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Hecke s theory of modular forms and Dirichlet series 2
Revised Edition Bruce C. Berndt Digital Instant
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Author(s): Bruce C. Berndt, Marvin Isadore Knopp
ISBN(s): 9789812792372, 9812792376
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Year: 2008
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Hecke’s Theory of Modular Forms
and Dirichlet Series
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NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
World Scientific
Bruce C Berndt
University of Illinois at Urbana-Champaign,USA
Marvin I Knopp
Temple University,USA
Hecke’s Theory of
Modular Forms and
Dirichlet Series
6438 tp.indd 2 11/19/07 3:19:37 PM

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HECKE’S THEORY OF MODULAR FORMS AND DIRICHLET SERIES
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1

November 17, 2007 11:23 WSPC/Book Trim Size for 9in x 6in bcb
In Memory of Hans Rademacher, the Father of our
Mathematical Family
v

vi Hecke’s Theory of Modular Forms and Dirichlet Series
Erich Hecke

Preface in Two Acts with a Prelude,
Interlude, and Postlude
Prelude
Thirty-seven years have elapsed between the first version and the present
version of this monograph. We begin with the first author’s slightly edited
preface from his first version. We then provide a lengthier second preface
composed by the second author.
The Original Preface
These notes are part of a course on modular forms and applications
to analytic number theory given by the first author at the University of
Illinois at Urbana-Champaign in the spring of 1970. The existing accounts
[47], [48], [87] of Hecke’s theory of modular forms and Dirichlet series are
somewhat concise. Therefore, it has been our intention to present a more
detailed account of a major portion of this material for those who are
unfamiliar with this beautiful theory. Readers already familiar with Hecke’s
theory will find little that is new here.
The first author is especially grateful to Ronald J. Evans for providing
a new proof of a fundamental region for Hecke’s modular groups, which we
present here. We express our thanks also to Elmer Hayashi for a detailed
reading of the manuscript and to Harold Diamond for several suggestions.
Bruce Berndt, May, 1970 & May, 2007
Interlude
The first author mailed a copy of his notes on Hecke’s theory of modular
forms and Dirichlet series to Dr. Jürgen Elstrodt, who at that time was at
Universität München. He responded with about a dozen pages of detailed
vii

viii Hecke’s Theory of Modular Forms and Dirichlet Series
comments, which, after an undeservedly quick reading, were deposited in
the first author’s file cabinet for approximately thirty-five years, until they
were dusted off and sent to the second author for incorporation in the new
version. We hope that it is not too late to thank Elstrodt for his kind
suggestions and patience.
The Second Preface
In the spring of 1971, I received the following letter, dated June 17.
Since it is brief, I quote it in full.
Under separate cover, I am sending you a copy of some lec-
ture notes, “Hecke’s theory of modular forms and Dirichlet
series.” I would appreciate any comments, corrections, crit-
icisms, or suggestions that you may have. Thank you very
much.
Most sincerely, (signed) Bruce
To establish the context of this letter, I recall that in the spring of 1938
Erich Hecke gave an important series of lectures at the Institute for Ad-
vanced Study, Princeton, on his correspondence theory published in 1936.
The notes from these lectures, taken by Hyman Serbin and produced in
planographed form by Edwards Brothers of Ann Arbor, received only lim-
ited circulation. To my knowledge there are only a few copies extant in
mathematics libraries (for example, the University of Illinois at Urbana-
Champaign) and private collections of professional mathematicians.
In 1970 Berndt produced a set of lecture notes based upon Hecke’s notes,
but with the addition of many details omitted from Hecke’s original notes.
The more extensive notes, too, had only limited circulation.
For the past thirty-five years I have employed both sets of notes to in-
troduce graduate students to the Hecke theory and the broader theory of
modular/automorphic forms. During this time my Ph.D. students and oth-
ers frequently asked why Berndt’s notes had never been published. Because
we are convinced that the reactions of these students reflect a genuine use-
fulness of these notes to the mathematical community, we have undertaken
the task of publishing this book based upon them, corrected and modified
where necessary, and expanded to include some of the many new develop-
ments in the theory during the past decades, as well as relevant earlier work
not previously included. We stress that the Hecke correspondence theory
has remained an active feature of research in number theory since the 1930s

Preface ix
and, in fact, its importance is perhaps better understood today than it was
in 1936.
The first six chapters of this book follow the organization of Berndt’s
original notes, hence that of the first part of Hecke’s notes as well. Beyond
this, we have added two completely new chapters based upon work done
since 1970 and upon earlier work not originally understood to lie within the
circle of ideas surrounding Hecke’s correspondence theorem.
Chapter 7 features Bochner’s important generalization of Hecke’s corre-
spondence theorem and some closely related results. Chapter 8 is devoted
to the great variety of identities related to the Hecke correspondence the-
ory (but not explicitly present in that theory) that have been developed
over the years. Among others, these identities are due to S. Ramanujan,
N. S. Koshliakov, G. N. Watson, A. P. Guinand, K. Chandrasekharan,
R. Narasimhan, and Berndt. Some antedate Hecke’s work, while others are
more recent.
Marvin Knopp, April, 2007
Postlude
We are grateful for the comments made by our students over the past
several decades. More recently, Shigeru Kanemitsu and Yoshio Tanigawa
offered several additional remarks and references. We thank Hilda Britt for
expertly typing most of our manuscript and Tim Huber for his graphical
expertise.

Contents
Preface in Two Acts with a Prelude, Interlude, and Postlude vii
1. Introduction 1
2. The main correspondence theorem 3
3. A fundamental region 15
4. The case λ > 2 23
5. The case λ < 2 35
6. The case λ = 2 69
7. Bochner’s generalization of the main correspondence
theorem of Hecke, and related results 87
8. Identities equivalent to the functional equation and to the
modular relation 115
Bibliography 129
Index 135
xi

Chapter 1
Introduction
The classical theta function, defined for Im τ > 0 by
θ(τ) =
∞
X
n=−∞
eπin2
τ
,
satisfies the modular transformation law
θ(−1/τ) = (τ/i)
1
2 θ(τ). (1.1)
Perhaps the best-known way to derive the functional equation of the Rie-
mann zeta function ζ(s),
π−s/2
Γ(s/2)ζ(s) = π(s−1)/2
Γ({1 − s}/2)ζ(1 − s), (1.2)
is by way of (1.1) [107, p. 22]. Conversely, it is not difficult to show that
(1.2) implies (1.1), but this derivation requires the use of the Phragmén-
Lindelöf Theorem [105, §5.65]. In 1921 H. Hamburger [38] showed that,
under certain auxiliary analytic conditions, ζ(s) is essentially the only so-
lution to the functional equation (1.2). For a more transparent proof, see
C. L. Siegel’s paper [99], [100, pp. 154–156]. More specifically, they proved
that if f(s) is a Dirichlet series satisfying the aforementioned auxiliary re-
strictions, and if
R(s) = π−s
Γ(s)f(2s), R(s) = R

1
2
− s

, (1.3)
then f(s) is a constant multiple of ζ(s). See also [107, pp. 31–32]. That
f(2s), as opposed to f(s), appears in (1.3) guarantees a priori that the
inverse Mellin transform of R(s), an exponential series, has its coefficient
sequence supported on integral squares, and thus it has the general shape of
1

2 Hecke’s Theory of Modular Forms and Dirichlet Series
θ(τ)−1. The proof of Hamburger’s Theorem is then completed by showing
that this inverse Mellin transform is in fact a constant multiple of θ(τ) − 1.
Of even greater interest within the context of the present work is a
second, distinct version, due to Hecke, of the Hamburger theorem. This
version is, in fact, a direct consequence of a general correspondence theo-
rem proved by Hecke in 1936 [47] (the “main correspondence theorem” of
Chapter 2, below) and the fact that, under certain conditions of regularity,
θ(τ) is the only solution to (1.1) that is periodic (with period 2). For further
details concerning the two formulations of Hamburger’s theorem, see the
introduction to Hecke’s final published paper [49], [62, esp. pp. 201–207],
and the Application following Remark 7.4.
Throughout the sequel we let τ = x + iy and s = σ + it with x, y, σ,
and t real. We denote the upper half-plane, {τ : y 0}, by H. The set of
all complex numbers will be denoted by C, the set of all real numbers by
R, the set of all rational numbers by Q, and the set of all rational integers
by Z. We adopt the following argument convention: for w ∈ C, w 6= 0, and
k ∈ R, wk
is defined by
wk
= |w|k
eik arg w
, −π ≤ arg w π. (1.4)
The summation sign
P
with no indices always means
∞
P
n=1
. We write
R
(c) for
R c+i∞
c−i∞ , where c is real and the path of integration is the straight
line from c−i∞ to c+i∞. We often use the symbol A to denote a positive
constant, not necessarily the same with each occurrence.

Chapter 2
The main correspondence theorem
Before proving the main theorem we first establish a couple of lemmas.
Lemma 2.1. Let ϕ(s) =
P
ann−s
. Then, ϕ(s) converges in some half-
plane if and only if an = O(nc
), for some constant c, as n tends to ∞.
Proof. First, assume that an = O(nc
) as n tends to ∞. Let σ ≥ c + 1 +
for some constant 0. Then,
X
ann−s
≤ A
X
nc−σ
≤ A
X
n−1−
∞.
Therefore, ϕ(s) converges for σ ≥ c + 1 + .
Conversely, if ϕ(s) converges for s = s0 = σ0 + it0, then ann−s0
tends
to 0 as n tends to ∞. In particular, an = O(nσ0
).
Lemma 2.2. Let λ 0 and c ≥ 0. Suppose that
f(τ) =
∞
X
n=0
ane2πinτ/λ
is analytic on H.
(i) If an = O(nc
), then f(τ) = O(y−c−1
), uniformly for all x, −∞
x ∞, as y → 0+.
(ii) If f(τ) = O(y−c
) as y 0 tends to 0, uniformly for all x, then
an = O(nc
) as n tends to ∞.
3

Proof of (i). For u ≥ 0, uc
exp(−2πuy/λ) achieves its maximum at
u = cλ/2πy. Thus, as y 0 tends to 0,
|f(τ)| ≤
∞
X
n=0
|an|e−2πny/λ
≤ A
X
nc
e−2πny/λ
≤ A
Z cλ/2πy
0
uc
du +
Z ∞
cλ/2πy
uc
e−2πuy/λ
du
!
+ O({cλ/2πy}c
e−2π(cλ/2πy)y/λ
)
= O(y−c−1
) + O

y−c−1
Z ∞
0
uc
e−u
du

+ O(y−c
)
= O(y−c−1
).

Proof of (ii). Let τ0 ∈ H. Then by Fourier’s formula for the coefficients
of a Fourier series,
an =
1
λ
Z τ0+λ
τ0
f(τ)e−2πinτ/λ
dτ
= O
Z τ0+λ
τ0
y−c
e2πny/λ
|dτ|
!
,
as y 0 tends to 0. If we set y = 1/n, we find that an = O(nc
) as n tends
to ∞.
Theorem 2.1. Let {an} and {bn}, 0 ≤ n ∞, be sequences of complex
numbers such that an, bn = O(nc
), as n tends to ∞, for some c ≥ 0. Let
λ 0, k ∈ R, and γ ∈ C. For σ c + 1, put
ϕ(s) =
X
ann−s
and ψ(s) =
X
bnn−s
.
Define, for σ c + 1,
Φ(s) = (2π/λ)−s
Γ(s)ϕ(s) and Ψ(s) = (2π/λ)−s
Γ(s)ψ(s).
For τ ∈ H, let
f(τ) =
∞
X
n=0
ane2πinτ/λ
and g(τ) =
∞
X
n=0
bne2πinτ/λ
.

The main correspondence theorem 5
Then the following two assertions are equivalent.
(i) f(τ) = γ(τ/i)−k
g(−1/τ).
(ii) Φ(s) + a0/s + γb0/(k − s) has an analytic continuation to the entire
complex plane that is entire and bounded in every vertical strip. Moreover,
Φ(s) = γΨ(k − s). (2.1)
Remark 2.1. Our formulation of Theorem 2.1 deviates from Hecke’s
original statement [47], [48] of his correspondence theorem in two ways.
In Hecke’s work there are not two, but only a single Dirichlet series; that
is, ψ(s) = ϕ(s). Also, our boundedness condition in (ii) replaces a corre-
sponding hypothesis of Hecke, who assumes that (s − k)ϕ(s) is an entire
function of finite genus, that is to say, there exists an M 0 such that
|(s − k)ϕ(s)| ≤ exp{|s|M
}, for all s in C. In Chapter 7 we discuss the
extent to which the Dirichlet series ϕ(s) and ψ(s) can differ, and how this
difference affects the theory developed in our Chapters 4–6.
We turn to the matter of the differing conditions on boundedness.
Clearly, they are equivalent within the framework of the Hecke correspon-
dence theorem. For, condition (ii) is equivalent to (i) in our Theorem 2.1,
while the original Hecke version of (ii) (assuming (s − k)ϕ(s) is entire of
finite genus) is likewise equivalent to (i) [47], [48]. On the other hand, these
conditions on boundedness are not equivalent outside of the context of the
correspondence theorem. To see this, recall that
Φ(s) = (2π/λ)−s
Γ(s)ϕ(s)
and that Γ(s) has the following growth properties:
(a) 1/Γ(s) is entire of finite genus;
(b) Γ(s) is bounded in vertical strips (truncated when necessary to avoid
the poles of Γ(s)).
Using these facts, we can reduce the equivalence of the two boundedness
assumptions to: h(s) is bounded in vertical strips if and only if h(s) is of
finite genus. But this equivalence fails in both directions, since
(c) h1(s) = exp(es
) is bounded in vertical strips, but not of finite genus;
(d) h2(s) = exp(−s2
) is of finite genus, but not bounded in vertical
strips.

Proof of Theorem 2.1. First assume that (i) is valid. From Euler’s
integral representation of the Γ-function, for σ c + 1,
Φ(s) =
X
an
Z ∞
0
(2πn/λ)−s
us−1
e−u
du
=
X
an
Z ∞
0
us−1
e−2πnu/λ
du.
Since σ c + 1, we may invert the order of summation and integration by
absolute convergence to obtain
Φ(s) =
Z ∞
0
us−1
X
ane−2πnu/λ
du
=
Z ∞
0
us−1
(f(iu) − a0)du
=
Z 1
0
+
Z ∞
1

us−1
(f(iu) − a0)du
=
Z ∞
1
u−s−1
f(i/u)du − a0/s +
Z ∞
1
us−1
(f(iu) − a0)du.
Using (i), we find that, for σ sup(c + 1, k),
Φ(s) = γ
Z ∞
1
u−s−1+k
(g(iu) − b0)du +
Z ∞
1
us−1
(f(iu) − a0)du
− a0/s − γb0/(k − s). (2.2)
Since f(iu) − a0, g(iu) − b0 = O(exp{−2πu/λ}) as u tends to ∞, it follows
by analytic continuation that Φ(s)+a0/s+γb0/(k−s) is an entire function.
It is easily seen that Φ(s) + a0/s + γb0/(k − s) is bounded in every vertical
strip by taking absolute values in (2.2). If we replace s by k − s in (2.2)
and use a formula analogous to (2.2) for Ψ(s), we immediately find that
Φ(k − s) = γΨ(s).
Conversely, we now assume (ii). By the Cahen-Mellin formula [75,
pp. 97–98], for x, d 0,
e−x
=
1
2πi
Z
(d)
Γ(s)x−s
ds. (2.3)
Upon letting x = 2πny/λ with n, y 0, we find that
e−2πny/λ
=
1
2πi
Z
(d)
Γ(s)(2πny/λ)−s
ds.

The main correspondence theorem 7
Multiplying both sides by an and summing on n, we deduce that, for d
c + 1,
f(iy) − a0 =
X
an
1
2πi
Z
(d)
Γ(s)(2πny/λ)−s
ds (2.4)
=
1
2πi
Z
(d)
Φ(s)y−s
ds,
where the inversion in order of summation and integration is justified by
the absolute and uniform convergence of ϕ(s) on the line σ = d.
We next move the path of integration to the line σ = −d. We shall
do this by integrating around a rectangle with vertices ±d ± iT, T 0,
applying the residue theorem, and then showing that the integrals along
the horizontal sides tend to 0 as T tends to ∞. Now, by Stirling’s formula
[26, p. 224],
|Γ(σ + it)| ∼ (2π)
1
2 |t|σ− 1
2 e−π|t|/2
, (2.5)
as |t| tends to ∞, uniformly on any fixed interval, σ1 ≤ σ ≤ σ2. By
hypothesis, Φ(s) is bounded in every vertical strip. It follows that
ϕ(s) = O(|t|
1
2 −σ
eπ|t|/2
), (2.6)
as |t| tends to ∞, uniformly for −d ≤ σ ≤ d. On the line σ = d, clearly,
ϕ(s) = O(1), (2.7)
as |t| tends to ∞. From (2.1) and (2.5), we find that, on the line σ = −d,
ϕ(s) = O(Γ(k − s)ψ(k − s)/Γ(s)) = O(|t|k+2d
), (2.8)
as |t| tends to ∞, since ψ(k − s) = O(1). Thus, from (2.6)–(2.8), we see
that the hypotheses of the Phragmén-Lindelöf Theorem for a vertical strip
[105, p. 180] are satisfied. Thus,
ϕ(s) = O(|t|A
), (2.9)
uniformly on −d ≤ σ ≤ d. Hence, from (2.5) and (2.9) it is easily seen that
the integrals on the horizontal sides approach 0 as T tends to ∞. Therefore,
we have
f(iy) − a0 =
1
2πi
Z
(−d)
Φ(s)y−s
ds − a0 + γb0y−k
.

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Persons have practiced the foregoing idea and succeeded in
accomplishing their purpose and have, no doubt, been highly
gratified that their horse was so intelligent, yielding a quiet
obedience to their efforts in a very short time; now had the owner
known that a foundation for a complete education was properly laid,
how easily could he have built a superstructure thereon that would
have been permanent and beneficial during the life of the animal.
Men are often conceited and think that because they have
experienced no difficulty in the past in breaking and handling their
horses, therefore all will be sunshine in the future. I would advise a
careful perusal of my work, and, by so doing, those of the character
above described will have their conceit supplanted as they gain the
knowledge of a practical system of properly educating the horse.
HOW TO LAY A HORSE DOWN.
HOW TO LAY A HORSE DOWN.

Place a rope or rein around the horse’s body, forming a surcingle;
pass the other end under the tail and bring it back, tying it to the
part around the body, thus forming a surcingle and crooper; also put
a ring, say an inch in diameter, in the surcingle on the back; then
strap up the nigh fore-leg as follows: take a quarter-strap, pass it
two or three times around below the fetlock, then through the
keeper; bring the leg up and buckle close to the belly; place around
the neck a quarter-inch strong rope, loosely, fixing the knot so it will
not slip; bring the end down the near side of the head, through the
mouth, and back on the off-side through the ring in the surcingle;
now take a position on the nigh-side of the horse, commencing to
pull gently, allowing him to struggle a short time, after which he will
lie down quite easily, without sustaining any injury.
By adhering strictly to the instructions given, every person will
readily see, that the object in laying the horse down is to give him to
understand that you are master; and, after repeating this two or
three times, he will be perfectly satisfied of the fact. While down,
place a small pole between his legs, moving it about; if he shows
fear or resists, give him a sharp pull on the cord in his mouth by way
of correction. In other words, punish him for doing wrong, and
caress him for doing right, thus making him understand the
difference between right and wrong.
HOW TO GET A HORSE UP THAT THROWS
HIMSELF.

HOW TO GET A HORSE UP THAT THROWS HIMSELF.
Animals are often sulky, and quickly acquire the habit of lying
down. Balky horses, when urged to go, will lie down and refuse to
get up, and an ox will sometimes lie down in the furrow when before
the plough. When the habit is thoroughly settled, it becomes very
annoying to the owner or driver, who often resorts to severe means,
but fails to accomplish the end desired; therefore, to prevent
violence and ill-treatment, I give the easy and simple remedy
subjoined, which, when adopted, will be found to be practical and
never-failing:
Raise the animal’s head up, as illustrated in the foregoing plate,
and pour into his nostril a small quantity of water, not to exceed a
pint, from a pitcher or cup, and you will be amused by the pleasing
result: the animal will rise to his feet as quickly as it is possible for
him to do so; he believes himself to be drowning, and will extricate
himself with all speed.

Simple as is this expedient, it is yet unfailing in its efficacy; and
that which most commends it to the acceptance of kind-hearted men
is the absence of all cruelty in its application. No pain is caused, but
the unusual sensation, together with the necessity for air on the part
of the horse, banishes his former feeling of sulkiness or anger and
the yields to the almost irresistible impulse to spring to his feet and
free his nostrils of the water.
If any one who has never applied this remedy should doubt its
power, he only needs to try the experiment in a mild way on himself,
when he will realize its power upon the horse.
I believe it would be impossible to devise another method so free
from pain, so harmless to the horse, and yet so thoroughly
efficacious as is the one we have here given.
TO EDUCATE A COLT NOT TO BE AFRAID OF
HIS HEELS.

TO EDUCATE A COLT NOT TO BE AFRAID OF HIS HEELS.
Too much importance cannot be attached to the manner of
educating a horse’s heels, as it is in that point his greatest means of
defense and resistance lies, and most men make the mistake of
breaking one end of the horse, while they allow his hind parts to go
uneducated. The instructions I am about to give will, if properly
followed, insure success.
After laying your colt down, commence to handle his hind parts
and heels, being careful to hold the cord firmly in your left hand, so
that, if he attempts to get up, you can control him; then strike him
gently with a stick, and, if he should show fear, which he naturally
will, punish him in the mouth; then place the stick between his legs
and commence moving it around, and, if he makes no resistance,
remember to caress him; almost as much is accomplished by
caressing as by punishing.
The above instruction is equally applicable to a kicking horse, but
in his education he will require more lessons before the habit will be
entirely removed; still, kindness and a little patience will soon
accomplish all you desire.
Men in general exercise too little patience in the training of their
colts, and they frequently expect to accomplish more in a short
space of time than can possibly be performed. Yet the time really
required, when measured by days, is so short as to be really
surprising. Let us suppose that in training a colt one were to spend
two hours a day for ten days, which is the longest time that could
possibly be needed; compute the time at ten hours to the day, and
the whole amounts to but two days, at the end of which he would
have a well-educated animal. I doubt if a farmer or horse-raiser
could employ his time more profitably in any other way than in
thoroughly educating his colts, as he thus enhances their value, for
there is no sensible man who would not give ten dollars more for a
properly educated animal than for one improperly trained.

TO EDUCATE A COLT TO DRIVE BEFORE BEING
HARNESSED.
TO EDUCATE A COLT TO DRIVE BEFORE BEING HARNESSED.
Place on him the Bonaparte bridle, as shown in engraving, with
your cord in the left hand and whip in the right; the cord referred to
should be about eighteen feet long; now drive him around a circle to
the right about fifteen minutes; then drive him to the left about the
same time. You have now educated your colt to drive, and may with
safety put on your harness, observing to put the reins through the
shaft tugs at his side; then commence driving him carefully for some
fifteen minutes on a walk, turning him to the right and left as before
directed. Do not use the whip more than sufficient to give him a
knowledge of its use. Never drive a colt without blinders. It is better
to first hitch him to a sulky or a cart, and do not put on breechings,
but allow the cross-bar frequently to come against his heels, so that

he may never be afraid, or learn to kick. Never forget, when your
colt is obedient, to stop him, and walk up to and caress him.
I am unwilling to pass on to another illustration without more fully
impressing on the minds of those who raise or break colts the
necessity of kind and careful usage in educating their animals. Never
approach your colt quickly. Never, pull the halter or bridle off quickly.
Always handle the colt’s ears with great care. Never punish him on
the body with anything but a whip, and with it as seldom as
possible, as many colts become sulky and show signs of balking
when severely whipped. It is better that you should give your colt
two or three lessons each day, as heretofore directed, at intervals of
say two hours apart: by this means you do not overtax his brain, nor
cause him to get weary. In this, as in many other cases, the wisest
course is to “make haste slowly.”
HOW TO EDUCATE A COLT TO MOVE HIS BODY
WHEN HE MOVES HIS HEAD.

TO EDUCATE A COLT TO MOVE HIS BODY WHEN HE MOVES HIS HEAD.
Place on your bridle, then your harness; carry your reins through
the shaft tugs; take your position behind the horse (see engraving);
now commence to drive, turning him round frequently, first to the
right, then to the left, and he will quickly understand to move his
body when he moves his head. By this means you are educating to
the shafts, and educating not to be afraid of his heels, thus
thoroughly breaking your horse at both sides and both ends.
After your colt has been driven two or three times, as above
described, educate him to obey the word “whoa:” let him walk along
smartly, then speak plain, with audible voice, and say “whoa;” at the
same time pull on the reins with some force; when he stops, caress
him; repeat this a few times, and, in the short space of fifteen
minutes, you will have taught him the use of the word. Now your
horse is educated to drive and stop at the word of command.
The next thing in order is to teach him to back. To accomplish
this, grasp your reins firmly, and with a determined effort; speak
firmly, making use of the word “back,” at the same time pulling with
all your might; if he obeys the first time, step up and caress him; if
not, increase the power by inviting one or more of your friends to
assist on the reins, being fully determined to accomplish your
purpose. As soon as he obeys, don’t fail to caress him, and by this
process you will educate your horse to the word, which he will never
forget.
Your colt being educated, you may now hitch him up to a vehicle,
observing to drive him very slow, only on a walk, and after thus
driving him a few times, you can with certainty say that you have a
thoroughly educated horse, whose value will be greatly increased,
compared with the old or any other system of breaking the colt.
Always observing to drive your colt with blinders, only using the
whip enough to let him know the use of it. Be kind to your animal,
never using harsh means, and he will reward your kindness by
implicit obedience.

IMPROVED METHOD OF BITTING A COLT.
IMPROVED METHOD OF BITTING A COLT.
Use the Bonaparte bridle, placing a loop on the lower jaw; carry
the cord back on the neck, bringing the end of the cord down
through the loop on the jaw; then draw the horse’s head up in an
easy and graceful position, and tie a bow-knot. Should the horse
attempt to rear and go over backwards, give a little pull to untie the
cord, and the horse is saved from any accident. The object aimed at
in bitting a horse is to give an easy position, with a high and graceful
carriage of the head, and, in our efforts to do this, we must be
careful not to give him a dead bearing on the bit, or make him what
is usually known as a “lugger.” All the bitting rings which we have
examined, and especially those of English make, are objectionable,
as having a tendency to produce this result. The rig which we here
give you is entirely free from this objection, and is better calculated
to produce the desired result of ease and gracefulness than any ever

before presented to the public. Our rig, instead of bearing on the
jaw-bones whenever the horse presses his weight upon the bit,
producing a calloused jaw and indifference to the bit, contracts the
side muscles of the cheek on the molar teeth, with a pain the horse
cannot endure; he lifts his head, the bit falls on the side rein, and
the mouth is at once relieved. Practice has shown that horses bitted
with this rig soon acquire the habit of gently and gracefully raising
the head with that occasional toss, or upward and downward
motion, and playing with the bit, which is the perfection of beauty in
a carriage horse, while standing in the harness.
It is not possible for a horse with our rig to become a “lugger.”
This bit never bears upon the jaw-bone with more than a light
pressure, and when he attempts to rest his head upon the bit, the
pressure on the teeth causes him to desist and elevate his head. He
soon dreads to rest upon the bit, and of his own free will, without
the force of the rein, carries it up with freedom and ease.
EDUCATING THE COLT TO RIDE

EDUCATING THE COLT TO RIDE.
First put on the Bonaparte bridle, make a double half-hitch,
bringing it over the head, back of his ears, and, carrying it down to
the mouth, place it under the upper lip, taking the end of the cord in
your right hand, placing it on the horse’s rump; then place your left
hand in the mane; now spring partly on and off, as seen in the plate
on preceding page. Do this several times; if he moves, punish him in
the mouth, by means of the cord; if he does not move, when you
get off caress him; then go to the opposite side, repeating the same
several times, after which you may safely mount your horse, but be
particular not to remain on his back too long at a time, as the
strength of the animal is not yet sufficiently developed to bear a
protracted strain. Like the young of all animals, the colt has a great
deal of energy and spirit, but lacks the stamina to endure long-
continued exertion, nor can it be imposed on them without certain
injury.

Men do not act wisely by practicing the old system of riding colts,
viz., by mounting on the back with reins and whip in hand, and, so
soon as the colt jumps about or rears, applying the whip or heels;
for the reason that the animal does not know what you require of
him, but believes you design some injury, and therefore resists your
efforts to ride him. No wonder that he repeatedly throws his rider
and treats him as an intruder. He cannot be less terrified than a man
would be if a wild animal were to mount on his back. In order to
secure success in educating his colt to ride let the reader adopt the
foregoing instructions, and he will not have cause for regret, but will
find that he has gained more than the single point of riding, as he
has taught his colt that he does not intend to harm him, and that his
duty is to yield a cheerful obedience to his owner’s commands.
INSTRUCTIONS TO RIDE THE COLT.
INSTRUCTIONS TO RIDE THE COLT.

Take a small cord, ten to twelve feet long, divide it in the center;
then place the center back of the ears, cross it in the mouth, then
bring both ends along the neck to the withers, and tie a knot, thus
forming a powerful bridle, sufficient to ride the most vicious animal.
Sacred history contains the declaration that there is “the bridle for
the horse, the whip for the ass, and the rod for the fool’s back,” and,
while writing my book, I have often thought of the first portion of
that quotation. The power of the bridle in controlling the horse is
really wonderful, and the new forms of powerful bridles given in this
work enable the most timid rider to secure the mastery of the most
powerful animal. The one described above is excellent, and can
never fail to give satisfaction when it is used as directed.
There is no exercise so invigorating and scarcely any so delightful
as the manly one of riding the horse, yet three-quarters of the
pleasure of equestrianism depends on the early training of the horse
for this delightful exercise. The rider who feels that he has beneath
him an animal obedient to his slightest wish, and which responds to
a touch of the heel or the lightest pressure of the bit, moving to the
lifting or the falling of the bridle, such a rider feels almost as though
the horse on which he sits forms a portion of himself, and courses
onward with a delightful sense of power and freedom. Nearly all of
this excellence in a riding-horse depends on the way in which he has
been educated while young. Faults then acquired may be corrected,
it is true, in later years, yet it is far more desirable that they should
never have been formed, but, in place thereof, the qualities secured
which form the excellence of a horse.
I throw out these suggestions at this point, for I am now dealing
with the early education of the colt; later in the book I shall have to
speak more of faults to be corrected, and it is my wish to impress on
my reader the great importance of the kind of education which the
colt receives at his hands.
TO HALTER-BREAK A COLT, AND HITCH IN THE
STABLE.

TO HALTER-BREAK AND HITCH A COLT IN THE STALL.
Place the center of a sixteen-foot cord under the horse’s tail, and
bring it over and cross it on the back; then tie it firmly in front of the
breast (as seen by reference to illustration on preceding page); carry
the halter-strap through the manger, and bring it back under the
mouth; then tie the end of the strap to the cord in front of the
breast. The colt is now tied by the head and tail. While he yields
quietly to the confinement, he is comfortable and easy, but the
moment he begins to resist he punishes himself; this he soon learns,
and in a little time ceases the efforts which he finds to be productive
of pain. He will always remember the lesson, and give no further
trouble.
It may seem a trifling thing to have gained the result described
above; yet, as the earth is made up of grains of sand, so the
thorough education of the horse is the result of attention to a
multitude of small affairs, each one seeming to be of little

importance in itself, but which, in their total, make the difference
between a gentle and an unruly animal.
Too much care and attention cannot be bestowed on the colt while
you are giving him lesson after lesson, as he is susceptible of
impressions that will take weeks to overcome, provided you should
through neglect or carelessness omit to practice the instructions laid
down. Do not suppose that any of the directions given in this book
are unimportant. They are, one and all, the result of long experience
in the management and education of the horse, and each one, in its
place and relation to the general system, is as necessary, though
perhaps not as important, as any other. To be certain of reaching the
best results of the system, the reader must not pass over any of our
directions as unimportant or unnecessary, but accept each as a part
of the system which it has required years of time and thought to
bring to its present state of perfection.
TO EDUCATE A HORSE NOT TO KICK AT YOU
WHEN ENTERING THE STALL.

TO EDUCATE A HORSE NOT TO KICK AT YOU WHEN ENTERING THE STALL.
Place on the horse the Bonaparte bridle; then drive a staple at the
side of the stall, near the manger, three or four feet from the floor;
then attach another staple at the entrance of the stall, the same
distance from the floor; now pass the cord through both staples and
tie it. When you enter the stall, pull sharply on the rope; at the same
time use the words “go over.” The head of the horse will be drawn
towards you, and his heels to the opposite side. Thus you avoid all
danger, and will very soon educate your horse to abandon this bad
habit.
Vicious and annoying habits in horses often owe their origin to
bad management by their owner or groom. Allow me to instance a
few examples: A man walks into the stable and approaches his horse
in the stall, and, if he should move about quickly, the person springs
back from him, evidently showing his fear, which is at once noticed
by the horse, and taken advantage of; so that, after a repetition of
this two or three times, the animal fancies he is master, and uses his

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Hecke s theory of modular forms and Dirichlet series 2 Revised Edition Bruce C. Berndt

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