Henryk lwaniec
Graduate Studies in Mathematics Volume 17
[email protected] .J>f';o;tll\t.4;,,,,
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American Mathematical Society
Selected Titles in This Series 17 Henryk Iwanlec, Topics in classical automorphic forms, 1997
16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume 11: Advanced theory, 1997
15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997
14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces. 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups. 1996 9 Dino Lorensinf, An invitation to arithmetic geometry, 1996 8 Winfrled Just and Martin Weese, Discovering modern set theory. 1: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantsen, Lectures on quantum groups. 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock. 1994
3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994
2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity. 1993 1
Ethan Akin, The general topology of dynamical systems. 1993
Topics in Classical
Automorphic Forms
Topics in Classical
Automorphic Forms Henryk Iwaniec
Graduate Studies in Mathematics Volume 17
American Mathematical Society Providence, Rhode Island
Editorial Board James E. Humphreys (Chair) David J. Saltman David Sattinger Julius L. Shaneson 1991 Mathematics Subject Classification. Primary 32Nxx.
Library of Congress CataloginginPublication Data Iwaniec, Henryk. Topics in classical automorphic forms / Henryk Iwaniec.
p. cm.  (Graduate studies in mathematics ; v. 17) Includes bibliographical references and index. ISBN 0821807773 (hardcover : alk. paper) 1. Automorphic forms. I. Title. II. Series. QA243.I95 1997 9724332
512'.7dc21
CIP
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Contents
Preface
Chapter 0. Introduction Chapter 1. The Classical Modular Forms 1.1. Periodic functions 1.2. Elliptic functions 1.3. Modular functions 1.4. The Fourier expansion of Eisenstein series 1.5. The modular group 1.6. The linear space of modular forms Chapter 2. Automorphic Forms in General 2.1. The hyperbolic plane 2.2. The classification of motions
2.3. Discrete groups  Fuchsian groups 2.4. Congruence groups 2.5. Double coset decomposition 2.6. Multiplier systems 2.7. Automorphic forms 2.8. The etafunction and the thetafunction
xi
1
3 3 6 10 13 15
18
23 23
27 29
34 37 40 42
44
vii
viii
Contents
Chapter 3. The Eisenstein and the Poincare Series 3.1. General Poincare series 3.2. Fourier expansion of Poincare series 3.3. The Hilbert space of cusp forms
47
Chapter 4. Kloosterman Sums 4.1. General Kloosterman sums 4.2. Kloosterman sums for congruence groups 4.3. The classical Kloosterman sums 4.4. Powermoments of Kloosterman sums 4.5. Sums of Kloosterman sums 4.6. The Salie sums
57
Chapter 5. Bounds for the Fourier Coefficients of Cusp Forms 5.1. General estimates 5.2. Estimates by Kloosterman sums 5.3. Coefficients of cusp forms with theta multiplier 5.4. Linear forms in Fourier coefficients of cusp forms 5.5. Spectral analysis of the diagonal symbol
69
Chapter 6. Hecke Operators 6.1. Introduction 6.2. Hecke operators Ttz 6.3. The Hecke operators on periodic functions 6.4. The Hecke operators for the modular group 6.5. The Hecke operators with a character 6.6. An overview of newforms 6.7. Hecke eigencuspforms for a primitive character 6.8. Final remarks
91
Chapter 7. Automorphic Lfunctions 7.1. Introduction 7.2. The Hecke Lfunctions 7.3. Twisting automorphic forms and Lfunctions 7.4. Converse theorems
47 49 52
57 58
59 61
65 66
69 72
75 81
83
91
92
94 98 101
107 108 118
119 119 120 124 126
Contents
Chapter 8. Cusp Forms Associated with Elliptic Curves 8.1. The HasseWeil Lfunction 8.2. Elliptic curves Er
8.3. Computing \(p) 8.4. A Hecke Grossencharacter 8.5. A theta series 8.6. The automorphy of f
ix
133
133 136 138 141
142
143
Chapter 9. Spherical Functions 9.1. Positive definite quadratic forms 9.2. Space spherical functions 9.3. The spherical functions reconsidered 9.4. Harmonic analysis on the sphere
147
Chapter 10. Theta Functions 10.1. Introduction 10.2. An inversion formula 10.3. The congruent theta functions 10.4. The automorphy of theta functions
165
10.5. The standard theta function
147 150 155
159
165
166 168 175 176
Chapter 11. Representations by Quadratic Forms 11.1. Introduction 11.2. Siegel's mass formula 11.3. Representations by Eisenstein series and cusp forms 11.4. The circle method after Kloosterman 11.5. The singular series 11.6. Equidistribution of integral points on ellipsoids
179
Chapter 12. Automorphic Forms Associated with Number Fields 12.1. Automorphic forms attached to Dirichlet Lfunctions 12.2. Hecke Lfunctions with Grossencharacters 12.3. Automorphic forms associated with quadratic fields 12.4. Class group Lfunctions reconsidered 12.5. Lfunctions for genus characters 12.6. Automorphic forms of weight one
203
179 180 185
190 196 199
203 206 211 215
219
224
x
Contents
Chapter 13. Convolution Lfunctions 13.1. Intro duction 13.2. Ran kinSelberg integrals 13.3. Selb erg's theory of Eisenstein series 13.4. Stat ement of general results 13.5. The scattering matrix for F0 (N) 13.6. Func tional equations for the convolution Lfunctions 13.7. Met aplectic Eisenstein series 13.8. Sym metric power Lfunctions
231
Bibliography
255
Index
257
231 232 235
240 240 243 246 248
Preface
Automorphic forms are present in almost every area of modern number theory. They also appear in other areas of mathematics and in physics. I have lectured on these topics many times at Rutgers University with only slight overlapping of content, and I still have new material to teach that is important. It is indeed a vast territory which cannot be grasped by any one person. While research publications on automorphic forms are rapidly increasing in quantity and quality, the demand for textbooks, particularly on the graduate level, is also growing. There are fine books on this subject such as [Lan], [Miy], [Sh2], but still more are needed, especially those which favor analytic methods. The present book is based on my lecture notes (almost verbatim except for Section 5.5) from a graduate course which I delivered in the Fall of 1994 _
and in the Spring of 1995 at Rutgers. The course was formulated, as the title implies, to acquaint our new students with the subject matter from various perspectives. Thus I have not followed direct or traditional paths, but rather I have frequently ventured into areas where different ideas and methods mix and interact. To cover a lot in a limited time, some material is necessarily presented as a survey. For example, the numerous connections
of automorphic forms with Lfunctions of number fields are discussed in Chapter 12 without details. However we do provide complete proofs of the most basic results in the earlier sections. An experienced reader will find some of our arguments to be nonstandard. It would be pointless to argue which approach is better, since our choice was made simply for the purpose of showing different possibilities. For example, our presentation of the theory of Hecke operators in Chapter 6
xi
xii
Preface
is completed for primitive characters quickly by establishing the multiplicityone principle using Gauss and Ramanujan sums instead of lengthy considerations of inner products. Of course, this is only a special case (the whole space is spanned by newforms), but it is an important case.
We pay great attention to detail in the subjects of theta functions and representations by quadratic forms (Chapters 10 and 11) because these are not sufficiently covered in textbooks, despite having a long history of research.
Because the original notes where written as the course was progressing, inevitably some redundancy has occurred. Nevertheless we have decided not to eliminate this redundancy, because it offers the option of selective reading. For example, our account of the ShimuraTaniyama conjecture for special curves (the congruent number curves) is selfcontained in Chapter 8, even though one could instead appeal to the later chapters on general theta functions.
Sergei Gelfand, Peter Sarnak and others have convinced me that these lecture notes might be useful for a large number of graduate students, and they have urged me to publish them. I would like to thank them for their encouragement. I am grateful to W. A. Gonzalez, C. L. Hamer and C. J. Mozzochi for helping me in the technical preparation of the original notes. Special thanks are expressed to T. Khovanova for corrections and improvements which she contributed when editing these notes for publication. Henryk Iwaniec
Chapter 0
Introduction The concept of an automorphic function (the name was given by F. Klein, 1890) is a natural generalization of a periodic function. Let X be a locally compact space acted on discontinuously by a group IF. Then a function f : X j C is automorphic with respect to IF if
f(yx)= f(x)
for all ryEI'. In other words, f lives on the quotient space r \ X (the space of orbits). A typical habitat for automorphic functions is the homogeneous space X = G/K of a Lie group G, where K is a closed subgroup. In this case X is a Riemannian manifold, so the differential calculus on X is available. If (0.1)
an autom.orphic function f is a common eigenfunction of the whole °a:Tgebra
D of invariant differential operators on X (D is a commutative algebra; it contains he LaplaceBeltrami operator), (0.2)
Df=)f(D)f forallDED,
then f is called an automorphic form. More generally, one considers automorphic functions for which the transformation rule (0.1) is modified by a differential factor and a suitable multiplier system. Those functions which live on X = SL2 (R) /SO2 (R) and which are complexanalytic (X is identified with the upperhalf plane M C C and the differential equations (0.2) are the CauchyRiemann equations) are called classical automorphic forms. In this book our main interest will be in the classical automorphic forms with respect to a congruence group, because of their importance for applications to arithmetic. On a few occasions we shall encounter the realanalytic Eisenstein series, and for clarity, we introduce basic concepts in the context of a general group acting discontinuously on the upperhalf plane. 1
2
Introduction
There are various books on spectral aspects of automorphic functions, but none covers and treats in detail as much as the expansive volumes by Dennis Hejhal [Hell. I recommend them to anyone who is concerned with research. In these books one also finds a very comprehensive bibliography. Those who wish to learn about the theory of automorphic forms on other symmetric spaces in addition to the hyperbolic plane should read Audrey Terras [Te]. A broad survey with emphasis on new developments is given by A. B. Venkov [Ve].
Chapter 1
The Classical Modular Forms
We begin by giving a glimpse of how automorphic forms emerged on the hyperbolic plane as descendants of those living on flat spaces.
1.1. Periodic functions Take X = R and IF = Z, so R/Z is the circle. A function f : IR a C is periodic of period 1 if
f (x + n) = f (x)
for all n E Z.
To construct a periodic function one can simply take any f on the segment [0, 1) and extend its values uniquely to R by requiring periodicity. For example {x} = x  [x] (the fractional part of x) is obtained from f (x) = x on [0, 1). Another construction uses the averaging method. Let f : 1[8 + C be any function of rapid decay at ±oo so that the series (1.1)
g(x) = > f(x + n) nEZ
converges absolutely. Clearly the function g is periodic of period 1. The exponential function (1.2)
e(x) = e272x = cos 27rx + i sin 27rx 3
1. The Classical Modular Forms
4
plays a special role in the linear space of periodic functions. Any periodic and piecewise continuous function f : III f C has the Fourier series representation (1.3)
f (x) = 1] a,e(nx) nEZ
with coefficients given by
f
an =
(1.4)
f (x)e(nx)dx.
If f is continuous, the Fourier series (1.3) converges absolutely, hence uniformly.
Exercise. Prove that the Fourier expansion for the fractional part of x is given by the series
{x} =
(1.5)
1
2
°O sin 27rnx
E n=1
7rn
which is boundedly convergent (the partial sums are uniformly bounded), but not absolutely. Precisely, we have N
{x} = 2 
sin 27 nx
+
0((1 +
IIxIIN)1),
ra=1
where jjxjj denotes the distance from x to the nearest integer and the implied constant is absolute.
Developing (1.1) into Fourier series, we find by unfolding the integral (1.4) that an = J
f(x)e(nx)dx = f(n)
D where R Y) denotes the Fourier transform of f (x) on ]I8; (1.6)
f(y) =
I1f6_xY
)dx.
Therefore the Fourier expansion (1.3) appears as (1.7)
E f (x + n) = > f (n)e(nx). nEZ
nEZG
1.1. Periodic functions
5
If we set x = 0, this becomes the Poisson summation formula
1: f(n)=1: f(n).
(1.8)
nEZ
nEZ
The above argument is valid if f and f have rapid decay at ±oo. As an example, take the function h(x) = e,x2, which is selfdual with respect to the Fourier transform, namely h(y) = h(y). Therefore, changing variables, for any v > 0 we have the Fourier pair f (x) = e"x2", f (y) _ ny2v' v 2e . In this case (1.7) yields eir(x+n)2v
(1.9)
= v 2
nEZ
E nEZ
e7rn2v'
e(nx).
Exercise. Using (1.8), derive the following formula of Dirichlet: N 2 e E (9) n (mod N) \
(1.10)
eNV lv,
where EN = 1 1 z
, 2
which is valid for any positive integer N. Here are some suggestions. Assume for simplicity that N is odd. First prove that the sum (1.10), can it G(N), satisfies the asymptotic formula G(N) ' ENVrN as N  oo, which is easier to do because G(N) can be smoothed before applying the Pois
son summation. Then show by elementary arguments that G(N2) = N and G(N3) = NG(N). Hence the asymptotic: formula G(N) " ENVW improves itself to the exact formula (1.10) by considering the sequence Nk with k = 1, 2, ... (see [Dav] for a direct derivation).
The theory of periodic functions generalizes with obvious modifications to any given dimension. The space X = Rk is acted on by I' = 7Lk as integral vector translations. The Poisson summation formula asserts that
f(n) _
(1.11) nEZk
f(n) nEVC
where
(y) = ff(x)e(_xY)dx Rk
and xy = xlyl +
+ xkyk is the scalar product. To ensure convergence it suffices that f (x) is in the Schwartz class, i.e., all its partial derivatives
satisfy
f(j)(x) 0 is arbitrary, the implied constant depending on A and (j) _ (j1,
,jk)
1. The Classical Modular Forms
6
Exercise. Show that if f (x) is a distance function, then so is f (y); more precisely, if
f (x) = g(IxI) then
f(y) = h(IyI) where h(y) is a Hankel type transform of g(x) given by (1.12)
h(y) = 27ry1
2 fg(x)J_i(2xY)xdx
and JQ(x) is the Bessel function. Hence the Poisson summation formula becomes cc
(1.13)
oc
E rk(m)g(m) = 1] rk(m)h(m) rn=O
mn.=0
where rk(m) denotes the number of representations of m as the sum of k squares;
rk(m) = I {(ni, ... , nk) E Zk : ni + ... + nk = m}I.
1,.2. Elliptic functions The next generation of periodic functions lives in the space of complex numbers X = C. This is not simply the former case in two dimensions. Here C is considered as a Riemannian manifold with complex structure. Consequently the theory of meromorphic functions can be employed, which is indeed a very powerful tool. For instance, one can establish interesting relations, even of arithmetic type, by examining the location of poles and zeros.
We shall use the standard complex number notation z = x + iy. The group Z acts on C giving periodic functions in the horizontal direction, such as
e(z) = e27rz4 = e(x)e2'ry.
To produce periodicity in any two described directions, consider a group A = w1Z + w2Z
where W1, w2 are complex numbers linearly independent over 1[8 so that C = w11l + w2lR.
1.2. Elliptic functions
7
W1+ W2
Figure 1. A parallelogram of a lattice
Thus A is a discrete, free abelian subgroup of C of rank 2, i.e., A is a lattice (see Figure 1). A function f : C > C is elliptic with respect to A if f is meromorphic on C  f is periodic with periods in A, (1.15)
f (u + w) = f (u)
for all w E A.
Clearly for the latter condition it suffices that f has two periods w1i w2 which generate A. An elliptic function can be regarded as a meromorphic function on the torus C/A. The elliptic functions for a lattice A form a field, say E(A). If f has no poles then it is holomorphic and bounded on C, so by Liouville's theorem f is a constant function. Choose a fundamental parallelogram for the lattice A, say
P = t1w1 + t2w2 + µ,
0 < ti t2 < 1,
with fi E C such that f has neither poles nor zeros on the boundary DP. Integrating along the boundary, we get by Cauchy's theorem (1.16)
E resf(w) = 0 WEP
because the integrals along the opposite sides cancel out by periodicity. Hence there is no elliptic function having exactly one simple pole (mod A). The order of f at w is the integer m = m f (w } such that f(u) = (u  w)mg(u)
where g(u) is meromorphic with g(w) 0, oc. Since f (u) is Aperiodic, so is mf(w). Notice that mf(w) 0 for at most a finite number of points in a
1. The Classical Modular Forms
8
given parallelogram. Applying (1.16) to the elliptic function f'(u)/ f (u), we get
E m f(w) = 0.
(1.17)
w (mod A)
Similarly integrating uf'(u)/f(u) along the boundary 8P we get
E mf(w)w  0 (mod A).
(1.18)
w (mod A)
Define the order of f to be the sum of orders of zeros (mod A) or the negative of the sum of orders of poles (mod A), i.e., (1.19)
rf=
E max(mf(w), 0) = 
E min(mf(w), 0). w (mod A)
w (mod A)
It follows from (1.16) that there is no elliptic function of order 1. From (1.17) applied to f (u)  c it follows that any complex number c is a value of f (u) for exactly r f points (mod A) counted with multiplicity. The simplest elliptic function which is not constant has one pole of order two with zero residue at the origin. Such a function can be constructed by averaging over the lattice; for example, one obtains (1.20)
P(u) = u2 + E' ((u 
w)2
 w2)
wcA
where i indicates that w = 0 is skipped in the summation. Notice that P(u) is even, i.e., yo(u) = p(u). Since p(u) has order 2 we infer that (1.21)
yo(u) = P (W)
u  ±w (mod A).
Moreover yo(u)  p(w) has a double zero at u = w exactly when w  w (mod A). There are three such points (mod A), namely wl/2, w2/2 and (J1 + w2) /2 (the point w = 0 is not because it is a pole of p). The numbers (1.22)
rrwlll
_
(W2\l
_
Cwt +W21 2
ll
are all distinct because of (1.21); thus the associated discriminant (1.23)
0 = 16 (el  e2)2 (e2  e3)2 (e3  el)2
h, not zero. The function yo(u) is called the Weierstrass function.
1.2. Elliptic functions
9
The Weierstrass function is basic in the sense that it generates the subfield C (p) of all even elliptic functions in E(A). Indeed if f c E(A) is even, then by comparing singularities (the zeros and the poles) we find that
fi
f(u)=c
'(w))"`r(w)
(p(u) 
w (mod A)
where c is a constant. Here the product takes only one singular point of f out of each pair ±w (mod A). Differentiating p(u) we obtain the elliptic function (u) _ 2 E(u  C.J)3
(1.24)
wEA
which is odd, p'(u) _ p'(u). Therefore the field of all elliptic functions for a lattice A is generated by p and p', E(A) = C (p, p')
(1.25)
.
The functions g(u), p'(u) are linearly independent over the constant field C, but they are algebraically related through the Weierstrass equation (1.26)
89 (u)2 = 4 (6o(u)  el) (p(u)  e2) (P('u)  e3)
with el, e2, e3 given by (1.22). To derive this equation recall that P(u)  el has exactly one double zero at u = wi/2, which is also a simple zero of p'(u). The same holds true for the other factors. Therefore the ratio .f (u) =
01
(u)2 (p(u)
 el) 1 (P(u)  e2)1 (8J(u)  e3)1
is an elliptic function whose only possible pole (mod A) is at the origin. But
u = 0 is a regular point of f (u) since
(u)

2u3 and yo(u)  u2 as
u > 0. Therefore f (u) is constant, in fact f (u) = 4 as claimed. From (1.20) one derives easily that oc
(1.27)
P(u) = u2 + E (m + 1)G7n+2urn rn.=1
where (1.28)
Gk
El L4)k. wEA
10
1.
Harmonic Analysis on the Hyperbolic Plane
Figure 2.. Geodesics in H.
The hyperbolic circles (loci of points at a fixed distance from a given point in IHI) are represented by the euclidean circles in liii (of course, not with
the same centers). There are various interesting relations in the hyperbolic plane. For instance, the trigonometry for a triangle asserts that sin a sinh a
_
sin /3
_ sing
sinh b
sinh c
and
sina sin,3 coshc= cosa cos,3 +cos7, where a, p, y are the interior angles from which the sides of length a, b, c are seen, respectively. The latter relation reveals that the length of sides depends only on the interior angles. More counterintuitive features occur with the area. To define area, one needs a measure. The riemannian measure derived from the Poincare differential ds = y11dzI on IH[ is expressed in terms of the Lebesgue measure simply by (1.12)
duz = y2 dx dy.
It is easy to show directly by (1.8) and (1.11) that the above measure is Ginvariant.
Theorem 1.3 (Gauss defect). The area of a hyperbolic triangle with interior angles a, ,Q, y is equal to (1.13)
7ra0 y.
1.3. Modular functions
11
that two pairs of complex numbers (linearly independent over R) determine the same lattice if they are connected by a iinirnodular transformation wi = awl + bw'2 w2 =cwl+dw2
where a, b, c., d E Z with ad  be = ±1. This is so because the matrix of coefficients } = C a d) is invertible; indeed y1 = ± ( d ab) has integer entries. We can interchange the generators w1, w2, if necessary, so that Imwl/w2 > 0, i.e. we can require the point z = w,/w2 to be in the upper halfplane
IHI={z=x+iy:xEIR,yEI18+}.
(1.36)
Then the action of unimodular transformations on lattices A = w1Z + w2Z corresponds to the action of the modular group (1.37)
SL2(Z) _
a
b I
: a, b, c, d E Z, ad  be = 1 I
on IHI by the linear fractional transformations (1.38)
yz = a + d
if y  +
a
d
Suppose f (u; wl, w2) is an elliptic function in u for a lattice A = w1Z + Suppose w2Z which is homogeneous in r and A of degree k E Z, i.e.,
f (au; Awl, ,\w2) = A" f (u; WI , L02) for any A E C, ,\
0. Then we can write /
f (u, wl, w2) = w2
kf
uw w2 w'2
where f (v; z) is a function on C x H. As a function of z it has the property (1.39)
f (yz) = (cz + d)' f (z)
for all y c SL2(Z).
A function f : IHI > C satisfying the transformation rule (1.39) is called a modular function of weight k. Note that k must he even, otherwise f (z) vanishes identically. Clearly any modular function is obtained from a homogeneous function of lattices in the above Fashion.
1. The Classical Modular Forms
12
In particular we obtain Gk = Gk (w1, w2) = w2 kGk (w1/W2) where (1.40)
Gk(z) _ E (mz + n)k (r..n)#(0,0)
provided k > 2 for convergence. This is a modular function of weight k, called the Eisenstein series. Arranging terms in accordance with the greatest common divisor d = (m, n), we can pull out from Gk(z) the following constant factor: (1.41)
((k) = E c'O
dk
d=1
This is a special value of the Riemann zetafunction. For k even we have (1.42)
((k) =
 (2k!
k
Bk
where Bk are the Bernoulli numbers; they are defined as the coefficients in the power series k
e'x1
k=0
Then we define the normalized Eisenstein series Ek(z) = Gk(z)/2( (k), which is
(1.43)
Ek(z) = 2 E (mz + n) k. (rn,n)=1
From the discriminant function (1.34) we get a modular function of weight twelve: (1.44)
A (z) = 92(Z)3 2793 (Z)
2
where 92(z) = 60G4(z) and 93(z) = 140G6(z). Therefore the function '1.45)
j(z) _
(1292(z))30(z)1
zas weight zero. The finction j (z) plays an important role in the theory of elliptic curves.
A modular function which is meromorphic on IHI and at oc is called a :Nodular form (later we shall require somewhat stronger analytic conditions). The requirement that f (.z) be meromorphic at z = oo needs a few words
of explanation. The map z H e(z) transforms C onto C* = C \ {0}, and
1.4. The Fourier expansion of Eisenstein series
13
since e(z) is invariant under the translation Tz = z + 1 this map establishes an analytic isomorphism between the strip C/T and the punctured plane C*. Hence a meromorphic function f (z) on IH[ which is Tinvariant (periodic of period 1) can be written as f (z) = g(e(z)), where g(q) is meromorphic in
C*. We say that f (z) is meromorphic at z = oc if g(q) is meromorphic at q = 0. Then the power series expansion of g(q) at q = 0, say
g(q) _
angn,
0,
ant
n=m
reads as the Fourier expansion of f (z) at z = oo. cc
f (z) = >] ane(nt). n=m
The smallest integer m = mf(oe) with am 0 is called the order of f at 00, and the complex numbers an with n > m are called the Fourier coefficients of f at oo. Similarly we define the order of f at points w E IHI, say m = m f(w), by examining the Taylor expansion 00
f (z) _ 1: an(z  w)n,
ant
0.
n=m
1.4. The Fourier expansion of Eisenstein series We begin by the wellknown product representation for the sine function (which is easy to establish by comparing zeros and applying Liouville's theorem)
x sin7rz=7rz11
z
(1n ) (1
n=1
z n
Take the logarithmic derivative 7
cos 7rz
sin 7Z
+E( zn +_z+n ) z 1
1
1
On the other hand, this is equal to 7ri
e(z) + 1 e(z)  1
= 7ri +
27ri e(z)  1
00
e(dz).
= 7ri  27ri d=O
1. The Classical Modular Forms
14
'Ale differentiate these expansions k  1 times, getting 00
(z  n)k
(1.46)
00
29ri k
=
n=oc
(k  1)1
E dk
le(dz)
d=1
for any k > 2. Hence we derive the Fourier expansion of Gk(z) as follows 00
00
(mz + n)k = 2((k) + 2 E
Gk(z) = (m,n)54(O,O)
= 2((k) +
(mz + n) k
m.=1 n=o0 (k2
oo 1)iE k
x
Edk le(dmz),
d,rn.= 1
and by collecting terms with dm = n we arrive at
Gk(z) = 2((k) + 2
(1.47)
(27ri) k
akl (n)e(nz).
F (k)
n=1
Here a5 (n) is the arithmetic function given by
a,(n) = Eds.
(1.48)
din
For the normalized Eisenstein series Ek(z) = Gk(z)/2((k) we get (by (1.42))
Ek(z) = 1  2kBk 1 Eak_1(n)e(nz).
(1.49) Since
o0
[1+240a3(n)e(nz)]
92(x) = (12)
n=1 00
s
1  504 93(x)
(216
a5(n)e(nz)
n=1
we find that in the Fourier expansion of 0(z) = g2(z)3 2793 (Z) 2 the constant terms cancel out, and 0(z) = (2;r)12
(1.50)
T(n)e(nz) n=1
where T(1) = 1. The arithmetic function T(n) is called the Ramanujan function; it possesses fascinating properties, which will be revealed throughout this book.
Exercise. Using the formula (1.46) for k = 2, derive the following Fourier expansion of the Weierstrass function: 8J (u; z) _
(5i
emu) 2

2
+ 167x2
00 (dsin2(du)) e(nz). n=1 \ din
1.5. The modular group
15
1.5. The modular group The modular group SL2(Z) is the first discrete subgroup of SL2( R ) which interests arithmeticians. We begin by proving
Theorem 1.1. The modular group is generated by two matrices
i),
T= (1
(1.51)
S=
Proof. The action of S on IHI (the inversion; is involutary, more precisely S2 = 1. We have
(a
S
d) _ (ac
bd)
so S acts by interchanging the rows up to the sign. The matrix T''2 _ (1 acts by translation, a
b
_
(c d)(
1
)
a+cn b+dn d
c
If c # 0 this has an effect of reducing the left upper entry to 0 < a < Icl by a suitable choice of n E Z. Applying both operations repeatedly, we end up Then applying with a matrix having c = 0, which must be of type Tm we arrive at ±1. Finally we can change 1 to 1 by applying S2. Remark. The procedure described in the proof of Theorem 1.1 follows the steps of the continued fraction expansion of
Theorem 1.2. The set (1.52)
D = { z = x + iy : xj
1 }
is a fundamental domain for the modular group F = SL2(Z), i.e. it has the following properties:
 D is a domain in IH,  every orbit of F has a point in D or on the boundary aD,  distinct points in D are not in the same orbit of F.
Proof. By (1.51) each orbit Fz has a point with largest height (the imaginary part). Such a point, say z, has the property that
Icz+dl >1for all ry= (*
d) EF.
1. The Classical Modular Forms
16
00
P
p,
i
1
1/2
1/2
Figure 2. A fundamental domain for the modular group
This maximal property is preserved by translations (which are represented by matrices with c = 0), so we can choose a maximal point of the orbit in the strip jxj We shall show that the domain r2.
D'={zEIH[:1xI1forallc,dwith c$0} coincides with D. Indeed, D' C D by choosing c = 1, d = 0. Conversely, if zEDandc 0 then cz + dl2 = c2 z12 + 2cdx + d2 > c2  jcdj + d2 , 1, so D C D'. From the construction of D' it follows that distinct points of D' are not equivalent and the closure D' = D' U 3D' contains points of every orbit. Thus D = D' is a fundamental domain. Let D be the standard fundamental polygon (1.52). We shall show that the equivalent points of the boundary 3D are exactly those pairs of points which interchange upon reflection in the line x = 0, and they are identified by the transformations T and S. Indeed, if both z and yz are on 3D then 1 = Icz + d2 = (cx + d)2 + c2y2 '> c2  jcdj + d2 , 1; hence either c = 0, d = ±1 or c = +1, d = 0, giving T or S respectively. Observe that D has a parabolic vertex at oo, an elliptic vertex at i of order m(i) = 2, and two equivalent elliptic vertices at p = 1 2z ' and p' = 1+2
of order m(p) = m(p') = 3 (see Figure 2 and Section 2.2 for a general definition).
Theorem 1.3. Let f $ 0 be a modular form of weight k , 0. Denote by m f (w) the order off at w. If w is an elliptic fixed point (of the modular grcup) denote its order by m(w), and put m(w) = 1 otherwise. We have
L m(w)
w (mod Pl
12
1.5. The modular group
17
Figure 3. The cut polygon
Proof. Cut off the upper part of the standard fundamental polygon D along a horizontal segment of very large height. Make very small circular cuts at the elliptic vertices and at any singular point of f which lies on the boundary aD in such a way that the equivalent points have congruent cuts. Let P denote the resulting cut polygon (see Figure 3).
Integrate fl f along the boundary aP of the cut polygon, getting by Cauchy's theorem
12,7rif f , (z)dz = 8P
'rnf (w). wEP
Note that at(w) = 1 for any w E P. On the horizontal segment (we can assume that it does not contain singular points) wve see from the Fourier expansions 00
f(z)
ane(nt),
n=m
f'(z) _ n=m
where a,,,, 0 0 for m = m f (oo) that
f,
f (z) _
or bee(Ez) 1=0
with bo = 2lrimf(oo). Hence the integral over the horizontal segment is equal to mf(oo). The two integrals along equivalent vertical segments with congruent cuts at singular points cancel out by periodicity. At every elliptic vertex w = p, i, p' we have ' (z) = 'nf( U)
f
+ h(2)
1. The Classical Modular Forms
18
where h(z) is holomorphic in a small disc centered at w. Hence the integral over the arc around w tends to p(w)m f(w) as the radius tends to zero, where 21rp.(w) is the arc angle. We have and µ(p) = l1(P) =
1 µ(Z) = 12 = m(i)
1
=
6
1
2m(p)
It remains to evaluate the integral along the arc of the circle Izi = 1 at the bot ,om of P with the elliptic points p, i, p' and other singular points being removed. The middle point i splits the arc into two equivalent arcs, say A and A' = SA. We have
f(Sz) = zkf(z) kzkl.f f'(Sz)z2 = (z) + zkf,(z)
f
(SZ)z2 =
k Z
+
f' (z).
Hence 2iri A'
f
(z)dz =
27ri
f
.f'(Sz)dSz =
If k A
A
and
kdz _ k
1
1
f (z)dz = 27ri I
27ri AUA'
f'
(z + f (z)) dz,
27ri
z
12
A
because the are A has onetwelfth the length of the whole circle jzI = 1. Gathering together the above evaluations, one obtains the formula (1.53).
Remark. It would have been more appropriate to perform the computat ions in geodesic polar coordinates at elliptic vertices, to avoid taking limits.
1.6. The linear space of modular forms From now on we require any modular form to be holomorphic in H and at 00 (before, at co we required only meromorphy). The Eisenstein series Ek(z)
is a modular form of weight k for every even k > 2, and the discriminant function 0(z) is a modular form of weight 12 in this stronger sense. But j(z) fails to be holomorphic at z = cc, though it is holomorphic in H because A (z) does not vanish. Let Mk denote the linear space of modular forms of weight k > 0 which are holomorphic in H and at cc. Clearly modular forms of different weights
1.6. The linear space of modular forms
19
are linearly independent over C, so the space of all modular forms is the direct sum of the Mk,
M = ®Mk. k>O
The whole space M can also be considered as a, graded algebra with respect to the inclusions MkMe C Mk+P
We shall use the formula (1.53) to examine the structure of M as an algebra over (C. Since f E M is holomorphic, all terms of (1.53) are non
negative. First let us consider each space Mk for a few small, even values of k separately.
Case k = 0. Then m f(z) = 0 for all z, so f (z i is constant, and (1.54)
M0 = C.
Case k = 2. Then m f(z) = 0 for all z, so f (z''i is constant, and necessarily zero, and (1.55)
M2 = 0.
Case k = 4. Then m1(p) = 1 and mf(z) = 0 for'z
p(mod F). This means f (z) has a simple zero at p and no other zeros mod F. In particular, one knows that G4(z) has a simple zero at p acid no other zeros. Therefore for any f (z) E M. we can find a constant c such that f (z) cG4(z) vanishes at oc, so it must vanish identically, i.e., f (z) = cG4(z). This proves (1.56)
M4 = G4(z)(C.
Case k = 6. Then m f(i) = 1 and m1(z) = 0 :'or z t i (mod F). The same argument as before shows that f (z) = cG6(z) and (1.57)
M6 = G6(z)C.
Case k = 8. Then m f (p) = 2 and m f (z) = 0 for z f (z) = cG4(z)2 and (1.58)
M8 = G4(z)2(C.
p (mod r). Hence
1. The Classical Modular Forms
20
Case k = 10. Then m f (i) = 1, m f (p) = 1 and m f (z) = 0 for other inequiv
alent points. Hence we argue as before to show that f (z) = cG4(z)G6(z) and Mlo = G4(z)G6(z)C.
(1.59)
Case k = 12. Recall that 0(z) does not vanish in H and it has a simple zero at z = oc (see (1.23) and (1.50)). Since G12(oo) 0 there exists a constant c such that f (z)  cG12(z) vanishes at oc, so f (z)  cG12(z)
A(z)
E MO = C
and f (z) = b0 (z) + cG12(z). This proves that .M12 = 0(z)C ® G12(z)C.
(1.60)
Similarly by induction we prove that (1.61)
Mk = 0(z)Mk12 © Gk(z)C,
if k > 12.
From the above results we conclude
Theorem 1.4. The dimension of the space Mk is given by (1.62)
dim Mk =
[kj 12]
if k  2 (mod 12)
[k/'12] + 1
if k 0 2 (mod 12).
Next we show that any f E Mk is a polynomial in G4 and G6. This is already proved for k < 10. If k > 12 we subtract from f a monomial in G41 G6 of special type, namely we consider
f
where 4a + 6b = k,
with a suitable constant c such that the resulting function vanishes at oo. Then divide by 0, getting a function in Mk12, and the proof is completed by induction. We shall also prove that G4, G6 are algebraically independent. Suppose to the contrary that P (C47 G6) = 0
for some polynomial P 0. Since there is no linear relation between functions of distinct weights, the monomials in P (G4i G6) have the same weight.
1.6. The linear space of modular forms
21
Therefore the above algebraic relation must contain one of the following type:
G4 +G6Q(G4,G6) := 0 Gm
+G4Q(G4,G6)0 Q (G4, G6)  0
where Q is a polynomial 0 0 of degree smaller than that of P. But the first equation is false because G6(i) = 0 while G4 ;i) # 0 (see the above cases k = 4 and k = 6 respectively). Similarly the econd equation is false. The third equation yields a new algebraic relation of smaller degree; whence one completes the proof by induction.
Theorem 1.5. The Eisenstein series G4, G6 are algebraically independent and generate the space of all modular forms, that is M = C[G4i G6]
 the polynomial ring.
A modular form f E Mk is called a cusp form if it vanishes at oo, i.e., its Fourier expansion has no constant term,
f (z) _
ane(nz). n=1
For example, A(z) is a cusp form of weight 12. The Eisenstein series Ek(z) is not a cusp form. The linear space Sk of cusp forms together with Ek(z)C gives (1.63)
Mk = Ek(z)C ®SA
if k > 2. It follows from (1.62) that there are no cusp forms of weight k S 10, and for k > 12 the space Sk has dimension one less than Mk.
Since Ek(z)Et(z)  Ek+t(z) vanishes at z = oc, it has to be a cusp form or vanish identically. From this observation we obtain the relations E42(z)
E4(z)E6(z)
E8(z), Elo(z),
E6(z)E8(z) = E4(z)Elo(z)
E'14(z),
E6 (Z)2
c:,(z).
 E12(z)
Comparing the Fourier coefficients (see (1.49) and (1.50)) we infer from the first relation that, (1.64)
cr (n) = o,3 (n) + 120 T 0 0.
We denote by GL,; (18) the group of 2 2 real matrices of positive determinant. Observe that a linear fractional transformation g determines the
matrix (n d) E GL.; (f8) up to a scalar bec a,use the matrices (a ) with a : 0 give the identity transformation. Dividing by a scalar, we can represent g by a matrix of determinant 1. The group SL2 (JR) of real matrices of determinant 1 contains two elements 1 = (1 } and 1 = (1 1) which act trivially. and the factor group 1
SL2(R)
is identified
1}
the linear fractional transformations. 23
2. Automorphic Forms in General
24
The linear fractional transformations are conformal mappings (preserve angles), and they transform a circle into a circle subject to the convention that a straight line is a circle passing through oo. Of course, the center of s. circle may not be mapped onto the center, save for g which represents a translation. For any g = (* d) E SL2(R) we introduce the function (2.2)
j9(z) = cz + d.
This satisfies the chain rule of differentiation (2.3)
igh(z) = jg(hz)jh(z)
We have (2.4)
gz  gw = (z  w)j9(z)1j9(w)1;
whence (2.5)
j9(z)2. clzgz =
Naturally I jg(z) I 2 = I cz + dl 2 is called the deformation factor of g at z.
Suppose g = (a d) E SL2(R) with c 54 0. Putting w = oo in (2.4), we get
gz= a
 c(cz + d)' c 1
(9a) (+) By (2.4) we also see that I gz  gwl = I z  wI if both points z, w are on the curve (2.6)
CQ = {z E 111
: Ijg(z)I =1},
which is the circle centered at d/c of radius IcI1. One can distinguish the exterior from the interior of Cg by considering all points of deformation
< 1 as the exterior. It is easy to see that g maps Cg to Cgi and reverses the interior of Cg onto the exterior of C9i (see Figure 4). We shall call Cg the isometric circle of g because g acts on Cg as a euclidean isometry (it preserves the distance). It follows from the formula (2.7)
Ij9(:)I2 In. qz = lm z
2.1. The hyperbolic plane
25
C9
C91
Figure 4. Isometric circles
that the complex plane C splits into three invariant subspaces of the group SL2(1R), namely the upper halfplane (2.8)
lE
z=x+iy:xEJ,yEW'},
the lower halfplane IElI (complex conjugate) and the real line IR = IR U {oo} (the common boundary of IEiT and H). Moreover we have (combine (2.5) with (2.7)) (2.9)
(Im gz)lidgz = (Im z.
dzJ
which shows that the differential form (2.10)
ds2
=
y2(dx2 + dye)
on TEE is invariant under the group GL2 (IR). With the metric derived from this differential the upper halfplane becomes a Riemannian manifold. This means that the distance function on IHZ is given by
p(z, w) = n n) (x '(t)2 + y,(t)2) y(t)idt JO
where L = [x(t), y(t)] ranges over smooth curves in IHI joining z with w. More explicitly. we have the relation (2.11)
coshp(z, w) = 1 + 2u(a, w)
with (2.12)
u(z, w) =
wl _ 41m  . Im i, z
2. Automorphic Forms in General
26
0
Figure 5. Geodesics in IHI
The upper halfplane I11 equipped with the above metric is the Poincare
model of a hyperbolic plane (curvature 1). The invariance of ds2 implies that the linear fractional transformations are hyperbolic isometries of H. In addition to these isometries we have the reflection in the imaginary line (2.13)
which reverses the orientation (thus it is not analytic).
Exercise. Show using the above asserted properties that the whole group of isometries of IH[ is generated by the following ones: z  z + t (translation). t G R, (dilation), p e RR', z * pz z 1/z (inversion'. z a (reflection). Show that the hyperbolic lines (geodesics in IFI[) are represented by the euclidean semicircles and half lines orthogonal to I 2, iii g i s e lli ptic J a + d l < 2.
2.3. Discrete groups  Fuchsian groups The group M2 (IR) of 2 x 2 real matrices is a vector space with a norm given by
IIgII =a2+b2+c2+d2
if(a b) c
d
Besides its usual properties this norm satisfies IlghtI < IIgillihil The norm gives a metric topology to SL2 (R) through the embedding into M2 (R) A subgroup r c SL2 (]R) is discrete if the induced topology in r is discrete, i.e. .
the sets {yEr:flyII< r} are finite for any r> 0. Let X be a topological space (Hausdorff) and r be a group of homeomorphisms of X acting on X (transitively). We say that r acts on X discontinuously if the orbit
rx={'yx:yEr} has no limit point in X. In particular the stability group
ry={yer:yx=x} of a point x E X is finite. Moreover, for any x E X there exists a small neighborhood U of x such that
{yEF:yUnU
O}=rx.
If two points z, w are not requivalent, i.e. are not on the same orbit, then there are neighborhoods U and V of z, w respectively such that yU n V = 0 for all y E F. We now return to the hyperbolic plane X = IHI. A subgroup IF c PSL2(R) acting discontinuously on H is called a Fuchsian group. In this case Poincare showed that a subgroup of SL2 (lid) is discrete if and only if it acts discontinuously on H (when considered as a subgroup of PSL2(R)).
2. Automorphic Forms in General
30 is
Remark. A discrete subgroup of SL2(R) acts on the whole of C, but not. necessarily discontinuously.
There is a multitude of Fuchsian groups, a lot more by comparision with the discrete groups of motion on R2. The essence of this fact is that PSL2(]R) is nonabelian. Let us mention the following result off. Nielsen: if F C PSL2 (IR) is nonabelian and hyperbolic (it contains only hyperbolic
elements and the identity), then P acts discontinuously on IHI, so it is a discrete group.
Proposition 2.1. Let r be a Fuchsian group. For any z E C the stability group Fz is cyclic (but not necessarily finite if z E R).
A Fuchsian group r is said to be of the first kind if every point on the boundary 8IHI = IR is a limit (in the Ctopology) of an orbit Fz for some z E H. Clearly, any subgroup of finite index of a Fuchsian group of the first kind is also a Fuchsian group of the first kind. But a Fuchsian group of the first kind cannot be too small. Obviously, it cannot be cyclic, and a fortiori the stability group Fz of a point z E C is not of the first kind. A Fuchsian group can be visualized by its fundamental domain. A set D C IHI is called a fundamental domain for F if  D is a domain in IHI,
 distinct points in D are not equivalent,  any orbit of r contains a point in the closure D (with respect to the Ctopology). There are many ways of constructing fundamental domains. Suppose r
is a Fuchsian group of the first kind and w E IHI is not fixed by any y E F other than the identity motion. Then the set
D = {zEH: p(z,w) F \ H is continuous; it is a connected Hausdorff space, and with properly chosen analytic charts it becomes a Riemann surface. Specifically, given a point z E H, consider discs U,z C H sufficiently small so
that for all 7 E F,'y Fz 'yUz = Uz for all 'y c rz.
yU,z fl U,z = O
The stability group Fz is cyclic of finite order, say m > 1. Choose Tz E SL2(C) which maps z to 0 and transforms Uz onto the unit disc
U= {zEC:IzI 0:
(*
I E oa'rub
c
J
Let c(a, b) denote the smallest element of C(a, b), and put c(a) = c(a, a). That c(a) exists is seen from the construction of the standard polygon for the group oa 1rua; c(a)1 is the radius of the largest isometric circle. Hence c(a, b) also exists; in fact one can show that (2.34)
c(a, b)2 >, max{c(a), c(b)} = cab, say.
Since the standard polygon contains the semistrip {z : 0 < x < 1, y > c(a)1} whose area is c(a), it follows that c(a) is bounded by the area of the fundamental domain, c(a) < ID1.
(2.35)
Proposition 2.8. For any X > 0 we have c1 {ci(mod c) : (c d) E oa1FUb }
(2.36)
2, then we have .
where (g; m1, , mj; h) is the signature of F. Hence by the GaussBonnet formula (2.17) we get the bound . . .
dimMk(I') < (47r)1IDlk+1. In full generality one can prove that (2.68)
dimMk(F,19) < c(k + 1)(IDI + 1)
where c is an absolute constant.
2.8. The etafunction and the thetafunction We shall conclude our general considerations by giving two examples of automorphic forms of weight 1/2 to present two interesting multiplier systems. The first one is constructed from the discriminant function (2.69)
0(z) = (27r)12e(z)fl(1  e(nz))24, 1
2.8. The etafunction and the thetafunction
45
which is a cusp form of weight 12 with respect to the trivial multiplier on the modular group. Since 0(z) does not vanish in IH[, one can define for every k > 0 a holomorphic function f (z) = (27r)k0(z)k/,2,
which is a cusp form of weight k, but, of course, its multiplier system is no longer trivial. For k = we obtain the etafunction 2
i(z) = e (_z 2) fl(1  e(nz)).
(2.70)
oo
Dedekind determined the multiplier system for q(z). Precisely, we have 7)((z) = 19(y)jy(z)27)(z)
(2.71)
if y c SL2(7G)
where 19(y) = e(1/4)19(y) for any y, 19(y) = e(b/24) if y = (1 b) and a + d  3c
t9(y) = e
24c

1
2
s(d c)
)
a
if
c
bl ,c>0.
d
Here s(d, c) denotes the Dedekind sum
s(d' c) _ O 0.
The Fourier expansion of 77(z) is a lacunary series e(n2z/24)
77 (z) =
n=±1 (mod 12)

E nf5 (mod 12)
e(n2z/24).
2. Automorphic Forms in General
46
The inverse
rj(z)1 has the Fourier expansion given by 00
00
e(z/24)77(z)1 = fl(1 
e(nz))1
= > p(n)e(nz) 0
1
where p(0) = 1 and p(n) is the number of unrestricted partitions of n > 0 into sums of positive integers. The Fourier coefficients of are quite i7(z)1 has weight z but it is not holomorphic at oo). large (note that G.H. Hardy and S. Ramanujan invented the circle method, by means of which they established the asymptotic formula 77(z)1
p(n)  (4,43n)  1 exp (r 2n/3)
as
n  oo.
A close relative of the Dedekind etafunction is the classical thetafunction defined by its Fourier series:
8(z) = E e(n2z).
(2.72)
00
It is an automorphic form (not a cusp form) for the group ro(4) of weight k = 1/2 with respect to a multiplier system slightly different from that of the etafunction. Precisely, we have (see Theorem 10.10) (2.73)
8('Yz)
=
Ed
(C) d,7.y(z)ao(z)
if y = ( c db) E ro(4)
where (2.74)
Ed =
51
ifd = 1(mod 4)
i ifd  1(mod 4) and (a) denotes the extended quadratic residue symbol; namely, it is the Jacobi symbol if 0 < d  1 (mod 2) extended to all d  1 (mod 2) by (2.75)
d (0
(2.76)
d)
if c # 0
d
Icl 1
{0
ifd=fl otherwise.
The thetafunction has the Jacobi product representation 00
(2.77)
0(z) = fl(1  e(nz))(1 + e((n + 1/2)z))2 1
which shows that 0(z) does not vanish in H.
Remark. The theta multiplier system coincides with the eta multiplier system on the subgroup r(24).
Chapter 3
The Eisenstein and the Poincare Series 3.1. General Poincare series A very important class of automorphic forms is constructed by the method of averaging. Suppose a is a cusp for r which is singular with respect to a multiplier system e9 of weight k. Let p : H ' C be a holomorphic function which is periodic of period 1. Define 7r: r x H  C by (3.1)
ir('y, z) = 9(y)'W (aa 1,'Y) joa 1ry(z)kp (Qa 17z)
Actually ir(y, z) depends only on the coset ray. To prove this, consider 7r (y', z), where 'y' = iy with rl E ra, so 17 = va/3va 1, where Q is an integral translation. We have p (ca ,7,z) = p (Qka "yz) = p (ca 1`yz)
ja.
(z)k
11,(z)k
j".
l.r(Z)k
t9('y') = i9(t17) = w(17,'Y)'9(7) by (2.53) and
w(71,1')w (cn 1,1'x) = w (aa , 7)
by (2.45). Collecting these results, we arrive at n(y, z) = ir(ry, z). This property allows us to write without ambiguity the infinite series
pn(z) = E irl'y, z) 7Ero\r 47
3. The Eisenstein and the Poincare Series
48
provided it converges absolutely. For example, if p(z) is bounded the series (3.2) is majorized by 17(z),_k
Ijao
k
(Im oalryz)'
= yk
,
yEro.\r
'YEr,\r
and this converges absolutely if k > 2 by virtue of Proposition 2.9. For any singular cusp b we deduce the following:
Palab(z) = jab(z)kPa(abz)
(Z) k E
=
7r ('Y,abz)
7Era\r iab
(Z) k
1
,
?f (aaryab 1 , abz)
7EB\aatrab =lab
(z)k
(aaryab 1) w (aa 1, aa'Yab 1) i7a61(abz)kp(7z)
7EB\ap lrab
Since ja(z)'kj7a1(az)k = w(rya1,a)j7(z)k, this gives
Pa1,b (z) = E
t9ab(7)j7(z)kp('Yz)
7EB\aa 1rab
where (3.4)
l9ab('y) = t9 (aa1'97b 1) w (aa
aaryab
w (ryab 1,ab)
Using the relation (2.41), one can derive a handful of expressions for i9ab(ry). For example, we have (3.5)
i9ab('y)W(aa,'Y) =19 (aaryab') to (aaryab n, 0'b)
Hence, in particular, for a = b the series (3.3) becomes (3.6)
Paiaa(z) _ E
'j'('Y)j7'(z)kp('Y'z)
' Er, 00 \r, where t9' is the multiplier system for the conjugate group I" = aal'aa given by (3.7)
19'(ry')w(aa, 7) = '9('y)w(ry, aa)
if ry' = as lryaa.
The function Pa(z) defined by (3.2) is called the Poincare series associated with the cusp a and the generating function p (a is required to be singular with respect to the multiplier system, and p is periodic such that the series (3.2) converges absolutely).
Proposition 3.1. The Poincar6 series Pa(z) is an autotnorphic form, P0Ir(z) = t9(T)Pa(z) if T E F. (3.8)
3.2. Fourier expansion of Poincare series
49
Proof. By conjugating the group we may assume that a = oo and aoo = 1, in which case the series (3.2) looks simpler, namely Poo(z) =
(3.9)
9(Y)jry(z)kP('Yz)
ryEra,\r
Hence for r E IF
Poo(rz) _
F, '9('Y)jy('rz)kP(YTZ)
.yEr.\r
= C`
j7r_i (rz)kp(yz).
ryEroo\r
Here we have
(yri) = w('y, T1)d(7)d(T1) = j.yr1(Tz)k = w(7,r1) jry(z)k jri
w('y,r1)w(,r, ri) 9(r),y(7),
(rz)k = w('y,T'),w(,r,r1) jr(Z)k j l(z)k,
and by these expressions we arrive at (3.8). Clearly Pa(z) is holomorphic in H. To prove the holomorphy at cusps we need to expand Pa(z) in Fourier series.
3.2. Fourier expansion of Poincare series Let a, 6 be singular cusps for a multiplier system t9 on r. We seek the Fourier expansion of Pa(z) at the cusp b, i.e. for the series (3.3). Applying the double coset decomposition (2.32), we split the series into Po1g (Z) = aabP(Z) +
l9ab(7)Ir(Z) 1; 7ES\oo 'rab/B
where the first term comes from the contribution of 7 = 1 (which exists only if a = b), and for any 'y = ( :)Ean'rcbwithc>Owehave
I.y(z) = > jyr(Z)'P(7rz) rEB nEZ nEZ
c z+n +d )
P
)
a
1
c
c(c(z + n) + d)
By Poisson's summation we get
I.y(z) = E J uEZ
(c(z + v) + d) kp I 00
a
 c(e(z + v) + d)) e(nv)dv.
3. The Eisenstein and the Poincare Series
50
In the sequel we specialize the generating function to p(z) = e(mz)
where m is a nonnegative integer. For this function we can compute the Fourier integral quite explicitly. First by a linear change of variable we obtain / ma C nd)\
1,(z) E e I nz + nEZ
where
Jc(m., n) =
roo+iy
J
/ m
(ev)ke I
\\
oo+iy
(TV
 nv I dv.
Notice that this integral does not depend on y by Cauchy's theorem. If n < 0, then, moving the horizontal line of integration upwards, we see that the integral vanishes,
3c(m, n) = 0
if n 0 but m = 0, then we have (see [GR], 8.315.1) (3.10)
nk1
27r
J.(0, n) = (ic)
T(k)
For n > 0 and m > 0 we have (see [GR], 8.412.2) (3.11)
Jc(m, n) =
21r
i
(n) a
Jkl
(4)
where J.y(x) is the Bessel function of order v, defined by 00
(3.12)
(1)t :C V I.2f J (x) = F Q!r(e+ 1 + v) (2) e=o
Exercise. Derive (3.11) from (3.10) using power series expansion for e(z).
Collecting the above computations, we obtain the desired Fourier expansion for the Poincare series generated by the function p(z) = e(mz), namely
P0!, (z) = Sahe(mz) + > e(nz) n=1
Spb(m, n; c)J,(m, n) n>n
3.2. Fourier expansion of Poincare series
51
where J (m, n) are given by (3.10)(3.11), and Sab(m, n; c) is the Kloosterman sum defined by (3.13)
''ab(l)e
Sab(m, n; c) _
/ ma + nd (
\
c
'Y=( d)EB\°O'rab/B
Recall that i9ab(y) is given in terms of the multiplier system by (3.5). Since there are no negative terms in the Fourier expansion at any cusp, it proves that the Poincare series is an automorphic form in our strict sense. For m = 0 we denote Pa(z) by (3.14)
Ea(z)
(7)w
_
a
1,'1) 700
y(z)k
'yEr,\r
which is called the Eisenstein series of weight k. This has the Fourier expansion 00
(3.15)
jab (z)kEa(O bz) = bab + > rlab(n)e(nz) n=1
with k
(3.16)
%b (n)
(2z)
k
1
T(k) 0 c ksab(0, ni c).
For m > 0 we denote Pa(z) by
(3.17)
Pam(z) _ E ('Y)'w (Qa 1, ry) joo ,Y(z)ke (mca 1ryz)
7Er,\r and call Pa,,,(z) the mth Poincare series of weight k. This has the Fourier expansion 00
(3.18)
jab(z)kPam(Qbz) =
ab(m,9b)e(nz) n=1
with (3.19)
pab (m n) r l
()
kI
/41r
k b abbmn + 27riE c1Sab(m, ni c)Jk1 l\
c>0
Since there is no constant term in the Fourier expansion (3.18), we obtain
Proposition 3.2. For m > 1 the Poincare series
is a cusp form.
3. The Eisenstein and the Poincare Series
52
3.3. The Hilbert space of cusp forms Let f, g be automorphic functions with respect to a multiplier system '0 of weight k for a group r. Thus fh7 = &(y)f,
g17 = 'd(7)g
for all y E r. Hence the expression (a measure on IIll) (3.20)
(f, g) (Z) = ykf (z)9(z)dµz
is rinvariant. Therefore we can define the inner product (due to H. Petersson) (3.21)
(f, g) =
J\a(f, g) (Z)
provided the integral converges absolutely. For f = g we set
r\
IIf112 = (f, f) = f Ykif(z)I2dµz. A cusp form f has exponential decay at cusps, so IIf II < oo. The linear space of cusp forms Sk(r, i9) equipped with the Petersson inner product is a finite dimensional Hilbert space.
The Poincare series Pa,,,(z) associated with a singular cusp a and a positive integer m belongs to Sk(r, z9). We shall compute the projection of
any f E Mk(r, i9) onto Pa,,, by the unfolding technique. Without loss of generality we can assume that a = oo and as = 1 (take the conjugate group a; lraa if necessary). In this case we obtain
(f,
_k
f \MYkf (z)
f
e(myz)dpz
(IM yz)k f (yz)e(myz)dµz \8i 7Er.\r 0o
I
= f f yk f (z)e(mz)dµz. 0
o
Inserting the Fourier expansion of f at cusp a = oo, say
f (z) _
=o
f (n)e(nz),
3.3. The Hilbert space of cusp forms
53
we get a contribution from the mth term only: 00
(f, Pr) = f (m) J
yk2 exp(41rmy)dy
= r(k  1)
0
(41rm)k1
f
In general, for a singular cusp this argument yields
Theorem 3.3. Let Pp,,, be the mth Poincare series attached to a singular cusp a for a multiplier system 0 of weight k > 2. Let f be an automorphic form with respect to the same multiplier system. Then (3.22)
(f, 1'asn) =
r(k 1)1 (4Trrrn) '
fa(m)
where fp(m) denotes the mth Fourier coefficient off at the cusp a, i.e. one has the expansion
fp(n)e(nz).
(z) _ n=0
Combining (3.22) with (3.19), we obtain Corollary 3.4. Let a, b be singular cusps for a multiplier system i9 of weight k > 2. For m, n positive integers we have (3.23)
(1 am, Pbn)
(4uj)k_1
c
Sab(m, n, c)Jk1
c>O
In particular, (3.24)
=
((4irm){1+2iri>c'Saa(mm;c)Jk.l k  1)
rli1 c
}
c>0
Corollary 3.5. Let a be a singular cusp for a multiplier system 17 of weight k > 2 on r. The space of cusp forms S1. (r, i9) is spanned by the Poincare series Pp,,, for m = 1, 2, 3, ... .
Proof. The linear space spanned by Pa,,, is closed in Sk(F, i9) because the whole space has finite dimension, and a function orthogonal to this subspace must be zero because all its Fourier coefficients vanish by virtue of (3.22). This proves Corollary 3.5. Since Sk(I', z9) is spanned by with in = 1,2,3,... and on the other hand s, (r, d) has finite dimension, it. follows that the Poiucar6 series
3. The Eisenstein and the Poincar6 Series
54
are linearly dependent. There are many open problems about Poincarb series such as:
Problem 1. Find all the linear relations between Pam(z). Problem 2. Find a basis of Sk(r, t9) consisting of the Poincar6 series. Problem S. Which Poincar6 series do not vanish identically? For the modular group it is known that the first Poincar6 series Pm(z) with m < dim Sk (r) span the space Sk(r). Recall that dim Sk(r) = k/12 +
0(1) by Theorem 1.4. R. Rankin has proved that Pm(z) does not vanish identically if m < c(e)k2f. C.J. Mozzochi extended this result for the group ro(q), showing that P,,,(z) # 0 if m < c(e)(kq)2. These results depend on Weil's bound for Kloosterman sums.
Exercise. Prove in general that Pam(z) does not vanish identically for any m < ckIDI, where IDS is the volume of the fundamental domain. Hint: estimate (3.24) using the trivial bound for Kloosterman sums (4.2), and apply (2.35). Choose an orthonormal basis of Sk(r,19), say .P, and expand Pam into this basis. By (3.22) we get (3.25)
P..(z) =

r(k  1) (41rm)k  1
f a(m)f (z)
fEF
where fa(m) denotes the mth Fourier coefficient of f at the cusp a. Comparing the nth Fourier coefficients at the cusp b on both sides of (3.25), we obtain by (3.19) Theorem 3.6. Let a, b be singular cusps for a multiplier system 19 of weight k > 2 on r. Let Jr be an orthonormal basis of Sk(r,19). Then for any positive integers m, n we have (3.26)
I'(k 1) (47r mn)k_1
fE
fa(m)fb(n)
=bmnaab+21rikEc 1Sah(m,n;c)Jk_1 c>O
(47 r
mn' Vr
.
e
Remark. For k = 2 the Poincar6 series (3.17) does not converge absolutely; nevertheless, the formula (3.26) remains valid (the series of Kloosterman sums and Bessel functions converges absolutlely).
3.3. The Hilbert space of cusp forms
55
Example 1. Let IF = SL2 (Z) and k be an even integer, k > 2. The Kloosterman sum for the trivial multiplier system and for the cusps at oo becomes the one which was originally introduced by Kloosterman [Klol, namely
ma + nd
S(m, n; c) _
(3.27)
eC
c
ad=1 (mo d c)
For in = 0 this degenerates to the Ramanujan sum 's(0' n; c)
d(mod c) e
\dc
/
E . ()a
6I(c,n)
where the star restricts the summation to the classes prime to the modulus. Hence for k > 2 slk = aki(n)/n k '((k) E C kS(0, n; c) _ ((k) c>0
61n
so by (3.16) the nth Fourier coefficient of the Eisenstein series of weight k is
77(n) =
(27ri)k
((k)r(k) Qk1(n)
This agrees with the previous result (1.49). Now suppose in > 0. Then the Poincar6 series
P,n(z) _
7y(z)ke(m7z)
ryer.\r
can be written more explicitly as
PMW = 2
(cz + d) ke I and (c,d)=1
 c(cz + d) )
The Fourier expansion (3.18) becomes 00
Pm(z) = E p(m, n)e(nz) n=1
where
(!)
p(m, n) = in
{6rnn +
27rik
E
c1S(in,
47r cmnl
n; c)JkI
,
c>o
and S(m, n; c) is the original Kloosterman sum (3.27). By the Selberg relation (4.10) for S(m, n; c) and by a similar relation (3.33) for b117 we deduce
that (3.28)
p(m, n) =
dk1 p (1, mnd2)
in1k
.
dl(m,n)
This formula belongs to the theory of Hecke operators (see Chapter 6).
3. The Eisenstein and the Poincarc Series
56
Example 2. Let r = SL2(Z) and k = 12. The space of cusp forms S12(r) is onedimensional, spanned by
O(z) = (27r) 12 E r(n)e(nz). 1
Therefore any Poincar6 series P7 (z) is a multiple of 0(z), say ,,,,(z) c(m)A(z). To compute the constant c(m) we appeal to (3.22), getting c(m)IIOII2 = (0, Pm) = 27rF(11)(2m)"r(m) Hence (3.29)
P,,,.(z) = 27rP (11 ) r(m) A (Z)
(2m)" II0II2
.
Applying (3.26), for f (z) = 0(z)/IIAII we obtain (3.30)
r(m)r(n) = v(mn) z 6m,n + 27r> c1S(m, n; c)J11
(4)
r>0
where v is an absolute constant. One can express this constant by the value at s = 1 of the symmetric square Lfunction associated with f ; namely, we get v = IIIII2/47r13r(11) =
by (13.62). In particular for n > 1 we have (3.31)
r(n) = 27rvn 21 > c'S(1, n; c)Jll
(47r c
c>o
Ramanujan Conjecture (proved by P. Deligne in 1974). Ir(n)I 1. In the spectral theory of automorphic forms one shows that Z(s) has meromorphic continuation to the whole complex splane. This is a very deep result (which is due to A. Selberg, [Se1]) and technically difficult to establish in full generality (cf. (1w2)). 57
4. Ifloosterman Sums
58
4.2. Kloosterman sums for congruence groups Kloosterman sums associated with a congruence group and the trivial multiplier system can be expressed in terms of the classical Kloosterman sum
ma + nd ad=_ 1(mod c)
c
Consider the Hecke congruence group r = ro(q) and the cusps 00, 0 (recall the last paragraph of Section 2.5). The sets of moduli are C(oo, oo) = C(0, 0) = {c = eq : e E N}
C(oo, 0) = C(0, oo) = {c = £ ,/ : e E N, (e, q) = 1} ,
and the Kloosterman sums are given by (see (2.39)) (4.5)
S.. (m, n; c) = Soo(m, n; c) = S(m, n; eq)
(4.6)
So. (m, n; c) = Sao(m, n; c) = S(mq, n; e)
where q denotes the multiplicative inverse of q modulo e, i.e. (4.7)
qq = 1 (mod e).
This notation for the multiplicative inverse will be used throughout the book. Be aware that q depends on the modulus e. Exercise. Let r = ro(q) with q = rs, (r, s) = 1. Consider two cusps, oo and 1 /r with the respective scaling matrices a. _ (1 t) and at /,. Show that the set of moduli are
C(1/r,1/r) = {c = 1q: e E N} C(oo, 1/r) = {c=ervrs :QE N, (e, s) = 1}
and the Kloosterman sums are given by S11,.,11,.(m, n; c) = e ((m  n) s)
S(m, n; eq)
Soo,1/,.(m, n; c) = e (mr) S(ms, n; er). s
4.3. The classical Kloosterman sums
59
4.3. The classical Kloosterman sums In this section we shall evaluate the Kloosterman sum (4.4) for special moduli. First we record the following elementary properties:
S(m, n; c) = S(n, m; c),
(4.8)
(4.9)
S(am, n; c) = S(m, an; c)
(4.10)
S(m, n; c) =
if (a, c) = 1,
dS(mnd2,1; cd1) dj(c,m,n)
The third property is due to A. Selberg [Se2]. In particular this gives (4.11)
S(m, n; c) = S(mn,1; c)
if (c, m, n) = 1.
Moreover we have (4.12)
S(m, n; c) = S(qm, 4n; r)S(rm, fn; q)
if c = qr with (q, r) = 1, where q, r are multiplicative inverses of q, r to moduli r, q respectively. The last property (twisted multiplicativity) allows one to reduce the problem of evaluating S(m, n; c) for any c to that for prime power moduli.
Lemma 4.1. If c = pea with a >, 1 and (c, 2n) = 1, then (4.13)
(2c )
S(n, n; c) = c2 (e
+e
2n)
)
Proof. Put d = a(l + bpa), where a ranges mod pea, a is prime to p, and b ranges freely mod pa. Then every primitive class mod plc is covered pa times. We have d  a(l  bpa) (mod p2c), so
S(n n, p2a)=p a
E e (na \ ++ d+ n (a  d)b)/I
I:.
a (mod p 2a )
E*
a (mod pa)
L
(mod
p)
elna+d
`. P2a l
a=_a (mod pa )
The solutions to the last congruence are a = ±1 + tpa with t ranging freely modulo pa, and a = ±1  tpa. Hence we deduce (4.13).
4. Kloostermazi Sums
60
Lemma 4.2. If c = p2a+1 with a >, 1 and (c, 2n)) = 1, then (4.14)
S(n, n; c) = 2
( c } c2 Re e,e (?)
where (n) is the LegendreJacobi symbol and Ec = 1, i according to whether
c  1 or 1 (mod 4). Proof. Putting d = a (1 + bpa+i), we obtain as before a+a
e n p2a+1)
S(n, n; pea) _
.
a (mod pea+I )
a=a (mod pa)
The solutions are a = ±1 + tpa with t ranging freely mod
pa+i, and a =
±1  tpa ± t2p2a. Hence
e(n
S(n, n; pea+') = Me
2+ t2 2a ) pea
t (mod p°+i )
= 2Re pae
2n
9(n, p)
p2at t
where 9(n, p) is the Gauss sum 2
(4.15)
g(n,p) _ t (mod p)
e
(_) = EP (p) p2.
This formula will be established later in Lemma 4.8. Hence (4.14) is proved. We put both formulas (4.13) and (4.14) (which are due to H. Salie, 1936) in a unified form
Proposition 4.3. If c = pa with J3 > 2 and (c, 2n) = 1, then (4.16)
S(n, n; c) = 2 (c) cz Re Eye
2cn
One can generalize this result slightly for sums S(m, n; c) with c = plj, Q
2, such that (c, 2mn) = 1. We have S(m, n; c) = 0 unless m
(mod c), in which case (4.9) and (4.16) yield S(m, n; c) = S(Cn, Cn; c) = 2 ( In any case we conclude
 c)
c^ Re e,e
(261). c
C271
4.4. Powermoments of Kloosterman sums
61
Corollary 4.4. If c = p3 with Q > 2 and (c, 2mn) = 1, then IS(m, n; c) I 1, m, n we have (4.20)
I S(m, n; c)I
(m, n, c) Z c2,r (c)
where r(c) denotes the divisor function.
4.4. Powermoments of Kloosterman sums Today the Riemann hypothesis for curves, hence the Weil bound for Kloosterman sums, can be established by elementary means (due to S.A. Stepanov, W. Schmidt and E. Bouibieri), yet the arguments are quite involved. It. is much easier to estimate S(m, n; p) on average with respect to na, n. In this section we estimate a few of the powermoments defined by E,.
(4.21)
VA (p) _
Sk(a,1;p)
a (mod p)
By (4.9) we obtain
(p  1)Vk(p) = EY: Sk(a, b;p) abAO (niod p)
J:E
SA(a,b;p)
 (p 1)A
2(1)'`(p 1)
a,b (mod p)
= r'k(p)2r'  (p 
1)
4. Kloosterman Sums
62
where vk.(p) is the number of solutions to the system
(modp) 0
(modp).
For k = 1 we have no solutions, so vl (p) = 0 and V1(p) = 1.
(4.22)
For k = 2 the solutions are x1 = x2  0 (mod p), so v2(p) = p  1 and V2(p) =p2  p  1.
(4.23)
For k = 3 the solutions are x2 = xly, x3 = X1(1 +V) where xl $ 0 (mod p) and y2  y + 1 0 (mod p). The number of solutions in y is 1 + (=3), so
V3(p) = (p  1) (1 + (=3)). Hence p (4.24)
3) p2 + 2p + 1.
V3 (P) _ (
\ p
For k = 4 we write
V4 (P) _ EE 7I(u>v)2 u,v (mod p)
where rj(u, v) is the number of solutions to the system
x+yu (modp), xJ1+y1v (modp). If u=0then v0and r7(0,0)=p1. Ifu 0then v$0,x+y=u, xy  uv1, so x, y are solutions to the quadratic congruence z2
 tiz + uv1 = 0 (mod p).
ty1, so the number of solutions is The cliscriminant. is A = v2  42(A) y(u,v)=1++(14/uv)
p
p Hence
EY: uvOO (mod p)
?(U, V)2
?(U, 1)1
= (p  1)
u O (mod p)
(i+()) =(p1) E ml (mod P) = (p  1) 4L ::l + 1
4
(p  1) (2p  5),
4.4. Powermoments of Kloosterman sums
and v4(p) = (p (4.25)

63
1)2 + (p  1) (2p  5) = 3(p  1) (p  2), giving
V4(p) = 2p3  3p2  p  1.
By (4.25) we infer by dropping all but one term in (4.21) that (4.26)
IS(m,n;p)I 2p 2, whence (4.27)
IS(a,1;p)I > (2p  2)21
for some a (mod p), showing that the constant 2 in Weil's bound (4.19) cannot be replaced by any number < V2. Therefore some of the roots ap are closer to the real axis than to the imaginary one. Incidentally, one can show that S(m, n; p) does not vanish, for S(m, n; p) _ 1
(mod 7r)
where 7r = 1  (p is the prime factor of p in the cyclotomic field Q((p). In 1931 H. Salie and H. Davenport (independently) estimated the sixth powermoment (still elementarily) (4.28)
V6 (p) dµ(c) r l
l
r
e\xc/
y (mod c)
d1c
y24mn (mod d)
If (x, c) = 1 this simplifies to F(x) = g(n, c)
e
F%Zmn (mod c)
Taking x = 1, we obtain the following.
Lemma 4.9. If (c, 2n) = 1 we have (4.44)
E
T (m, n; c) = (c) g(c)
y2ctnn (mod c)
e (v).
Corollary 4.10. If (c, 2n) = 1 we have (4.45)
IT(m,n;c)l 0 for r which is singular at the cusp oo.
5.1. General estimates Let f be a cusp form for r with respect to the multiplier system 19. Thus f has the Fourier expansion
f (z) =
L a(n)e(nz).
Our problem is to estimate the coefficients a(n) = a f(n). Quite strong results are deduced rather easily from the observation that (5.2)
F(z) = y4lf(z)I
is a bounded function on the whole upper halfplane H. This is clear because F(z) is Fperiodic and has exponential decay at every cusp. Therefore, we 69
5. Bounds for the Fourier Coefficients of Cusp Forms
70
have f (z) G< yk/2
(5.3)
if z E H
where the implied constant depends on f. Conversely, if F(z) is bounded in H then for any a E SL2(R) we have I fro (z) I = y12F(az) 0 for a group r. Then f is a cusp form for r if and only if (IM z)1/2I If (z) I is bounded in the upperhalf plane.
By the Parseval formula and (5.3) we obtain 1
la(n)
I2e4any = f If 0
n
(z) I2dx