / 35

Modular forms
For an integer N ≥ 1, set Γ0(N) = { a b
c d

∈ SL2(Z) : N | c}.
Γ0(N) acts on the upper-half plane H = {z ∈ C : Im(z) 0}:
▶ Via a b
c d

· z = az+b
cz+d .
A cusp form of level N is a holomorphic map f : H → C such that:
1 f(γz) = (cz + d)2
f(z) for all γ = a b
c d

∈ Γ0(N).
2 Cuspidal: limz→i∞ f(z) = 0.
( 1 1
0 1 ) ∈ Γ0(N) ; have Fourier expansions f(z) =
P∞
n=1 an(f)e2πinz.
Given an elliptic curve E of conductor N, define an for all n ≥ 1 as
follows:
ap = ap(E) for all primes p.
anm = anam if n and m are coprime.
apr = apapr−1 − papr−2 for r ≥ 2 (ommit second term if p | N).
, 2025 21 / 35

Modularity
Theorem (Wiles, Taylor–Wiles, Breuil–Conrad–Diamond–Taylor
2001)
The function
fE(z) =
X
n≥1
ane2πinz
is the Fourier expansion of a cusp form of level N = cond(E).
, 2025 22 / 35

Heegner Points (K/Q imaginary quadratic)
Modularity =⇒ ∃ cusp form fE attached to E.
ωE = 2πifE(z)dz = 2πi
X
n≥1
ane2πinz
dz.
This is a differential form on H, invariant under Γ0(N).
Given τ ∈ K ∩ H, set Jτ =
Z τ
i∞
ωE ∈ C.
Well-defined up to ΛE =
nR
γ ωE | γ closed path in Γ0(N)H
o
.
Martin Eichler Goro Shimura
Theorem (Eichler–Shimura 1959)
There exists a complex-analytic group
isomorphism
η∞ : C/ΛE → E(C).
, 2025 23 / 35

Algebraicity of Heegner points
Theorem (Shimura, Gross–Zagier, Kolyvagin)
1 Pτ = η∞(Jτ ) ∈ E(C) has algebraic coordinates.
2 PK = Tr(Pτ ) is nontorsion ⇐⇒ ords=1 L(E/K, s) = 1.
3 If ords=1 L(E/Q, s) ≤ 1 then BSD holds for E(Q).
, 2025 24 / 35

An example: E : y2
+ y = x3
− x2
− 10x − 20 (“11a1”)
fE(z) = q − 2q2 − q3 + 2q4 + q5 + 2q6 − 2q7 − 2q9 − 2q10 + · · ·
fE is a modular form of level N = 11.
Consider the point τ = −3+
√
−2
11 .
Jτ =
P
n≥1
an
n e2πinτ ∼ 0.126920930427956 − 0.536079610338652 · i.
Pτ = η∞(Jτ ) ∼ (−3.00000 + 1.41421 · i, 3.00000 + 4.242640 · i).
Pτ is very close to the algebraic point of infinite order
(−3 +
√
−2, 3 + 3
√
−2) ∈ E(K).
, 2025 25 / 35

Darmon points (K/Q real quadratic)
Henri Darmon
If K = Q(
√
D) is real and p ∥ N is inert in K,
then ords=1 L(E/K, s) is odd as well. . .
. . . but Heegner points are not available.
▶ Note that in this case, K ∩ H = ∅!
In 2001, Darmon gave another analytic
construction:
▶ p-adic analytic (instead of complex analytic).
▶ The algebraicity of these points is still open
nowadays.
, 2025 26 / 35

Recall: how to construct H
On Q, consider the usual absolute value
|x| =
(
x x ≥ 0,
−x x 0.
Complete Q with respect to the metric given by | · | (we call this R).
▶ Have decimal expansions, e.g 2.147581534 . . .
▶ They are really power series
P
n≥n0
an10−n
.
Adjoin a root i of an irreducible quadratic to get C.
▶ We also have decimal expansions, e.g.
2.147581534 . . . + i · 3.6171346234 . . .
Finally, remove from C the real line R. We obtain two connected
components, and H is just one of them.
, 2025 27 / 35

How to construct its p-adic analogue
On Q, consider the p-adic absolute value |x|p:
pt m
n p
= p−t
if p ∤ mn, |0|p = 0.
▶ |3|3 = 1/3, |18|3 = 1/9, |2/3|3 = 3, . . .
Complete Q with respect to the metric given by | · |p (we call this Qp).
▶ Have p-adic expansions:
X
n≥n0
anpn
, an ∈ {0, . . . , p − 1}.
▶ For example, −1 = (p − 1) + (p − 1)p + (p − 1)p2
+ (p − 1)p3
+ . . .
Take an irreducible quadratic and adjoin a root α to get Qp2 .
▶ These too have p-adic expansions:
X
n≥n0
(an + αbn)pn
, an, bn ∈ {0, . . . , p − 1}.
The p-adic analogue to H is Hp = Qp2 ∖ Qp.
▶ More like an analogue of H together with the lower half-plane. . .
, 2025 28 / 35

Heegner points vs Darmon points
Recall K = Q(
√
D) real quadratic, p is inert in K.
▶ Note that X2
− D doesn’t have roots in Qp.
▶ Take Qp2 = Qp(
√
D), and note that K ,→ Qp2 .
Recall that the conductor of E is of the form pM with p ∤ M.
Theorem (Tate 1959)
There is a p-adic number qE ∈ Q×
p , and a p-adic
analytic isomorphism
ηp : Q×
p2 /qZ
E → E(Qp2 ).
Let
Γ =

γ = a b
c d

∈ SL2(Z[1/p]) | M | c .
E gives rise to a “rigid analytic” differential (1, 1)-form ωE on H × Hp.
▶ Invariant under Γ, so it “descends” to XΓ = Γ H × Hp.
, 2025 29 / 35

A null-homologous cycle in XΓ = Γ H × Hp
Consider τ ∈ Hp be a real quadratic point.
Let γ = γτ ∈ Γ be a generator of the stabilizer of τ in Γ.
Fact: Γab is finite.
▶ So if e = #Γab, then γe
is a product of commutators.
▶ Assume (for simplicity) that γe
= aba−1
b−1
, for some a, b ∈ Γ.
Consider the 1-cycle in XΓ = ΓH × Hp:
Θ = (γe
∞ → ∞) × τ,
where ∞ ∈ H is any choice of base point.
▶ Note that (γe
∞, τ) = γe
· (∞, τ), so it is closed.
Turns out that Θ is null-homologous: it is the boundary of a
2-chain.
, 2025 30 / 35

A 2-chain with boundary Θ
×
×
×
∞
a∞
∞
b∞
a−1
τ
τ
τ
∞
a∞ ab∞
aba−1
∞
aba−1
b−1
∞
∞
a∞
×
τ
×
τ
×
τ
×
τ
×
τ
×
τ
×
b−1
τ
×
τ
×
a−1
τ
×
a∞
×
∞
×
∞
×
b∞
a∞
ab∞
ab∞
aba−1
∞
aba−1
∞
aba−1
b−1
∞
aba−1
b−1
∞
∞
∞
a∞
∞
a∞
∞
b∞
∞
b∞
b−1
τ
τ
a−1
τ
τ
a−1
τ
τ
b−1
τ
τ
b−1
τ
τ
+
+ + + +
−
+
−
−
+
−
+
+
−
+
×
∞
b−1
τ
τ
×
a∞
τ
b−1
τ
+ ×
∞
τ
a−1
τ
+ ×
b∞
a−1
τ
τ
+ = ∞×
b−1
τ
τ
a−1
τ
a−1
b−1
τ
τ
a−1
τ
b−1
a−1
τ
b−1
τ
+ + + = ∞×
b−1
a−1
τ
a−1
b−1
τ
∂
, 2025 31 / 35

Conjecture
Recall
Θ = ∂ × {τ} − (∞ → a∞) × (b−1τ → τ) + (∞ → b∞) × (a−1τ → τ)

.
ωE = (1, 1) form on H × Hp attached to E.
Technical detail: Can define a “multiplicative integral” for ωE.
▶ Essentially, replace Riemann sums with products.
▶ Its p-adic logarithm recovers the usual integral.
Jτ = ×
Z a∞
∞
×
Z b−1τ
τ
ωE − ×
Z b∞
∞
×
Z a−1τ
τ
ωE ∈ Q×
p2 .
The 2-chain is not unique: it can be changed by any 2-cycle.
Jτ is well defined modulo elements in
Λ =

×
Z
×
Z
ξ
ωE : ξ ∈ H2 (ΓH × Hp, Z)

⊂ Q×
p2
, 2025 32 / 35

The conjecture
M. Bertolini H. Darmon
Theorem (Bertolini–Darmon 1997)
Λ =

×
Z
×
Z
ξ
ωE : ξ ∈ H2 (ΓH × Hp, Z)

= qZ
E.
Conjecture (Darmon 1999)
Consider the point
Pτ = ηp(Jτ ) = ηp ×
Z a∞
∞
×
Z b−1τ
τ
ωE − ×
Z b∞
∞
×
Z a−1τ
τ
ωE
!
∈ E(Qp2 ).
1 Pτ has algebraic coordinates.
2 PK = Tr(Pτ ) is nontorsion ⇐⇒ L′(E/K, 1) ̸= 0.
, 2025 33 / 35

Example: E : y2
+ xy + y = x3
+ x2
− 10x − 10 (“15a1”)
fE(z) = q − q2 − q3 − q4 + q5 + q6 + 3q8 + q9 − q10 + · · ·
fE is a modular form of level N = 15.
Take τ = −3+
√
13
2 , seen in Q52 .
We get γ = 10 −3
−3 1

.
We can calculate Jτ very efficiently, obtaining (set β = 1+
√
13
2 ∈ Q52 )
(3β+4)+(4β+1)5
2
+(2β+2)5
3
+(2β+4)5
4
+(3β+2)5
5
+(β+2)5
6
+(2β+2)5
7
+(β+4)5
8
+(2β+4)5
9
+· · · .
Pτ = η5(Jτ ) ∈ E(Q52 ) is

(3β + 2) + 4β · 5 + 4β · 5
2
+ 4β · 5
3
+ 4β · 5
4
+ · · · , (4β + 4) + 3 · 5 + 4 · 5
2
+ 4 · 5
3
+ 4 · 5
4
+ · · ·

.
This is 5-adically close to the algebraic point of infinite order

1 −
√
13, −4 + 2
√
13

∈ E(K).
, 2025 34 / 35

The Darmon Points Machine™
Darmon Points
E/F K/F quadratic
P
?
∈ E(Kab)
1999
, 2025 35 / 35

Non-archimedean
Archimedean
Ramification
Darmon Points
H∗
H∗
Modularity
2012
, 2025 35 / 35

Non-archimedean
Archimedean
Ramification
Darmon Points
H∗
H∗
2013
, 2025 35 / 35

Non-archimedean
Archimedean
Ramification
Periods Machine
H∗
H∗
2014
, 2025 35 / 35

Darmon – Vonk
H∗
H∗
Hilbert 12th
2017
, 2025 35 / 35

Darmon – Vonk
H∗
H∗
2020
Non-archimedean
Ramification
?
, 2025 35 / 35

Questions?
http://www.mat.uab.cat/˜masdeu/
, 2025 35 / 35

Rational points on curves -- BIMR 2025 --

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