ON SIEGEL MODULAR FORMS ON

Γ0(N)

Martin J. Dickson

School of Mathematics, University of Bristol

June 2015

A dissertation submitted to the University of Bristol

in accordance with the requirements of the degree

of Doctor of Philosophy in the Faculty of Science

Word count: 27883

Abstract

In this thesis we consider various aspects of the theory of Siegel modular forms on the

congruence subgroup Γ0(N). The basic motivation is that the theta function for higher

representation numbers of a quadratic forms define such modular forms.

In the first half of the thesis we focus on Eisenstein series. First we provide a descrip-

tion of the action of Hecke operators on (Klingen–)Eisenstein series of any degree, but

squarefree level. The tools used are a concrete description of the boundary of the Satake

compactification of Γ(n)0 (N)\Hn, and how the Hecke operators relate to the restriction-

to-the-boundary map. Next we provide some formulas for Fourier coefficients of Siegel–

Eisenstein series of degree two, squarefree level and trivial nebentypus, the key novelty

in these results being that they pertain to a full basis for the space of Siegel–Eisenstein

series. We also explain how these formulas can be used to give explicit, exact formulas

for average representation numbers of certain quadratic forms.

In the second half of the thesis we focus on cusp forms. We prove a result about equidis-

tribution of Satake parameters of Siegel cusp forms as one varies over eigenforms of

increasing weight and level. Along the way we obtain some estimates on the size of the

Fourier coefficients of Siegel cusp forms. In the final chapter we discuss L-functions at-

tached to Siegel cusp forms, and use the equidistribution result to compute the one-level

density attached to the low-lying zeros of the (spin) L-functions, again as one varies over

eigenforms of increasing weight and level.

i

Acknowledgements

Firstly I would like to thank my supervisor Dr. Lynne Walling for her constant support

and open-door policy, and for helping with my various questions about Eisenstein series

and quadratic forms. I would also like to thank Dr. Abhishek Saha for being so generous

with his time, and for helping me to understand the automorphic representation theory

used in the second half of this thesis.

I would like to extend my gratitude to the department at Bristol as a whole and the

number theory group in particular for providing an excellent environment to work in.

This applies especially to my fellow graduate students.

This thesis is dedicated to my parents. I would like to thank them, my sister Louise, and

the rest of my family for their love and support. Finally, I would like to thank Kate, for

everything.

ii

Author’s Declaration

I declare that the work in this dissertation was carried

out in accordance with the requirements of the Univer-

sity’s Regulations and Code of Practice for Research De-

gree Programmes and that it has not been submitted for

any other academic award. Except where indicated by

specific reference in the text, the work is the candidate’s

own work. Work done in collaboration with, or with the

assistance of, others, is indicated as such. Any views ex-

pressed in the dissertation are those of the author.

Martin J. Dickson

Date: June 2015

iii

Contents

Abstract i

Acknowledgements ii

Author’s Declaration iii

Notation vi

1 Introduction 1

1.1 Representation numbers and modular forms: an overview . . . . . . . . . 1

1.2 Representations and L-functions . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Background 12

2.1 Siegel modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Quadratic forms and theta series . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Action of Hecke operators on Klingen–Eisenstein series 25

3.1 Action of Hecke operators on Fourier expansions . . . . . . . . . . . . . . 28

3.2 The intertwining relations for Φ and T(n)j (p2) . . . . . . . . . . . . . . . . 31

3.3 The intertwining relation for Φ and T (n)(p) . . . . . . . . . . . . . . . . . 44

3.4 Review of the Satake compactification . . . . . . . . . . . . . . . . . . . 46

3.5 The Satake compactification of Γ(n)0 (N)\Hn . . . . . . . . . . . . . . . . . 49

3.6 Intertwining relations at arbitrary cusps for squarefree level . . . . . . . . 55

iv

3.7 Action of Hecke operators on Klingen–Eisenstein series . . . . . . . . . . 57

4 Fourier coefficients of level N Siegel–Eisenstein series 64

4.1 The action of Hecke operators on Siegel–Eisenstein series . . . . . . . . . 67

4.2 Calculation of the Fourier coefficients . . . . . . . . . . . . . . . . . . . . 69

4.3 Applications to representation numbers of quadratic forms . . . . . . . . 79

5 Equidistribution of Satake parameters attached to Siegel cusp forms 84

5.1 The representation attached to a Siegel modular form . . . . . . . . . . . 89

5.2 The equidistribution problem . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3 Bessel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4 Estimates for sums of Fourier coefficients of cusp forms . . . . . . . . . . 107

5.5 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6 L-functions, low-lying zeros, and Bocherer’s conjecture 128

6.1 Background on L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.2 Saito–Kurokawa lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.3 Low lying zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.4 Discussion of results and future research . . . . . . . . . . . . . . . . . . 144

Bibliography 148

v

Notation

If R is a ring then Rn×n denotes the set of n×n matrices over R, and Rn×nsym the subset of

symmetric ones. We will realise algebraic groups G as closed subgroups of GLm for some

integer m, and an index appearing in the name of such a group refers to the size of the

matrices involved, so for example Sp2n denotes the subgroup of symplectic matrices in

GL2n. We write ZG for the center of the algebraic group G.

We say that T = (tij) ∈ Qn×n is semi-integral if tii ∈ Z for 1 ≤ i ≤ n and tij ∈ 12Z for

all 1 ≤ i, j ≤ n. Similarly, we say that T is even integral if tii ∈ 2Z for 1 ≤ i ≤ n and

tij ∈ Z for all 1 ≤ i, j ≤ n. Thus if T is semi-integral, then 2T is even integral.

We use 1m to denote the m×m identity matrix, and 0m to denote the m×m zero matrix.

When we write a 2n×2n matrix γ as γ = ( A BC D ) we mean that A,B,C,D are all matrices

of size n. Occasionally we will be required to write matrices γ in block form but where

the blocks have different size; we will explain our notation for this as it arises.

When z is a complex variable, we write e(z) for e2πiz.

We use the terms (which are explained in the text) “Eisenstein series” and “Klingen–

Eisenstein series” interchangeably. In contrast, “Siegel–Eisenstein” series always refers to

a specific kind of Eisenstein series, namely those coming from the zero-dimensional cusps

of Γ(n)0 (N)\H∗n.

We write Z≥0 for the non-negative integers and Z≥1 for the positive integers, so we do

not need to define what we mean by N. Otherwise, we use the standard notation (e.g.

vi

R,Q,Zp,Qp, ...). We write R× for the unit group of a ring R.

If S is a finite set of primes and N is a positive integer, we write gcd(N,S) = 1 to mean

that no prime factor of N lies in S. We write (a, b) for the greatest common divisor of a

and b; the omission of the “gcd” in this notation should cause no confusion.

For N ∈ Z≥1, we write 1N for the trivial character modulo N , which satisfies 1N(n) = 1

when (n,N) = 1 and 1N(n) = 0 otherwise. We write 1 for the principal character (the

“trivial character modulo 1”), which satisfies 1(n) = 1 for all n ∈ Z.

If χ is a Dirichlet character modulo N then it factors as a product of local charac-

ters, i.e. χ =∏

p|N χpe . We will mostly use these factorisations when N is squarefree, so

χ =∏

p|N χp. In the squarefree case, for any divisor d of N we also write χd =∏

p|d χp.

The Vinogradov and Landau symbols have their usual usage: given two functions f and

g defined on R, the notations f(x) g(x) and f(x) = O(g(x)) are synonymous, and

mean that there exists a constant C such that |f(x)| ≤ C|g(x)| for all x ∈ R. If both

f(x) g(x) and g(x) f(x) then we write f(x) g(x).

vii

Chapter 1

Introduction

We begin this introductory chapter by providing a (hopefully) non-technical overview

of the connection between quadratic forms and modular forms. Although fascinating in

its own right, this is only a part of the vast theory of modular forms and higher rank

groups; consequently we say a little about these questions within that broader context,

with particular emphasis on bridging the gap between Fourier coefficients and Hecke

eigenvalues for Siegel modular forms. Finally, having identified guiding problems in the

area, we give a brief discussion of the results of the present thesis. A more detailed

introduction to Chapters 3-6 will be given at the corresponding points the text.

1.1 Representation numbers and modular forms: an

overview

Let Q be an integral quadratic form in m variables, that is a polynomial

Q(x1, ..., xm) =m∑i=1

tix2i +

∑i 6=j

ti,jxixj,

where ti, ti,j ∈ Z. For our present purposes it will be convenient for us to package this in-

formation in an m×m matrix, denoted T , with diagonal entries Ti,i = ti, and off-diagonal

1

entries Ti,j = 12ti,j, so that the quadratic Q is then given by the matrix multiplication

Q(x1, ..., xm) =(x1 . . . xm

)T

x1

...

xm

.1

A classic problem in number theory is to count the number of solutions of the equation

Q(x) = a (1.1)

where a ∈ Z≥1, x ∈ Zm. In this generality there could be infinitely many solutions, or

none at all. To rule out the former possibility we impose the condition that Q be positive

definite: that is Q(x) ≥ 0 for all x ∈ Rm, with equality if and only if x is the zero vector.

Then the number

r(Q, a) = #x ∈ Zm; Q(x) = a (1.2)

of solutions to (1.1) is finite. This is in fact a special instance of a more general represen-

tation problem. Suppose we have another integral quadratic form Q′ in m′ ≤ m variables,

given by a matrix T ′. We say that Q represents Q′ if there is a matrix X ∈ Zm,m′ such

that

tXTX = T ′. (1.3)

Similarly, we define

r(Q,Q′) = #X ∈ Zm,m′ ; tXTX = T ′, (1.4)

which is again finite (possibly zero) since Q is positive definite. Taking Q′(x) = ax2 this

recovers the definition of r(Q, a).

The numbers r(Q, a) have interested mathematicians since antiquity, and interest in

r(Q,Q′) goes at least as far back as Gauss. The ideal result in this direction is an exact

formula, for example a theorem of Jacobi says that

r(x21 + x2

2 + x23 + x2

4, a) =

8∑

d|a a if a is odd,

24∑

d|ad odd

d if a is even.

1This differs from the convention for normalisation of quadratic forms that we adopt in the main textby a factor of two.

2

There are many other examples of exact representation numbers in the literature (indeed

we will provide some more in Chapter 4), but it is fanciful to hope to be able to provide

such formulas in general.

In some cases there might be an obvious obstructions to (1.3), or even (1.1), having any

solutions at all. For example if Q = x21 + x2

2 and a ≡ 3 mod 4 then there can be no

solutions to (1.1). This is a “2-adic obstruction”, since we see that the equation can not

hold since it does not hold modulo 22. If there are no such obstructions then we say (1.1),

or more generally (1.3), is p-adically soluble; more formally we mean that (1.3) can be

solved with X ∈ Zm,m′p for all p. There is a similar notion over R, which is automatically

true for us since Q and Q′ are positive definite. When (1.3) is p-adically soluble for all

p (and soluble over R) we say that it is locally soluble. If (1.3) is locally soluble then it

at least has a chance of being globally soluble, i.e. can be solved with X ∈ Zm,m′ . Clearly

we always have the implication “globally soluble implies locally soluble”. We say that

the integral Hasse principle holds when the implication “locally soluble implies globally

soluble” also holds.

It is well-known that the integral Hasse principle does not always hold. This failure is

related to the genus of Q. We say that two quadratic forms Q1 and Q2 (given by matrices

T1 and T2) are isometric if they are related by an integral change of variables: T1 = tXT2X

with X ∈ Zm,m. We say Q1 and Q2 are in the same genus if one can solve T1 = tXT2X for

X ∈ Zm,mp , for each prime p,1 thus being in the same genus is a coarser notion than being

isometric. It is well-known that there are finitely many (isometry classes of) quadratic

forms Q = Q1, ..., Qh in the genus of Q. The integral Hasse–Minkowski theorem states

that if Q represents Q′ everywhere locally then there exists a form Qi in the genus of Q

which represents Q′ globally. The genesis of Siegel modular forms lies in a quantitative

version of this result, Siegel’s Hauptsatz, which we now describe.

Let us now write n = m′ for the number of variables of Q′. One begins by placing the

1The same condition over R is again automatic since Q1 and Q2 are positive definite.

3

representation numbers in a sort of generating function

θ(n)Q (Z) =

∑Q′

r(Q,Q′)eQ′Z,

where Q′ varies over all positive definite quadratic forms in n variables, Z is a complex

n× n matrix (we will assume that its imaginary part is positive definite, which will pro-

vide θ(n)Q (Z) with excellent convergence properties) and eQ′Z is a certain exponential

factor (we refer to §3.1 for the full definition, which is not important for now). Although

it is defined simply as a generating function for the sequence we are interested in, with

some small manipulations and a bit of Fourier analysis one can show that θ(n)(Q) sat-

isfies a certain transformation law under the change of variables Z 7→ −Z−1. In fact, it

defines what is called a Siegel modular form of degree n, weight m/2. These are func-

tions of the complex matrices Z which satisfy a transformation law under an action of

(a subgroup of) the sympletric group Sp2n(Z). It is convenient (although by no means

necessary) to restrict to the case when the number of variables m of Q is even, say

m = 2k, so as to work only with modular forms of integer weight. Then θ(n)Q defines an

element of a space of Siegel modular forms denotedM(n)k (N,χ), where N ∈ Z≥1 and χ is

a Dirichlet character modulo N , both of these being determined in a simple fashion by Q.

The quantitative integral Hasse–Minkowski theorem can then be described by forming

an average theta series, where each isomorphism class is weighted by the number of

automorphisms it has:

θ(n)gen(Q) =

1

w

h∑i=1

1

O(Qi)θ

(n)Qi.

Here O(Qi) is the size of the isometry group of Qi (which is finite since each Qi is

positive definite), and w =∑h

i=1 1/O(Qi). Since M(n)k (N,χ) is a vector space (and N

and χ depend only on the genus of Q) one easily sees that θ(n)gen(Q) is an element of

M(n)k (N,χ); Siegel’s Hauptsatz says that θ

(n)gen(Q) is in fact a rather special kind of el-

ement of M(n)k (N,χ), namely a Siegel–Eisenstein series. So, in particular, the average

representation numbers

rgen(Q,Q′) =1

w

h∑i=1

1

O(Qi)r(Qi, Q

′) (1.5)

4

are given by the Fourier coefficients of a Siegel–Eisenstein series.

Siegel’s original proof of this result proceeds via the circle method, and also provides a

formula for the quantity rgen(Q,Q′) in terms of a product over all places of Q of the

relative density of solutions of (1.3) modulo higher prime powers (the local densities). Of

course this term only depends on the genus of Q, and should be understood as the main

term for r(Q,Q′). In order to justify this, however, one must show that the error term

e(Q,Q′) = r(Q,Q′)− rgen(Q,Q′) is smaller than rgen(Q,Q′) in the limit H(Q′)→∞, for

some quantity H(Q′) adapted to the problem at hand.1 Assuming

e(Q,Q′) = o(rgen(Q,Q′)) (1.6)

for H(Q′) sufficiently large, one can then deduce that if H(Q′) is sufficiently large then

the integral Hasse principle holds. Moreover, in this situation, the number of represen-

tations is governed by the main term rgen(Q,Q′), justifying our statement that Siegel’s

Hauptsatz is a quantitative integral Hasse–Minkowski theorem.

The problem is then under which circumstances one can obtain estimates of the form

(1.6). The typical result is that if one assumes that Q has sufficiently many variables

relative to Q′ (i.e. assuming large “codimension”) then one can obtain (1.6), either using

techniques from the theory of modular forms, the circle method, or directly from quadratic

form theory. In this thesis we will adopt the viewpoint of modular forms for this prob-

lem, although it is worth mentioning that the some of the best results in fact come from

ergodic theory. The main result of the remarkable paper of Ellenberg–Venkatesh ([18])

states, under certain conditions, that if m ≥ m′ + 3 and Q′ is represented everywhere

locally by Q, then Q′ is represented globally by Q, whilst the other methods require

m ≥ 2m′ + 3. On the other hand, modular forms and the circle method seem are more

suited to providing asymptotic formulas, which do not currently follow from the ergodic

techniques.

1As a first approximation one might try to take H(Q′) to be the determinant of the Gram matrixassociated to Q′, although this turns out to be unwise.

5

Recall the spaceM(n)k (N,χ), which consists of all functions of the variable Z which satisfy

the transformation law that θ(n)(Q) does, as well as some mild analytic conditions. This

imposes an incredible rigidity in the situation: firstly, M(n)k (N,χ) is a finite dimensional

complex vector space1, and secondly M(n)k (N,χ) carries the action of an infinite family

of Hecke operators, which give arithmetic relations amongst the various r(Q,Q′).

Finite dimensional complex vector spaces are generally amenable to computation, so one

is tempted to study θ(n)Q by studying the space M(n)

k (N,χ) directly. The extra struc-

ture coming from the Hecke operators suggest moreover that it is pertinent to study

M(n)k (N,χ) as a Hecke module. In any case, every element of M(n)

k (N,χ) admits a sort

of Fourier expansion, and by comparing Fourier expansions one can therefore write the

r(Q,Q′) as a sum of Fourier coefficients for elements of a basis of M(n)k (N,χ). Roughly

speaking one would then like to write

θ(n)Q = E + F,

where E is an Eisenstein series, with Fourier coefficients that should be easy to compute,

and F is a whatever is left, whose Fourier coefficients should be small, so that this can

be understood as a “main term” and “error term” expression. In light of Siegel’s theo-

rem the Eisenstein series is none other than θ(n)gen(Q), which is a Siegel–Eisenstein series.

One can attempt to compute the Fourier coefficients from the local densities in Siegel’s

Hauptsatz, or attempt to compute the Fourier coefficients directly from the definition of

the Eisenstein series and write the theta series explicitly in terms of Eisenstein series.

This is possible when n = 1 (see e.g. [70]), but even computing the Fourier coefficients of

Eisenstein series is difficult when n > 1; we will describe this problem in more detail in

Chapter 4.

To understand the representation numbers of an individual quadratic form is to under-

stand how far they can deviate from the average result; here the methods of analytic

number theory and estimates on the Fourier coefficients come in to play. For n = 1 this

is well understood: the difference F = θ(n)Q − θ

(n)gen(Q) must be a cusp form, and its Fourier

1Zagier ([9]) cites this finite dimensionality as “the origin of the (unreasonable?) effectiveness ofmodular forms in number theory”.

6

coefficients are therefore subject to the optimal bounds coming from Deligne’s theorem.

When n > 1, however, this is no longer true: we can only say that F vanishes at all zero-

dimensional cusps, so it must be a sum of a cusp form and also various Eisenstein series

(which are of Klingen but not Siegel type) coming from higher dimensional cusps. This

greatly complicates the arithmetic and analysis of the situation. Thus as well as studying

cusp forms (for example obtaining bounds on their Fourier coefficients) it is also important

to understand the space of Klingen–Eisenstein series as fully as possible, which provides

the main motivation for our Chapter 3. Although explicit results for Klingen–Eisenstein

series are not particularly well-documented in the literature, the technical difficulties that

they impose for the theory of quadratic forms have been overcome by Kitaoka (see for

example [32]) in certain cases, and estimates on Fourier coefficients are able to provide

the expected asymptotic formula when m ≥ 2m′ + 3. Here it is necessary to take H(Q′)

a function of both the determinant of a Gram matrix for Q′ and the minimum of the

lattice corresponding to Q′, further highlighting the additional arithmetic complications

present in the problem of higher representation numbers.

1.2 Representations and L-functions

In the previous section we saw that the study ofM(n)k (N,χ) can lead to concrete results re-

garding representation numbers of quadratic forms. For this and other problems, however,

it is sometimes useful to consider Siegel modular forms within their larger framework of

modular forms on higher rank groups. Here the language of automorphic representations

is often used. In this case one passes from the Siegel modular form F , which transforms

under a subgroup of Sp2n(Z), to a function ΦF on the adelic points of the similtude sym-

plectic group GSp2n(A), and then one step further to the representation πF generated by

ΦF under the right regular action of GSp2n(A). The representation πF retains all of the

information about the modular form F although it is now packaged differently. For ex-

ample if F is an eigenform (i.e. an eigenfunction of the Hecke operators) then the Hecke

eigenvalues of F at p are repackaged in terms of the local representation πF,p coming

from the action GSp2n(Qp). The formation of the automorphic representation πF lands

within the Langlands framework and, following the general constructions of that theory,

7

one can associate a various L-functions to πF . For example one can form the “spin” and

“standard” L-functions, which recover two L-functions attached to eigenforms F which

are often studied in the classical language.

The automorphic representations πF and their associated L-functions are interesting ob-

jects in their own right. From this point of view, it is known in degree two (i.e. auto-

morphic representation of GSp4(A)) that one can attach, by results of Weissauer and

Taylor, a Galois representation to such a πF . This Galois representation appears in the

cohomology of the associated Shimura variety (if the weight k is ≥ 3), and one can prove

using cohomological methods that the Ramanujan conjecture holds for degree two Siegel

modular forms. In the classical language this translates into sharp estimates for the size

of Hecke eigenvalues of degree two Siegel modular forms. In degree greater than two there

has been much work towards the Ramanujan bounds (see e.g. [35] for a general overview

and results on T (p); [71] for results on Tj(p2)); however obtaining the optimal bounds is

still an open problem in degree greater than two.

In the case of n = 1 it is well-known that there is a close connection between Hecke eigen-

values and Fourier coefficients (of newforms, say). In particular, the sharp estimates for

the size of degree one Hecke eigenvalues, due to Deligne (weight k ≥ 2) and Deligne–Serre

(k = 1), also give sharp estimates for the size of the Fourier coefficients of cusp forms.

Moreover, Sato–Tate type theorems, which are normally phrased as regarding the distri-

bution of Hecke eigenvalues, could equally be understood in terms of Fourier coefficients

of cusp forms.

Unfortunately there is no simple passage between the Hecke eigenvalues and Fourier co-

efficients when n ≥ 2. In the classical language this appears simply from the formulas

for the action of the Hecke operators: the Hecke operators do not relate enough Fourier

coefficients for it to be obvious how to recover the Fourier coefficients from the Hecke

eigenvalues. In the automorphic language, as we shall see in the text (Chapter 5), the

Hecke eigenvalues determine a collection of local representations πF,p, at least when p

does not divide the level of F . Adopting the naıve expectation from the case n = 1,

8

one might hope that knowing πF,p at all but finitely many primes would be enough to

determine the automorphic representation (from which one could extract the form F )

uniquely, but this is known to be false; for example one can construct counter-examples

using Yoshida lifts, where the Hecke eigenvalues agree at all primes outside the level N but

differ at primes dividing N . It is still expected to be true that knowledge of all the πF,p is

enough to determine the automorphic representation uniquely, but this is still conjectural.

Of course if one knows all the Fourier coefficients then one can deduce the Hecke eigenval-

ues, so it should not be surprising that for Siegel modular forms the Hecke eigenvalues are

“more accessible” than Fourier coefficients, which will be a recurring theme throughout

this thesis. For example in Chapters 3, 5, and 6 the main results will essentially be about

Hecke eigenvalues. For the theory of quadratic forms it would be nice to have versions

of the results of Chapters 3 and 5 describing Fourier coefficients, but these seem to be

genuinely more difficult.

So whilst the automorphic representation theory provides powerful methods for studying

Hecke eigenvalues, it does not yet provide a straightforward way to study Fourier coef-

ficients. However, in the closing sections of this thesis we will discuss some conjectures

and results on the automorphic side which are attempting to bridge this gap.

1.3 This thesis

In Chapter 2 we provide some background on the classical theory of Siegel modular forms,

with particular attention paid toM(n)k (N,χ). Since various aspects of this (particular the

Siegel lowering operator and Eisenstein series) are only sketched in the literature, we take

a little more care than usual in introducing some of these notions.

Chapter 3 also begins with a fair amount of background. Our first goal is to describe how

the action of Hecke operators and the Siegel lowering operator intertwine with each other

(Theorem 3.1). Results of along these lines have appeared before and are originally due

to Zarkovskaja ([69]); we give a new version proved using the action of Hecke operators

9

on Fourier expansions, which is well-suited to studying M(n)k (N,χ).

Next we consider the geometry of Γ(n)0 (N)\Hn in more detail. This space is non-compact,

but a compactification can be formed by adding various lower-dimensional components

to the boundary. We recall the construction, and describe it in detail for our specfic case.

The picture can become rather complicated in general, as boundary components will in-

tersect each other in lower dimensions. However we shall assume that N is squarefree,

so it is possible to describe this rather explicitly (Theorem 3.17). Although we do not

require such detailed knowledge for the main applications we feel the inclusion of this

result nonetheless provides a more accurate depiction of the situation.

With a better understanding of the geometry we are able to describe the intertwining

relation between the Hecke operators and the lowering operator to any cusp (Theorem

3.2). This is the key result that allows us to deduce the main theorem of Chapter 3 (The-

orem 3.23), which describes how the Hecke operators (at least the ones not dividing the

level) act on the space of Eisenstein series inside M(n)k (N,χ). This is carried out under

the assumption that N is squarefree.

Next in Chapter 4 we provide some formulas for the Fourier coefficients of Siegel–

Eisenstein series. As well as assuming that the level N is squarefree, we now must also

assume that the transformation character χ is the trivial character modulo N and that

the degree n is equal to two. With these assumptions in place we are able to provide

formulas for the Fourier coefficients of a full basis for the space of Siegel–Eisenstein series

(Theorem 4.8), which appears to be the first result of this kind in the literature when

N > 1. We emphasize the importance of giving a Fourier coefficients for a full basis by

showing how, in tandem with Siegel’s Hauptsatz, these allow one to deduce explicit, ex-

act formulas for the genus-average representation numbers of quadratic forms (Corollary

4.10).

In Chapter 5 we shift gears and start talking about cusp forms and the automorphic

representations attached to them. The main result of this chapter is a (weighted) equidis-

10

tribution result about the Satake parameters attached to Siegel cusp forms of degree two

as one increases the weight and (not necessarily squarefree) level (Theorem 5.17). In or-

der to get to this we explain some of the background of the automorphic representation

and particularly the Bessel models available for automorphic representations for Siegel

cusp forms. These are a substitute for Whittaker models, which are not available for the

representations attached to Siegel cusp forms (i.e. those representations are not generic).

With the necessary theory in place the main technical input for this chapter boils down

to a statement about the Fourier coefficients of Poincare series: we require a quantitative

version of the statement that the T th Fourier coefficient of a degree two Poincare series

GN,k,Q is essentially δT=Q (Theorem 5.29).

In the final Chapter 6 we begin by describing the (Langlands method of) formation of

L-functions for Siegel modular forms. We then consider the distribution of the low-lying

zeros of the L-functions attached to degree two Siegel cusp forms of increasing weight

and level, with a certain arithmetic weighting (the “spectral weight”, coming from the

Petersson trace formula) introduced in Chapter 5. After some more background we prove

that, at least in terms of one-level densities, the low-lying zeros of the L-functions in this

weighted “family” are distributed according to a sympletic random matrix model. This is

in contrast to results in the literature showing that the unweighted versions are distributed

according to an even orthogonal random matrix model. We close by discussing the role

of the weighting in this discrepancy, more precisely we give a heuristic for how this can

be explained by a conjecture of Bocherer relating central values of twisted L-functions

and sums of Fourier coefficients of Siegel cusp forms. We also make some general remarks

about how this conjecture relates to the information barrier between Hecke eigenvalues

and Fourier coefficients of Siegel cusp forms which had important implications for the

arithmetic of quadratic forms.

11

Chapter 2

Background

In this section we give a brief introduction to the classical theory of Siegel modular forms.

2.1 Siegel modular forms

For any integer n ≥ 1 the algebraic group GSp2n is defined by

GSp2n = g ∈ GL2n; tgJg = µn(g)J for some µn(g) ∈ GL1,

where

J =

0n −1n

1n 0n

.

The map µn : GSp2n → GL1 is a homomorphism, and its kernel is by definition Sp2n. If

R is a subring of R then we write GSp+2n(R) for the subgroup of GSp2n(R) consisting of

those g with µn(g) > 0. A priori Sp2n is a subgroup of GL2n, however it is well-known

that in fact it is contained in SL2n.

Let Hn = Z ∈ Cn×nsym ; Im(Z) > 0 be Siegel’s upper half space of degree n. GSp+

2n(R)

acts on Hn by

(γ, Z) 7→ γ〈Z〉 = (AZ +B)(CZ +D)−1, (2.1)

where γ = ( A BC D ) ∈ GSp+

2n(R). For k an integer we can also define an action of GSp+2n(R)

on functions F : Hn → C by

(F |kγ)(Z) = µn(γ)nk/2j(γ, Z)−kF (γ〈Z〉), (2.2)

12

where

j(γ, Z) = det(CZ +D).

It will be useful for us to have an interpretation of this formula even when n = 0. In this,

H0 becomes a point, a complex valued function F : H0 → C is therefore constant, and we

identify F with the corresponding complex number. Finally for n = 0, any k, and any γ

the action F |kγ is taken to be trivial.

Let N be a positive integer. The principal congruence subgroup of level N is

Γ(n)(N) = γ ∈ Sp2n(Z); γ ≡ 12n mod N .

We say that a subgroup Γ(n) of Sp2n(Q) is a congruence subgroup if there exists N ∈ Z≥1

such that Γ(n) contains Γ(n)(N) as a subgroup of finite index.

Definition 2.1. Let n ∈ Z≥1, k ∈ Z≥1, Γ(n) ⊂ Sp2n(Q) be a congruence subgroup. A

holomorphic function F : Hn → C is a Siegel modular form of degree n and weight k for

Γ(n) if

F |kγ = F

for all γ ∈ Γ(n). When n = 1 we impose the additional condition that F be regular at

all cusps.1 We write M(n)k (Γ(n)) for the complex vector space of Siegel modular forms of

degree n and weight k for Γ(n).

In many cases it will be clear that we are working with modular forms of weight k, in

these cases we will simply write F |γ instead of F |kγ.

The main focus of this thesis concerns a particular congruence subgroup, namely the

Hecke congruence subgroup of level N ,

Γ(n)0 (N) =

A B

C D

∈ Sp2n(Z); C ≡ 0 mod N

.

Let χ be a Dirichlet character modulo N . We define a character of Γ(n)0 (N), also denoted

χ, by the rule χ (( A BC D )) = χ(det(D)) (or, equivalently, χ (( A B

C D )) = χ(det(A))). For the

1This is not necessary when n > 1, by the Koecher principle.

13

Hecke type congruence subgroup we allow a slight generalisation of the transformation

law in (2.1), involving transformation with character. Equivalently one could work with

modular forms on a smaller group Γ(n)1 (N) and define “diamond operators” to decompose

according to character of (Z/NZ)×, but we prefer the following definition:

Definition 2.2. Let n ∈ Z≥1, k ∈ Z≥1, N ∈ Z≥1, and χ be a Dirichlet character modulo

N . A holomorphic function F : Hn → C is a Siegel modular form of degree n, weight k,

level N , and character χ if

F |kγ = χ(γ)F

for all γ ∈ Γ(n)0 (N). When n = 1 we impose the additional condition that F be regular at

all cusps. We write M(n)k (N,χ) for the complex vector space of Siegel modular forms of

degree n, weight k, level N , and character χ.

Remark 2.3. Since −12n ∈ Γ(n)0 (N), we have that M(n)

k (N,χ) = 0 when χ((−1)n) 6=

(−1)nk; i.e. when n is odd the character and weight must be compatible.

Comparing Definitions 2.1 and 2.2, we seeM(n)k (N,1N) =M(n)

k (Γ(n)0 (N)). We denote this

space also byM(n)k (N). These modular forms with trivial character will be the main focus

in Chapters 4, 5, and 6. In Chapter 3 we allow arbitrary character modulo N . With the

view to studying quadratic forms it is only actually necessary to consider all quadratic

characters.

To define cusp forms, we require the Siegel lowering operator. Let 0 ≤ r ≤ n, and take

Z ′ ∈ Hr. Define

Φ(r)(F )(Z ′) = limλ→∞

F

Z ′ 0

0 iλ1n−r

, (2.3)

and for γ ∈ Sp2n(Q) define

Φ(r)γ (F )(Z ′) = Φ(r)(F |γ).

We also write Φ = Φ(n−1) and Φγ = Φ(n−1)γ . The cuspidal subspace ofM(n)

k (Γ(n)) is defined

to be

S(n)k (Γ(n)) =

F ∈M(n)

k (Γ(n)); Φγ(F ) = 0 for all γ ∈ Sp2n(Q).

This is a generalization of the familiar process of examining the values at the various

cusps from the case n = 1. However, the output is generally not a constant, but rather a

14

function on Hn−1. The cuspidality condition is that all the above functions be identically

zero. The same definitions apply to give the cuspidal subspace S(n)k (N,χ) ⊂M(n)

k (N,χ).

This defines cusp forms of degree n > 0. Whenever we write formulas which require an

interpretation of degree zero cusp forms, we understand thatM(0)k (Γ(0)) = S(0)

k (Γ(0)) = C,

and also M(0)k (N,χ) = S(0)

k (N,χ) = C.

The condition imposing cuspidality is equivalent to the (finite) condition whereby we

replace Sp2n(Q) with a system of representatives for Γ(n)\ Sp2n(Q)/Pn,n−1(Q), where for

n and r positive integers Pn,r is the parabolic subgroup

Pn,r =

A11 0 B11 B12

A21 A22 B21 B22

C11 0 D11 D12

0 0 0 D22

∈ Sp2n; ∗11 size r; ∗22 size (n− r)

. (2.4)

Let us also define a map ωn,r : Pn,r(Q)→ Sp2r(Q) by

ωn,r

A11 0 B11 B12

A21 A22 B21 B22

C11 0 D11 D12

0 0 0 D22

=

A11 B11

C11 D11

. (2.5)

Remark 2.4. Note that Φ(r)γ is not well-defined on the double coset Γ(n)γPn,r(Q). Indeed,

for γ′ ∈ Γ(n) and δ ∈ Pn,r(Q) written in the form of (2.4), we have

Φ(r)γ′γδ(F ) = det(D22)−kΦ(r)

γ (F )|ωn,r(δ).1 (2.6)

Note that since δ ∈ Pn,r(Q) we have det(D22) ∈ Q×. Thus the choice of representative is

not important for defining cuspidality, but more care will be required in other applications.

In Chapter 3 we will re-interpret these double cosets in terms of the boundary of the

Satake compactification of the complex analytic space Γ(n)\Hn. It will be seen that these

boundary components are also quotients of Siegel upper half spaces Hr by arithmetic

1The verification of this equation involves taking slightly more general limits than those in (2.3); thatthese more general limits give the same result follows from [34] §5 Proposition 1.

15

subgroups, and we will study the relationship between modular forms on the boundary

components and modular forms on Γ(n)\Hn. Particular attention will be paid to the

Satake compactification and the role of the Siegel lowering operator in the context of

M(n)k (N,χ).

Remark 2.5. Assume for simplicity that N is squarefree. In Chapter 3 we will make

certain choices of representations to define lowering operators

Φl :M(n)k (N,χ)→M(n−1)

k (N,χlχN/l),

the cuspidality condition then being that Φl(F ) ≡ 0 for each l. But if χ((−1)n−1) 6=

(−1)(n−1)k then Φl(F ) must be zero (c.f. Remark 2.3). In tandem with Remark 2.3 we

then see that for non-cusp forms to exist the condition χ(−1) = (−1)k must hold. It is

therefore natural to assume this in Chapters 3 and 4 where we study Eisenstein series. If

the weight and character do not satisfy this condition then there may exist non-zero cusp

forms, for example there is a non-zero element in S(2)35 (1).

Let Γ(n) ⊂ Sp2n(Q) be a congruence subgroup, and F ∈M(n)k (Γ(n)). It is well-known that

F admits a Fourier expansion supported on a lattice (which depends on Γ(n)) of positive

semi-definite matrices T ∈ Qn×nsym ,

F (Z) =∑T≥0

a(T ;F )e(tr(TZ)). (2.7)

When F ∈ M(n)k (N,χ) this is simply the lattice of semi-integral matrices. Note that

modularity of F implies that a(T ;F ) depends only on the SLn(Z) equivalence class of T ,

i.e.

a(T ;F ) = a(tGTG;F )

for all G ∈ SLn(Z). If we refer to a(T ;F ) when T is not semi-integral and positive semi-

definite then we understand that a(T ;F ) = 0.

It is easy to show that, for F ∈ M(n)k (Γ(n)), Φ(F ) = 0 if and only if the Fourier expan-

sion is supported on strictly positive definite matrices (i.e. a(T ;F ) 6= 0 =⇒ T > 0).

This gives the characterization that F ∈ M(n)k (Γ(n)) is cuspidal if and only if its Fourier

16

expansion at each cusp is supported on strictly positive definite matrices. This general-

izes the familiar condition of vanishing of constant terms at all cusps from the case n = 1.

Let Γ(n) be a congruence subgroup. There is a pairing 〈·, ·〉 defined on S(n)k (Γ(n)) (and

partially onM(n)k (Γ(n)), so long as one element is in S(n)

k (Γ(n))) called the Petersson inner

product. This is given by

〈F,G〉 =1

vol(Γ(n)\Hn)

∫Γ(n)\Hn

F (Z)G(Z) det(Y )kdµ(Z), (2.8)

where Z = X + iY is the variable in Hn, and dµ(Z) = dXdY/ det(Y )n+1 is the invariant

measure on Hn. The volume is of course computed with the same measure, and the fact

that Γ(n) is a congruence subgroup (and that vol(Sp2n(Z)\Hn) is finite) means that this

volume is finite. With this normalization factor in place, the value of 〈F,G〉 is unchanged

if one replaces Γ(n) by a smaller congruence subgroup in the definition. The same formula

also defines an inner product on (part of) M(n)k (N,χ).

We write S(n)k =

⋃Γ(n) S(n)

k (Γ(n)), where the union is over all congruence subgroups; or

equivalently S(n)k =

⋃N≥1 S

(n)k (Γ(n)(N)). The normalising factor therefore ensures that

(2.8) is well-defined on S(n)k .

2.2 Quadratic forms and theta series

Let L be a rank 2k lattice endowed with a quadratic form Q : L → Z. Then Q also

defines a symmetric bilinear form on L by the formula

B(x, y) = Q(x+ y)−Q(x)−Q(y)

which is Z-valued and moreoever satisfies B(x, x) ∈ 2Z for all x ∈ L. Conversely given

such a bilinear form B we can define a quadratic form by the rule Q(x) = 12B(x, x).

This sets up a bijection, so that specifying a Z-valued quadratic form Q is equivalent to

specifying a Z-valued bilinear form B such that B(x, x) ∈ 2Z for all x ∈ L. We shall refer

to a lattice endowed with either of these equivalent structures as an even lattice. Picking

a basis (e1, ..., e2k) for L we can map B to the matrix S = (B(ei, ej)); this is the Gram

17

matrix associated to the basis (e1, ..., e2k). By our assumptions on B, the Gram matrix is

even integral. If we choose a different basis then we obtain a different matrix, which has

the form tGSG where G ∈ GLn(Z).

For such an even lattice L we form the degree n theta series by

θ(n)L (Z) =

∑X∈Z2k,n

eπi tr(tXSXZ)

=∑T≥0

T even integral

rS(T )eπi tr(TZ),

where S is any Gram matrix for L, and rS(T ) = #X ∈ Z2k,n; tXSX = T is the

number of representations of the n-variable quadratic form T by S. This in the form of

the Fourier expansion of a Siegel modular form (c.f. (2.7)). It is well-known that θ(n)L is

indeed a Siegel modular form, namely θ(n)L ∈M

(n)k (N,χ) where the level N is the level of

the the lattice L (equivalently the smallest integer N such that NS−1 is an even integral

matrix) and the character is

χ =

((−1)k det(S)

·

).

Note that if (−1)k det(S) is a (global) square then this character is trivial.

Since both the level and character are genus-invariants of the quadratic form, it makes

sense to define the genus theta series θ(n)gen(L)(Z), which is done as follows: let L =

L1, L2, ..., Lh be the inequivalent lattices in the genus of L, write O(Li) for the size of the

isometry group of Li, w =∑h

i=11

O(Li), and put

θ(n)gen(L)(Z) =

1

w

h∑i=1

1

O(Li)θ

(n)Li.

Let S = S1, S2, ..., Sh be Gram matrices for L = L1, L2, ..., Lh respectively. Then

θ(n)gen(L)(Z) =

∑T≥0

T even integral

rgen(S)(T )eπi tr(TZ),

where

rgen(S)(T ) =1

w

h∑i=1

1

O(Si)rSi(T )

18

measures the average number of representations of T by the genus of S.

Siegel’s Hauptsatz says that θ(n)gen(L) ∈ M

(n)k (N,χ) actually lies in the subspace of Siegel–

Eisenstein series (see §2.3). This remarkable theorem was originally proved using the circle

method; and in the process of doing this Siegel showed that the Fourier coefficients of

the genus-average theta series rgen(L)(T ) (equivalently the Fourier coefficients of a certain

Siegel–Eisenstein sereis) can be expressed as a product of p-adic densities of solutions to

the representation problem.1 Weil famously remarked that the symplectic group appears

deus ex machina in this argument, and gave a rather different proof of Siegel’s Hauptsatz

using his eponymous representation to transfer automorphic forms on orthogonal groups

to automorphic forms on symplectic groups, and the corresponding generalisation is called

the Siegel–Weil formula. The relationship between quadratic forms and Eisenstein series

is central to the first half of this thesis, particular Chapter 4.

2.3 Eisenstein series

Let us fix a representative γ for the double coset

Γ(n)γPn,r(Q) ⊂ Γ(n)\ Sp2n(Q)/Pn,r(Q),

where 0 ≤ r ≤ n. Set

Γ(r)γ = ωn,r(γ

−1Γ(n)γ ∩ Pn,r(Q)).

Let F ∈ S(r)k (Γ

(r)γ ), and define

E(n)γ (Z;F ) =

∑M∈(γ−1Γ(n)γ∩Pn,r(Q))\γ−1Γ(n)

j(M,Z)−kF (π(M〈Z〉)). (2.9)

Here π : Hn → Hr is the map

π

Z11 Z12

Z21 Z22

= Z11. (2.10)

This is compatible with the action of Pn,r(Q), in the sense that if δ ∈ Pn,r(Q) and Z ∈ Hn

then

π(δ〈Z〉) = ωn,r(δ)〈π(Z)〉. (2.11)

1From this one begins to see why genus-averages enter the picture.

19

Let us consider when the terms of this sum are independent of the choice of coset

representative. So suppose that in some summand we replace M by δM , where δ ∈

γ−1Γ(n)γ ∩ Pn,r(Q). Then we require that

j(δM,Z)−kF (π(δM〈Z〉)) = j(M,Z)−kF (π(M〈Z〉)).

Using the cocycle relation j(δM,Z) = j(δ,M〈Z〉)j(M,Z), (2.11), and the modularity of

F this becomes

j(δ,M〈Z〉)k = j(ωn,r(δ), π(M〈Z〉))k.

Writing δ as in (2.4) this is seen to be equivalent to

det(D22)k = 1, for all δ ∈ γ−1Γ(n)γ ∩ Pn,r(Q). (2.12)

When we talk about Eisenstein series for Γ(n), we understand that Γ(n) satisfies the condi-

tion (2.12). In the cases when Γ(n) ⊂ Sp2n(Z) it is not difficult to see that det(D22) = ±1,

so (2.12) reduces to the (possible) condition that k be even. In particular, for M(n)k (N)

the condition (2.12) already holds under the assumptions we make based on Remark 2.5.

In general, when (2.12) holds, it can be shown (c.f. [34] §5 Theorem 1) that (2.9) defines

an element of M(n)k (Γ(n)) provided that k > n+ r + 1.

In §3 we will also require a version of (2.9), and the corresponding condition (2.12), for

modular forms with character. However we postpone the definition for now, since it re-

quires knowledge of the character that F transforms with, and for this one needs a more

detailed study of the boundary of Γ(n)0 (N)\Hn.

When r = 0 we may omit F from the notation and simply write E(n)γ (Z); by this we mean

that F has been taken as the function which is constantly one. We call these Eisenstein

series arising in the cases r = 0 the Siegel–Eisenstein series.

Remark 2.6. Similarly to Remark 2.4, this Eisenstein series depends on the choice of

representative for Γ(n)γPn,r(Q). More precisely, if γ′ ∈ Γ(n) and δ ∈ Pn,r(Q) is written as

in (2.4), then

E(n)γ′γδ(Z; det(D22)−kF |ωn,r(δ)) = E(n)

γ (Z;F ).

20

When the Eisenstein series converge, they provide sections to the Siegel lowering opera-

tors. Let us fix representatives γi for Γ(n)\ Sp2n(Q)/Pn,r(Q). Then

Φ(r)γi

(E(n)γj

(·;F ))(Z ′) =

F (Z ′) if i = j,

0 otherwise.

(2.13)

In fact it is not difficult to see from this that every element of M(n)k (Γ(n)) (and similarly

M(n)k (N,χ)) can be written as a sum of Eisenstein series and cusp forms. We will discuss

this in more detail in Chapter 3.

In contrast to the case n = 1 it is not straightforward to give formulas the Fourier

coefficients of Eisenstein series, even at full level. On the other hand, Siegel’s Hauptsatz

tells us that they are arithmetically interesting. The problem of computing the Fourier

coefficients of Siegel–Eisenstein series will be discussed in Chapter 4, where we also include

some explicit applications of Siegel’s theorem.

2.4 Hecke operators

Let

∆(n)0 (N)

=

A B

C D

∈ GSp+2n(Q) ∩ Z2n×2n; C ≡ 0 mod N ; gcd(det(A), N) = 1

.

This is a subsemigroup of GSp2n(Q) which contains Γ(n)0 (N). We extend our character χ,

currently defined on Γ(n)0 (N), to a character of ∆

(n)0 (N) by defining

χ(δ) = χ(det(A))

for δ = ( A BC D ) ∈ ∆

(n)0 (N). We write H(n)(N) for the Hecke algebra of the Hecke pair

(Γ(n)0 (N),∆

(n)0 (N)). One can check that if γ ∈ Γ

(n)0 (N) and α ∈ ∆

(n)0 (N) are such that

αγα−1 ∈ Γ(n)0 (N), then

χ(αγα−1) = χ(γ). (2.14)

21

Focusing on a prime p we have the local Hecke algebra H(n)p (N), which is the ring of

Z-linear combinations of double cosets Γ(n)0 (N)MΓ

(n)0 (N), where

M ∈g ∈ ∆

(n)0 (N); µ(g) is a power of p

.

Assume p - N and define the following elements of H(n)p (N):

T (n)(p) = Γ(n)0 (N)

1n

p1n

Γ(n)0 (N),

T(n)j (p2) = Γ

(n)0 (N)

1j

p1n−j

p21j

p1n−j

Γ(n)0 (N), for 0 ≤ j ≤ n.

We will soon define an action of these double cosets on modular forms, when it will be

seen that T(n)0 (p2) acts as a scalar. The remaining operators are more interesting, although

one of them is somewhat redundant:

Lemma 2.7. The ring H(n)p (N) is commutative, and generated by T (n)(p) and T

(n)j (p2)

for 1 ≤ j ≤ n− 1. Moreover, for any p - N , the natural map H(n)p (N)→ H(n)

p (1) defined

by

Γ(n)0 (N)αΓ

(n)0 (N) 7→ Sp2n(Z)α Sp2n(Z)

is an isomorphism.

Proof of Lemma 2.7. See [2] Chapter 3 Lemma 3.3, Theorem 3.7, and Theorem 3.23.

Before proceeding let us make some remarks on Hecke operators for different congruence

subgroups, which are also relevant for understanding the references given for the proof of

Lemma 2.7. Set

∆(n)(N) =

M ∈ GSp+2n(Q) ∩ Z2n×2n;M ≡

1n 0n

0n µn(M)1n

mod N

.

This is the Γ(n)(N)-analogue to ∆(n)0 (N) introduced above, and indeed one can define

analogues for any congruence subgroup. In (3.4) of Chapter 3 of [2] the “q-symmetry”

condition is introduced, where their q is our N . Both Γ(n)(N) (by definition) and Γ(n)0 (N)

22

(by [2] Chapter 3 Lemma 3.5) satisfy this condition, so by [2] Chapter 3 Theorem 3.3 the

Hecke rings are isomorphic. The upshot is that, at least when p - N , we can talk about

T (n)(p) and T(n)j (p2) acting on either M(n)

k (Γ(n)(N)) or M(n)k (N) without any confusion.

For most of this thesis we will talk about Hecke operators acting on Γ(n)0 (N)-type modular

forms, but in §5 we will require that there is no confusion when we pass between these

and Γ(n)(N)-type modular forms.

We let an element Γ(n)0 (N)αΓ

(n)0 (N) ∈ H(n)

p (N) act on M(n)k (N,χ) by writing

Γ(n)0 (N)αΓ

(n)0 (N) =

⊔v

Γ(n)0 (N)αv

and defining

F |Γ(n)0 (N)αΓ

(n)0 (N) = µ(α)

nk2−n(n+1)

2

∑v

χ(αv)F |kαv. (2.15)

This is extended linearly to an action of H(n)p (N)⊗Z C. One easily checks that this defi-

nition is independent of the choice representatives αv, and (using (2.14)) that the image

does indeed land inside M(n)k (N,χ).

The action on modular forms for Γ(n)(N) is then as one would expect: we let an element

Γ(n)(N)αΓ(n)(N) act on M(n)k (Γ(n)(N)) by writing

Γ(n)(N)αΓ(n)(N) =⊔

Γ(n)(N)αv

and defining

F |Γ(n)(N)αΓ(n)(N) = µ(α)nk2−n(n+1)

2

∑v

F |αv.

When p | N we can define Hecke operators using the same double coset, although care is

required since the double coset decomposition differs according to whether p | N or not.

In Chapter 3 we denote these by the same letter T (n)(p) and T(n)j (p2) for ease of notation.

However in Chapter 4 we change this convention and instead denote these operators by

U (n)(p) and U(n)1 (p2), as we will work with varying levels and it is potentially confusing

whether p | N or not.

Since the modularity subgroup of some F ∈ Sk is not uniquely determined, we introduce

the following definition: we say that F ∈ Sk is p-spherical if there exists N with p - N

23

such that F ∈ Sk(Γ(N)). The action of T = Sp2n(Z)α Sp2n(Z) ∈ H(n)p on such an F is de-

fined via the isomorphism of Lemma 2.7 (which could equivalently be stated with Γ(n)(N)

replacing Γ(n)0 (N), as we noted in the ensuing discussion). Thus H(n)

p acts on p-spherical

elements of Sk, and the action is completely determined by the generators in Lemma 2.7.

This action of Hecke operators on modular forms of unspecified, but p-spherical, levels

will be used in §5.

On the normalization of the Hecke operators. Our notation here is mostly based on that

of [46], which deals with n = 1. Our normalisation of the slash operator in the Siegel case

differs from the classical Andrianov notation because we include the factor of µn(γ)nk/2 to

force scalar matrices to act trivially (this is a natural normalisation to use if one wishes

to study the associated automorphic representation to a Siegel modular form, as well

will do in §5). However, this effect is compensated for in our normalisation of the Hecke

operators: our Hecke operators are normalised as in the Andrianov notation (which is also

the Miyake normalisation when n = 1), the only caveat being that we have interchanged

the roles of j and n− j in T(2)j (p2, χ).

Note that [72] uses a definition of Hecke operators that is equivalent to our double coset

definition except that the representative matrices differ by a factor of p. This makes no

difference because we both normalize the slash operator so that scalars act trivially.

24

Chapter 3

Action of Hecke operators on

Klingen–Eisenstein series

In this chapter we work from the basic principle that modular forms of smaller degree

forms are “simpler”, and that the Siegel lowering operator should therefore allow us to re-

duce questions about non-cuspidal forms of degree n to simpler questions about modular

forms of small degree. The particular application we have in mind is that of computing

Hecke action: we want to understand the action of Hecke operators on Eisenstein series

of degree n as well as we understand the action of Hecke operators on cusp forms of lower

degree.

In order to do so explicitly, however, one requires an explicit relation between the action

of Hecke operators at degree n and degree n− 1. This is particularly pertinent for Siegel

modular forms on the congruence subgroup Γ(n)0 (N) ⊂ Sp2n(Z) because of the connections

with the arithmetic theory of quadratic forms. For F ∈M(n)k (1) it is not difficult to show1

that

Φ(F |T (n)(p)) = (1 + pk−n)Φ(F )|T (n−1)(p).

Slightly more complicated, but still completely explicit, relations were found for the re-

maining Hecke operators Tj(p2) acting on M(n)

k (1) in [40]. As noted in [40] there is also

a version of intertwining relationship due to Zarkovskaja ([69]) which holds in more gen-

1See for example [19] Satz IV.4.4, but beware the differences in normalisation.

25

erality, but takes place on the Satake-transform side and is thus not explicit enough for

our purposes. The first main result of this paper is a completely explicit forms of the

intertwining relations for arbitrary level and character1:

Theorem 3.1. Let n, k, and N be positive integers, let χ be a character modulo N such

that χ(−1) = (−1)k, and let F ∈M(n)k (N,χ). Then

Φ(F |T (n)(p, χ)) = c(n−1)(χ)Φ(F )|T (n−1)(p, χ),

Φ(F |T (n)j (p2, χ)) = c

(n−1)j,j (χ)Φ(F )|T (n−1)

j (p2, χ)

+ c(n−1)j,j−1 (χ)Φ(F )|T (n−1)

j−1 (p2, χ)

+ c(n−1)j,j−2 (χ)Φ(F )|T (n−1)

j−2 (p2, χ)

where

c(n−1)(χ) = (1 + χ(p)pk−n),

c(n−1)j,j (χ) = χ(p)pj+k−2n,

c(n−1)j,j−1 (χ) = χ(p2)p2k−2n + χ(p)(pj+k−2n − pj+k−2n−1) + 1,

c(n−1)j,j−2 (χ) = χ(p2)(p2k−2j+1 − p2k−2n−1),

with the understanding that T(n−1)j (p, χ) is the zero operator for j ∈ −2,−1, n.

Our argument is based on the action of Hecke operators on Fourier expansions. The same

style of argument works in all cases: it is straightforward for T (n)(p, χ) but far more in-

volved for T(n)j (p2, χ). We therefore provide full details in the latter case in §3.2 and some

indications of how one can argue similarly for the former in §3.3. Some readers may prefer

to first read §3.3 for the outline of the argument without the technicalities of §3.2. From

the definitions in §2.4 we see that there is nothing to prove for j = 0; we will deduce

the relations for T(n)j (p2, χ) when j > 0 from analogous relations for a set of averaged

operators T(n)j (p2, χ).

Of course Theorem 3.1 only refers to the output at a single cusp, whereas from §2.1 we see

that we should really be examining the behaviour of F at all (n−1)-cusps simultaneously.

It is therefore necessary to consider the question of intertwining between the action of

Hecke operators and restrictions to other cusps. In this consideration we restrict to the

1Which satisfies the natural condition explained in Remark 2.5.

26

case when N is squarefree. We begin by providing a description of the Satake compactifi-

cation Γ(n)0 (N)\H∗n of Γ

(n)0 (N)\Hn when N is squarefree. The compactification is obtained

by adding quotients of Hr (which are essentially r(r+ 1)/2-dimensional manifolds) to the

boundary. We describe these in detail, and how they intersect each other in lower dimen-

sional components (see Theorem 3.17 for a precise statement). Using this description,

we may parameterise the r-cusps of Γ(n)0 (N)\H∗n with sequences (ln−r, ..., l1) of divisors

of N which are pairwise coprime. Given such a sequence, we define l0 = N/ln−r . . . l1. In

particular, an (n − 1)-cusp corresponds to a divisor l1 of N ; we write Φl1 for the map

restricting to that cusp.1 With our definitions, Φ1 is the usual lowering operator Φ. Using

an argument similar to one used in [3] for modular forms of degree 1 we obtain relations

which differ to those of Theorem 3.1 only in the characters:

Theorem 3.2. Let n and k be positive integers, let N be a squarefree positive integer,

let χ be a character modulo N such that χ(−1) = (−1)k, let p - N be prime, and let

F ∈M(n)k (N,χ). Then

Φl(F |T (n)(p, χ)) = χl1(pn)c(n−1)(χl1χl0)Φl1(F )|T (n−1)(p, χl1χl0),

Φ(F |T (n)j (p2, χ)) = χl1(p

2n)[c

(n−1)j,j (χl1χl0)Φ(F )|T (n−1)

j (p2, χl1χl0)

+c(n−1)j,j−1 (χl1χl0)Φ(F )|T (n−1)

j−1 (p2, χl1χl0)

+c(n−1)j,j−2 (χl1χl0)Φ(F )|T (n−1)

j−2 (p2, χl1χl0)],

where c(n−1), c(n−1)j,j , c

(n−1)j,j−1 and c

(n−1)j,j−2 are as in Theorem 3.1, and the same convention

that T(n−1)j is the zero operator for j ∈ n,−1,−2.

Finally we use Theorems 3.17 and 3.2 to describe the action of the Hecke operators on

the full space of Eisenstein series. We continue to work with N squarefree and p - N

prime. Since we are working with “good” Hecke operators it is not difficult to show, using

the normality of these Hecke operators with respect to the inner product (2.8), that the

Klingen lift of a degree r cuspidal eigenform to a degree n modular form is again an

eigenform. Since there are various ways that this process can be normalised (c.f. Remarks

2.4, 2.6) one must take care in what they mean by Klingen lift. Following the ideas from

Theorem 3.2 we will perform the Klingen lift in a way that is amenable to computation.

1This depends on a choice of coset representative (Remark 2.4), see §3.6 for our precise definition.

27

We work iteratively, and keeping track of the action of the Hecke operators at each stage

we are eventually able to provide formulas for the degree n Hecke eigenvalues in terms

of the Hecke eigenvalues of the degree r cusp form. These formulas are specific to the

r-cusp which we lift from. Lifting a basis of cuspidal eigenforms from each r-cusps, for all

0 ≤ r < n, our lifting process provides a basis of eigenforms for the space of Eisenstein

series. Moreover, we are able to provide formulas for the Hecke eigenvalues of this basis.

The end result is Theorem 3.23.

3.1 Action of Hecke operators on Fourier expansions

Let F ∈ M(n)k (N,χ), so it has a Fourier expansion as in (2.7). It will be convenient

for us to introduce another indexing set for the Fourier expansion. Let Λ be an even

lattice; attached to Λ we have a collection of even integral Gram matrices tGTG; G ∈

GLn(Z), as in §2.2. We will assume that χ(−1) = (−1)k; the modularity of F then

implies a(12T ;F ) = a(1

2tGTG;F ).1 It then make sense to define a(Λ;F ) = a(1

2T ;F )

where T is the Gram matrix for any basis of Λ. Now varying Λ over all even lattices we

obtain all possible (classes of) T , so allowing Λ to vary thus in the following sum we have

F (Z) =∑

Λ

a(Λ;F )eΛZ. (3.1)

Here

eΛZ =∑

G∈O(Λ)\GLn(Z)

e(tr(tGTGZ)),

where O(Λ) is the orthogonal group of the lattice Λ. If we refer to a(Λ;F ) when the

quadratic form on Λ is not integral then we understand a(Λ;F ) = 0.

The formula for the action of Hecke operators on Fourier expansions was found in [26]. It is

most conveniently stated using the indexing of Fourier coefficients by lattices Λ as above,

and moreover it is easiest to work not with the operators Tj(p2) but rather with certain

averaged versions, which we now introduce. We will use these operators extensively in

§3.2, but they will not appear anywhere else in this thesis. To define them, fist let(nr

)p

1To circumvent the assumption χ(−1) = (−1)k one may work with oriented lattices. However we areinterested in Eisenstein series in this chapter, so the point is moot.

28

be the Gaussian binomial coefficient, i.e.(n

r

)p

=r∏i=1

pn−i+1 − 1

pr−i+1 − 1.

Then, for F ∈M(n)k (N,χ),

F |T (n)j (p2, χ) := p(n−j)(n−k+1)χ(pn−j)

j∑t=0

(n− tj − t

)p

F |T (n)t (p2, χ). (3.2)

In order to state the action of these operators on Fourier expansions we first introduce

some useful notation:

Definition 3.3. Let Λ be a lattice, and p be a prime. Let Ω be a lattice such that pΛ ⊂

Ω ⊂ Λ. By the invariant factor theorem we can write

Λ = Λ0 ⊕ Λ1,

Ω = Λ0 ⊕ pΛ1.

We call the tuple (rk(Λ0), rk(Λ1)) the p-type of Ω (in Λ). Similarly, let Ω be a lattice such

that pΛ ⊂ Ω ⊂ p−1Λ. By the invariant factor theorem we can write

Λ = Λ0 ⊕ Λ1 ⊕ Λ2,

Ω = p−1Λ0 ⊕ Λ1 ⊕ pΛ2.

We (again) call the tuple (rk(Λ0), rk(Λ1), rk(Λ2)) the p-type of Ω (in Λ).

Theorem 3.4 (Hafner–Walling, [26]). Let F ∈M(n)k (N,χ) have Fourier expansion (3.1),

and write

(F |T (n)(p, χ))(Z) =∑

Λ

a(Λ;F |T (n)(p, χ))eΛZ.

Then

a(Λ;F |T (n)(p, χ)) =∑

pΛ⊂Ω⊂Λ

A(Ω,Λ;F |T (n)(p, χ))

with A(Ω,Λ;F |T (n)(p, χ)) defined as follows: let (m0,m1) be the p-type of Ω in Λ, and set

E(n)(Ω,Λ) = m0k +m1(m1 + 1)

2− n(n+ 1)

2;

and if Ω has quadratic form Q let Ω1/p denote the same lattice with the quadratic form

x 7→ 1pQ(x) (which may not be integral); then

A(Ω,Λ;F |T (n)(p)) = χ([Ω : pΛ])pE(Ω,Λ)a(Ω1/p;F ).

29

Theorem 3.5 (Hafner–Walling, [26]). Let F ∈M(n)k (N,χ) have Fourier expansion (3.1),

let 1 ≤ j ≤ n, and write

(F |T (n)j (p2, χ))(Z) =

∑Λ

a(Λ;F |T (n)j (p2, χ))eΛZ.

Then

a(Λ;F |T (n)j (p2, χ)) =

∑pΛ⊂Ω⊂ 1

pΛ

A(Ω,Λ;F |T (n)j (p2))

with A(Ω,Λ;F |T (n)j (p2) defined as follows: let (m0,m1,m2) be the p-type of Ω in Λ, and

set

Ej(Ω,Λ) = k(m0 −m2 + j) +m2(m2 +m1 + 1)

+(m1 − n+ j)(m1 − n+ j + 1)

2− j(n+ 1);

in the notation of Definition 3.3 let αj(Ω,Λ) denote the number of totally isotropic sub-

spaces of Λ1/pΛ1 of codimension n− j; then

A(Ω,Λ;F |T (n)j (p2)) = χ(pj−n[Ω : pΛ])pEj(Ω,Λ)αj(Ω,Λ)a(Ω;F ).

Let us finally record two simple results that we will frequently use. The first is that the

well-known fact that the reduction modulo p map SLn(Z) → SLn(Z/pZ) is surjective.

The second is the following simple corollary:

Lemma 3.6. Let G =(H 0B Im

)∈ SLn(Z/pZ), where H ∈ SLn−m(Z/pZ). Let H ∈

SLn−m(Z) such that H mod p = H. Then we can take the lift G ∈ SLn(Z) of G to be of

the form(H 0B Im

).

Proof. Let B be any lift of B, and consider G =(H 0B Im

). Then G ∈ SLn(Z), and G mod

p = G.

30

3.2 The intertwining relations for Φ and T(n)j (p2)

For this section fix n a positive integer and 1 ≤ j ≤ n, and for ease of notation drop the

character from the Hecke operator notation, so that T(n)j (p2) = T

(n)j (p2, χ).1 Let

F (Z) =∑

Λ

a(Λ;F )eΛZ ∈ M(n)k (N,χ).

Applying the Hecke operator T(n)j (p2) then the Siegel lowering operator Φ we obtain

Φ(F |T (n)j (p2))(Z ′) =

∑Λ′

∑pΛ⊂Ω⊂ 1

pΛ

A(Ω,Λ;F |T (n)j (p2))eΛ′Z ′ (3.3)

where Λ varies over all rank n lattices of the form Λ′⊕Zxn, endowed with bilinear form B

obtained by extended the bilinear form B′ of Λ′ by the rule B(xn, y) = 0 for all y ∈ Λ. On

the other hand, if we apply Φ first then T(n−1)j (p2) (where we are now assuming j ≤ n−1

as well) we obtain

(Φ(F )|T (n−1)j (p2))(Z ′) =

∑Λ′

∑pΛ′⊂Ω′⊂ 1

pΛ′

A(Ω′,Λ′; Φ(F )|T (n−1)j (p2))eΛ′Z ′. (3.4)

Proposition 3.11 in the sequel is an intertwining relation for the operators Φ and T(n)j (p2).

We will prove this by comparing Fourier coefficients in (3.3) and (3.4). We therefore fix

a single lattice Λ′ of rank n − 1 endowed with a bilinear form B′. We write Λ for the

lattice Λ′ ⊕ Zxn which is endowed with the bilinear form B extending B′ as above. A

preliminary step in comparing the Fourier coefficients at Λ′ in (3.3) and (3.4) is to know

which lattices pΛ ⊂ Ω ⊂ 1pΛ project on to a given pΛ′ ⊂ Ω′ ⊂ 1

pΛ′. This is the content of

Lemmas 3.7 and 3.9:

Lemma 3.7. There is a one-to-one correspondence between:

• lattices Ω such that pΛ ⊂ Ω ⊂ 1pΛ with p-type (t, s− t, n− s),

• the following data:

– an s-dimensional subspace ∆1 of Λ/pΛ. Let ∆1 be the preimage of this in Λ,

1Unfortunately dropping the character form the Hecke operator introduces a clash with our notation

for Hecke operators on Mk(Γ(n)0 (N)) (i.e. trivial character). However we feel that this is justified as

it makes the following argument more readable. After this section and §3.3 we will always keep thecharacters in our Hecke operator notation.

31

– a t-dimensional subspace ∆2 of ∆1/p∆1, linearly independent of the subspace

pΛ of ∆1/p∆1.

Proof. Suppose we are given Ω with pΛ ⊂ Ω ⊂ p−1Λ and p-type (t, s− t, n− s). By the

invariant factor theorem we can write

Λ = Λ0 ⊕ Λ1 ⊕ Λ2,

Ω =1

pΛ0 ⊕ Λ1 ⊕ pΛ2,

where rk(Λ0) = t, rk(Λ1) = s− t, rk(Λ2) = n− s. Let ∆1 = Λ∩Ω = Λ0⊕Λ1⊕ pΛ2. Then

∆1 = ∆1 + pΛ ⊂ Λ/pΛ has dimension s. Also, pΩ ⊂ ∆1, and ∆2 = pΩ + p∆1 ⊂ ∆1/p∆1

has dimension t, and is linearly independent of pΛ ⊂ ∆1/p∆1.

Conversely, suppose we pick a subspace ∆1 ⊂ Λ/pΛ of dimension s; let ∆1 be its preimage

in Λ. Pick a basis (y1, ..., ys) for ∆1 and extend to a basis (y1, ..., yn) of Λ/pΛ. Note that

(x1, ..., xn) is also a basis for Λ/pΛ, so there existsG1 ∈ GLn(Z/pZ) such that (y1, ..., yn) =

(x1, ..., xn)G1. Replacing y1 by det(G1)−1y1 we may assume G1 ∈ SLn(Z/pZ). Since the

projection map SLn(Z) → SLn(Z/pZ) is surjective, we can pick G1 ∈ SLn(Z) reducing

modulo p to G1. Let (y1, ..., yn) = (x1, ..., xn)G1, so (y1, ..., yn) is a basis for Λ with yi

reducing modulo p to yi and now

∆1 = Zy1 ⊕ ...⊕ Zys ⊕ Zpys+1 ⊕ ...⊕ Zpyn. (3.5)

Note that, in ∆1/p∆1, pΛ = pΛ + p∆1 has basis (pys+1, ..., pyn). Now pick a sub-

space ∆2 ⊂ ∆1/p∆1 linearly independent of pΛ. Let (z1, ..., zt) be a basis for ∆2. Since

the set z1, ..., zt, pys+1, ..., pyn is linearly independent, we can extend it to a basis

(z1, ...zs, pys+1, ..., pyn) for ∆1/p∆. For future reference, call this extension step (*). From

(3.5) we have that (y1, ..., ys, pys+1, ..., pyn) is a basis for ∆1/p∆1. So, modifiyng z1 if

necessary as above, there is G2 ∈ SLn(Z/pZ) such that

(z1, ...zs, pys+1, ..., pyn) = (y1, ..., ys, pys+1, ..., pyn)G2.

In fact, we see G2 =(H 0B In−s

)for some H ∈ SLs(Z/pZ). Pick a lift H ∈ SLs(Z)

of H. Using Lemma 3.6, choose a lift G2 ∈ SLn(Z) of G2 of the form (H 0B I ). Let

32

(z1, ..., zs, pys+1, ..., pyn) = (y1, ..., ys, pys+1, ..., pyn)G2; thus (z1, ..., zs, pys+1, ..., pyn) is a

basis for ∆1, the zi reduce modulo p∆1 to zi, and the preimage of ∆2 in ∆1 is

Zz1 ⊕ ...⊕ Zzt ⊕ Zpzt+1 ⊕ ...⊕ Zpys ⊕ Zp2ys+1 ⊕ ...⊕ Zp2yn.

Recall G2 = (H 0B I ). Then G′2 =

(H 0pB I

)∈ SLn(Z) as well, and we have

(z1, ..., zs, ys+1, ..., yn) = (x1, ..., xn)G1G′2.

Thus (z1, ..., zs, ys+1, ..., yn) is a basis for Λ, and we can consider the lattice

Ω = Z

(1

pz1

)⊕ ...⊕ Z

(1

pzt

)⊕ Zzt+1 ⊕ ...⊕ Zzs ⊕ Zpys+1 ⊕ ...⊕ Zpyn.

Note that this construction is independent of the choice of (y1, ..., yn) and (z1, ..., zt).

We have therefore constructed maps between the two pieces of data, and they are easily

seen to be inverse to each other.

Corollary 3.8. The number of lattices Ω with pΛ ⊂ Ω ⊂ p−1Λ and p-type (t, s− t, n− s)

is(ns

)p

(st

)ppt(n−s).

Proof.(ns

)p

counts the number of s-dimensional subspaces of Λ/pΛ, and(st

)ppt(n−s)

counts the number of t-dimensional subspaces of ∆1/p∆1 linearly independent of pΛ ⊂

∆1/p∆1.

Lemma 3.9. Let Ω′ be a lattice with pΛ′ ⊂ Ω′ ⊂ p−1Λ′ and p-type (l, r−l, n−r−1). Recall

that Λ = Λ′ ⊕ Zxn. Then under the projection Λ → Λ′ the lattices Ω with pΛ ⊂ Ω ⊂ 1pΛ

that project on to Ω′ are classified as follows:

(A) one lattice with p-type (l + 1, r − l, n − r − 1), which (following the proof) we will

denote Ω(1).

(B) pl lattices with p-type (l, r − l + 1, n − r − 1), which we will denote Ω(2)((αi)1≤i≤l)

where αi ∈ Z/pZ for 1 ≤ i ≤ l.

(C) pl+r lattices with p-type (l, r − l, n− r), which we will denote Ω(3)((αi)1≤i≤r) where

αi ∈ Z/p2Z for 1 ≤ i ≤ l and αi ∈ Z/pZ for l + 1 ≤ i ≤ r.

33

(D) for each of the pr−l − 1 non-zero vectors u′ ∈ Λ′1/pΛ′1, pl lattices with p-type (l +

1, r − l − 1, n− r), which we will denote Ω(4)(u′, (γi)1≤i≤l).

Moreover, let Ω be such a lattice projecting on to Ω′. Write

Λ′ = Λ′0 ⊕ Λ′1 ⊕ Λ′2, and Ω′ =1

pΛ′0 ⊕ Λ′1 ⊕ pΛ′2, (3.6)

Λ = Λ0 ⊕ Λ1 ⊕ Λ2 and Ω =1

pΛ0 ⊕ Λ1 ⊕ pΛ2. (3.7)

Then we have the following characterisation of Λ1/pΛ1 in each case:

(A) For Ω = Ω(1), Λ1/pΛ1 = Λ′1/pΛ′1.

(B) For any Ω = Ω(2)((αi)1≤i≤l), Λ1/pΛ1 = Λ′1/pΛ′1 ⊕ (Z/pZ)xn.

(C) For any Ω = Ω(3)((αi)1≤i≤r), Λ1/pΛ1 = Λ′1/pΛ′1.

(D) For Ω = Ω(4)(u′, (γi)1≤i≤l), Λ1/pΛ is a codimension one subspace of Λ′/pΛ′ which

does not contain u′.

Proof. We follow the construction of Lemma 3.7. First pick the subspace ∆1, there are

two possibilities:

1. xn ∈ ∆1. We may assume ys = xn, and choosing our the lifting matrix G1 with the

aid of Lemma 3.6, we may also assume that ys = xn, so that

∆1 = Zy1 ⊕ ...⊕ Zys−1 ⊕ Zxn ⊕ Zpys+1 ⊕ ...⊕ Zpyn.

Here each yi ∈ Λ. Recall that Λ = Λ′ ⊕Zxn. For s+ 1 ≤ i ≤ n write yi = y′i + αixn

where y′i ∈ Λ′. Since xn ∈ ∆1 we may assume αi = 0 for s + 1 ≤ i ≤ n (we could

also do this for 1 ≤ i ≤ s, but it is convenient not to for now). Thus yi = y′i ∈ Λ′

and we have

∆1 = Zy1 ⊕ ...⊕ Zys−1 ⊕ Zxn ⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n.

We now pick ∆2:

34

(a) xn ∈ ∆2. We may assume zt = xn, and choosing our lifting matrix G2 (or, more

precisely, H) appropriately we may also assume that zt = xn. This constructs

the lattice

Ω(1) = Z

(1

pz1

)⊕ ...⊕ Z

(1

pzt−1

)⊕ Z

(1

pxn

)⊕ Zzt+1 ⊕ ...⊕ Zzs ⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n.

Since the zi are in Λ we can write zi = z′i+αixn where z′i ∈ Λ′. Since (1/p)xn ∈

Ω(1) we may assume all αi = 0. Thus our lattice is

Ω(1) = Z

(1

pz′1

)⊕ ...⊕ Z

(1

pz′t−1

)⊕ Z

(1

pxn

)⊕ Zz′t+1 ⊕ ...⊕ Zz′s ⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n

and this projects to

Ω(1)′ = Z

(1

pz′1

)⊕ ...⊕ Z

(1

pz′t−1

)⊕ Zz′t+1 ⊕ ...⊕ Zz′s ⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n.

(b) xn /∈ ∆2. Let z1, ..., zt be a basis for ∆2 and recall py′s+1, ..., py′n is a basis for

pΛ ⊂ ∆1/p∆ as in Lemma 3.7; and moreover that z1, ..., zt, py′s+1, ...py′n is

linearly independent. There are two possibilities:

i. z1, ....zt, py′s+1, ..., py′n, xn is linearly independent. So when we extend to

a basis (z1, ..., zs, py′s+1, ..., pyn′) for ∆1/p∆1 at step (*) in the proof of

Lemma 3.9, we can include xn in this extension, say zs = xn. Choosing

the lifting matrix G2 appropriately we may assume zs = xn as well. Then

we have the lattice

Ω(2) = Z

(1

pz1

)⊕ ...⊕ Z

(1

pzt

)⊕ Zzt+1 ⊕ ...⊕ Zzs−1

⊕ Zxn ⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n.

Again write zi = z′i + αixn where z′i ∈ Λ′. Since xn ∈ Λ′ we may assume

αi = 0 for t + 1 ≤ i ≤ s − 1, and αi ∈ 0, ..., p − 1 for 1 ≤ i ≤ t. Hence

our lattice is

Ω(2)((αi)1≤i≤t) = Z

(1

p(z′1 + α1xn)

)⊕ ...⊕ Z

(1

p(z′t + αtxn)

)⊕ Zz′t+1 ⊕ ...⊕ Zz′s−1 ⊕ Zxn

⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n

35

and, for any choice of (αi), this projects to

Ω(2)′ = Z

(1

pz′1

)⊕ ...⊕ Z

(1

pz′t

)⊕ Zz′t+1 ⊕ ...⊕ Zz′s−1

⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n.

ii. z1, ....zt, py′s+1, ..., py′n, xn is linearly dependent, so we have a relation

xn =∑aizi +

∑bipy′i. If all the ai are 0 then xn ∈ pΛ which is a con-

tradiction; and if all the bi are 0 then xn ∈ ∆2 which is also a contra-

diction. Modifying the basis z1, ..., zt for ∆2, we may therefore assume

xn = zt − pu′ for some non-zero pu′ ∈⊕n

i=s+1 Fppyi, or zt = xn + pu′. Ex-

tend to a basis (z1, ..., zs, py′s+1, ..., py′n) for ∆1/p∆ as in step (*) of in the

proof of Lemma 3.7. Pick some lift u′ of u′. Recall that (x1, ..., xn) is our

basis for Λ, and that u′ ∈ Λ′ where Λ = Λ′⊕Zxn, so (x1, ..., xn−1, xn+pu′)

is also a basis for Λ. Note that (xn + pu′) + p∆1 = zt. We can then choose

a lifting matrix appropriately with respect to this basis to ensure that

zt = xn + pu′ is a basis vector of Ω, so that our lattice is

Ω(4)(u′) = Z

(1

pz1

)⊕ ...⊕ Z

(1

pzt−1

)⊕ Z

(1

pxn + u′

)⊕

Zzt+1 ⊕ ...⊕ Zzs ⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n.

Write each zi = z′i + αixn where z′i ∈ Λ′. Note that for t + 1 ≤ i ≤ s we

have

z′i = (z′i + αixn)− αip(

1

pxn + y′n

)+ αipy

′n

so we can assume αi = 0 for t + 1 ≤ i ≤ s. Similarly we may assume

αi ∈ 0, ..., p− 1 for 1 ≤ i ≤ t. Then our lattice is

Ω(4)(u′, (αi)1≤i≤t−1)

= Z

(1

p(z′1 + α1xn)

)⊕ ...⊕ Z

(1

p(z′t−1 + αt−1xn)

)⊕ Z

(1

pxn + u′

)⊕ Zz′t+1 ⊕ ...⊕ Zz′s

⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n

36

and, for any choice of (αi), this projects to

Ω(4)(u′)′ = Z

(1

pz′1

)+ ...+ Z

(1

pz′t−1

)+ Zu′

+ Zz′t+1 + ...+ Zz′s + Zpy′s+1 + ...+ Zpy′n.

2. In contrast to 1. we now have xn /∈ ∆1. Pick a basis y1, ..., ys for ∆1. When we

extend to a basis for Λ/pΛ we may assume xn is included in that extension, say

yn = xn. Choosing our lifting matrix G2 with the aid of Lemma 3.6 we may assume

yn = xn. Follow through the rest of the construction as in Lemma 3.9, we construct

the lattice

Ω(3) = Z

(1

pz1

)⊕ ...⊕ Z

(1

pzt

)⊕ Zzt+1 ⊕ ...⊕ Zzs

⊕ Zpys+1 ⊕ ...⊕ Zpyn−1 ⊕ Zpxn.

Write each zi = z′i + αixn, yi = y′i = αixn where x′i, y′i ∈ Λ′. Since pxn ∈ Ω(3),

we may assume αi = 0 for i ≥ s + 1, αi = 0, ..., p − 1 for t + 1 ≤ i ≤ s and

αi ∈ 0, ..., p2 − 1 for 1 ≤ i ≤ t. Then we have

Ω(3)((αi)1≤i≤s) = Z

(1

p(z′1 + α1xn)

)⊕ ...⊕ Z

(1

p(z′t + αtxn)

)⊕ Z(z′t+1 + αt+1xn)⊕ ...⊕ Z(z′s + αsxn)

⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n−1 ⊕ Zpxn

and, for any choice of (αi), this projects on to

Ω(3)′ = Z

(1

pz′1

)⊕ ...⊕ Z

(1

pz′t

)⊕ Zz′t+1 ⊕ ...⊕ Zz′s ⊕ Zpy′s+1 ⊕ ...⊕ Zpy′n−1.

Now fix a lattice Ω′ with p-type (l, r− l, n−r−1). We consider in the following cases how

many lattices project on to Ω′, what their p-types are, and the structure of their Λ1/pΛ1

part in (3.7):

(A) Consider case 1(a). Here we see that, since Ω(1)′ has p-type (l, r− l, n− r− 1), Ω(1)

must have p-type (l+ 1, r− l, n− r− 1). Also, Ω(1) is uniquely determined by Ω(1)′.

Finally, by inspection we see that Λ1/pΛ1 = Λ′1/pΛ′1.

(B) Consider case 1(b)(i). Here we see that Ω(2)((αi)) must have p-type (l, r − l +

1, n − r − 1), and there are pl lattices with the same projection Ω(2)′. Moreover,

Λ1/pΛ1 = Λ′1/pΛ′1 ⊕ (Z/pZ)xn.

37

(C) Consider case 2. Here we see that Ω(3)((αi)) must have p-type (l, r − l, n− r), and

there are pr+l lattices with the same projection Ω(3)′. Moreover, Λ1/pΛ1 = Λ′1/pΛ′1.

(D) Consider case 1(b)(ii). Since Ω(4)(u′)′ has p-type (l, r − l, n − r − 1) we see that

Ω(4)(u′, (αi)) must have p-type (l + 1, r − l − 1, n − r). Also there are pl lattices

with the same projection Ω(4)(u′)′, and by inspection we see that for these lattices

Λ1/pΛ1 is a codimension 1 subspace of Λ′1/pΛ′1 which does not contain u′.

We now describe some cases when different choices of the vector u′ give different

lattices with the same projection. Following this, we will prove that, after taking

this in to account, we have constructed all lattices projecting on to Ω′. First note

that Ω(4)(u′1, (αi)) = Ω(4)(u′2, (βi)) if and only if (αi) = (βi) and u′1− u′2 ∈ pΛ′. Now

fix a basis for the projection

Ω′ = Z

(1

pw′1

)⊕ ...⊕ Z

(1

pw′l

)⊕ Zw′l+1 ⊕ ...⊕ Zw′r ⊕ Zpw′r+1 ⊕ ...⊕ Zpw′n−1.

Take u′ = a1w′l+1 + ...+arw

′r to be any vector such that u′ /∈ pΛ′. We easily see that,

for any choice of (αi), Ω(4)(u′, (αi))′ = Ω′. As u′ varies such that u′ + pΛ′ covers all

pr−l − 1 non-zero possibilities, we obtain pl(pr−l − 1) distinct lattices Ω(4)(u′, (αi))

all projecting on to Ω′.

We have now listed all possible rank n lattices projecting on to Ω′. Note that these lattice

are all distinct: indeed, the lattices within each case are distinct by construction, and

there can be no equality between two lattices in different cases since the p-type of their

projections are different. This completes the proof.

Remark 3.10. Let us demonstrate the consistency of the numbers from Lemma 3.9 by

counting the number M(t, s − t, n − s) of rank n lattices with p-type (t, s − t, n − s): on

the one hand this is equal to(ns

)p

(st

)ppt(n−s), by Corollary 3.8. On the other hand, using

Lemma 3.9, it is equal to

38

M(t, s− t, n− s)

=

(n− 1

s− 1

)p

(s− 1

t− 1

)p

p(t−1)(n−s) + pt(n− 1

s− 1

)p

(s− 1

t

)p

pt(n−s)

+ pt+s(n− 1

s

)p

(st

)ppt(n−s−1)

+ (ps − pt−1)

(n− 1

s

)p

(s

t− 1

)p

p(t−1)(n−s−1)

= pt(n−s)

[(n− 1

s− 1

)p

(s− 1

t− 1

)p

p−n+s +

(n− 1

s− 1

)p

(s− 1

t

)p

pt

+

(n− 1

s

)p

(st

)pps +

(n− 1

s

)p

(s

t− 1

)p

p−n+s(ps − pt−1)

].

It is then straightforward using the properties of the Gaussian binomial coefficient to prove

that the right hand side is equal to pt(n−s)(ns

)p

(st

)p.

Proposition 3.11. Let F ∈ Mk(N,χ), 1 ≤ j ≤ n, and let Λ be a Z-lattice with a

Z-valued quadratic form. Then

Φ(F |T (n)j (p2)) = Φ(F )|T (n−1)

j (p2) + c(n−1)j,j−1 Φ(F )|T (n−1)

j−1 (p2)

+ c(n−1)j,j−2 Φ(F )|T (n−1)

j−2 (p2)

where

c(n−1)j,j−1 = χ(p2)p2k−j−n + χ(p)pk−n + pn−j,

c(n−1)j,j−2 = χ(p2)(p2k−2j+1 − p2k−n−j).

We adopt the convention that T(n−1)j (p2) is the zero operator for j ∈ n,−1,−2.

Proof. Continue with the fixed lattice Λ′, and the lattice Λ = Λ′⊕Zxn with the quadratic

form extended as above. It suffices to show that∑pΛ⊂Ω⊂ 1

pΛ

A(Ω,Λ;F |T (n)j (p2)) =

∑pΛ′⊂Ω′⊂ 1

pΛ′

A(Ω′,Λ′; Φ(F )|T (n−1)j (p2))

+ c(n−1)j,j−1

∑pΛ′⊂Ω′⊂ 1

pΛ′

A(Ω′,Λ′; Φ(F )|T (n−1)j−1 (p2))

+ c(n−1)j,j−2

∑pΛ′⊂Ω′⊂ 1

pΛ′

A(Ω′,Λ′; Φ(F )|T (n−1)j−2 (p2))

39

Write π for map of Lemma 3.9 (i.e. the projection xn 7→ 0). For Ω′ a rank n − 1 lattice

set

B(Ω′,Λ′;F |T (n)j (p2)) =

∑Ω s.t.π(Ω)=Ω′

A(Ω,Λ;F |T (n)j (p2)).

So ∑pΛ⊂Ω⊂ 1

pΛ

A(Ω,Λ;F |T (n)j (p2)) =

∑pΛ′⊂Ω′⊂ 1

pΛ′

B(Ω′,Λ′;F |T (n)j (p2)),

and it suffices to show that

B(Ω′,Λ′;F |T (n)j (p2)) = A(Ω′,Λ′; Φ(F )|T (n−1)

j (p2))

+ c(n−1)j,j−1 A(Ω′,Λ′; Φ(F )|T (n−1)

j−1 (p2))

+ c(n−1)j,j−2 A(Ω′,Λ′; Φ(F )|T (n−1)

j−2 (p2))

(3.8)

for each pΛ′ ⊂ Ω′ ⊂ 1pΛ′.

Take such an Ω′, say with p-type (l, r − l, n − r − 1). Then the Ω such that π(Ω) = Ω′

are described by Lemma 3.9. Working from the notation of Lemma 3.9, let us write Ω(2)

for any lattice of the form Ω(2)((αi)), Ω(3) any lattice of the form Ω(3)((αi)), and Ω(4)(u′)

any lattice of the form Ω(4)(u′, (αi)). Then it is easy to see that

α(n)j (Ω(1),Λ) = α

(n−1)j−1 (Ω′,Λ′) (3.9)

and

α(n)j (Ω(3),Λ) = α

(n−1)j−1 (Ω′,Λ′). (3.10)

Indeed, by Lemma 3.9 we have, for Ω = Ω(1), Λ1/pΛ1 = Λ′1/pΛ′1. Thus α

(n)j (Ω(1),Λ) counts

the number of codimesnion n−j totally isotropic subspaces of Λ′1/pΛ′1. But α

(n−1)j−1 (Ω′,Λ′)

also counts the number of codimension (n−1)− (j−1) = n− j totally isotropic subspace

of Λ′1/pΛ′1. The same argument works for Ω(3).

For Ω = Ω(2) we have Λ1/pΛ1 = Λ′1/pΛ′1⊕ (Z/pZ)xn, and α

(n)j (Ω(2),Λ) counts the number

of codimension n−j totally isotropic subspaces of this space. Ω′ has p-type (l, r−l, n−r−1)

so Λ1/pΛ1 has dimension r − l + 1, so a codimension n − j subspace is a dimension

r− l+ 1− n+ j subspace. Recall that the line (Z/pZ)xn is isotropic. A totally isotropic

subspace of Λ1/pΛ1 of dimension r−l−n+j+1 is therefore either the direct sum (Z/pZ)xn

40

with a dimension r− l−n+ j subspace of Λ′/pΛ′ (of which there are α(n−1)j−1 (Λ′,Ω′)); or is

formed by picking a totally isotropic subspace of Λ′/pΛ′ of dimension r− l−n+ j+ 1 (of

which there are α(n−1)j (Λ′,Ω′)) and adding some αxn (α ∈ (Z/pZ)) to each basis vector.

We therefore have

α(n)j (Ω(2),Λ) = pr−l−n+j+1α

(n−1)j (Ω′,Λ′) + α

(n−1)j−1 (Ω′,Λ′). (3.11)

Finally, consider∑

u′ α(n)j (Ω(4)(u′),Λ). For Ω = Ω(4)(u′), Λ/pΛ is a codimension 1 subspace

of Λ′/pΛ′ which does not contain u′. α(n)j (Ω(4)(u′),Λ), which counts the number of totally

isotropic codimension n−j subspaces of Λ/pΛ, therefore counts totally isotropic subspaces

of Λ/pΛ of dimension r−l−n+j−1. Subspaces of this dimension in Λ′/pΛ′ are counted by

α(n−1)j−2 (Ω′,Λ′). Let V be a totally isotropic subspace of Λ′/pΛ′ of dimension r−l−n+j−1;

we will consider how many times V is counted in∑

u′ α(n)j (Ω(4)(u′),Λ). For a fixed choice

of nonzero u′ ∈ Λ′/pΛ′ we see that V is counted by α(n)j (Ω(4)(u′),Λ) if and only if u′ /∈ V .

So the number of times V is counted in∑

u′ α(n)j (Ω(4)(u′),Λ) is precisely the number

of nonzero vectors u′ ∈ Λ′/pΛ′ that are not contained in V . Since V has codimension

n− j + 1, the number of such u′ is pn−j+1 − 1. We therefore have∑u′

α(n)j (Ω(4)(u′),Λ) = (pn−j+1 − 1)α

(n−1)j−2 (Ω′,Λ′). (3.12)

Now the remaining quantities appearing in A(Ω,Λ;F |T (n)j (p2) depend only on the p-type

of Ω in Λ. Using this observation and the above computations together with the count of

Lemma 3.9 we can write

B(Ω′,Λ′;F |T (n)j (p2)) = A(Ω(1),Λ;F |T (n)

j (p2))

+ plA(Ω(2),Λ;F |T (n)j (p2))

+ pr+lA(Ω(3),Λ;F |T (n)j (p2))

+ pl∑u′

A(Ω(4)(u′),Λ;F |T (n)j (p2)).

(3.13)

Now the appearance to the subscript j on the right hand side of (3.11) suggests that we

should consider Ω(2) first: one easily computes from

Ej(Ω(2),Λ) = n− r − j − 1 + Ej(Ω

′,Λ′)

χ(pj−n[Ω(2) : pΛ]) = χ(pj−n+1[Ω′ : pΛ′]),

41

and (3.11) that

plA(Ω(2),Λ; T(n)j (p2)) = A(Ω′,Λ′; Φ(F )|T (n−1)

j (p2))

+ plχ(pj−n[Ω(2) : pΛ])pEj(Ω(2),Λ)α

(n−1)j−1 (Ω′,Λ′)a(Ω′; Φ(F )).

Substituting this in to (3.13) we have

B(Ω′,Λ′;F |T (n)j (p2)) = A(Ω′,Λ′; Φ(F )|T (n−1)

j (p2))

+ A(Ω(1),Λ;F |T (n)j (p2))

+ plχ(pj−n[Ω(2) : pΛ])pEj(Ω(2),Λ′)α

(n−1)j−1 (Ω′,Λ′)a(Ω′; Φ(F ))

+ pr+lA(Ω(3),Λ;F |T (n)j (p2))

+ pl∑u′

A(Ω(4)(u′),Λ;F |T (n)j (p2)).

(3.14)

From the formulas

E(n)j (Ω(1),Λ) = 2k − j − n+ E

(n−1)j−1 (Ω′,Λ′),

E(n)j (Ω(2),Λ) = −l − n+ k + E

(n−1)j−1 (Ω′,Λ′),

E(n)j (Ω(3),Λ) = −r − l + n− j + E

(n−1)j−1 (Ω′,Λ′),

and

χ(pj−n[Ω(1) : pΛ]) = χ(p2)χ(pj−n[Ω′ : pΛ′]),

χ(pj−n[Ω(2) : pΛ]) = χ(p)χ(pj−n[Ω′ : pΛ′]),

χ(pj−n[Ω(3) : pΛ]) = χ(pj−n[Ω′ : Λ′]),

together with (3.9) and (3.10) we easily compute

A(Ω(1),Λ;F |T (n)j (p2)) = χ(p2)p2k−j−nA(Ω′,Λ′; Φ(F )|T (n−1)

j−1 (p2)),

plχ(pj−n[Ω(2) : pΛ])pEj(Ω(2),Λ)α

(n−1)j−1 (Ω′,Λ′)a(Ω′; Φ(F ))

= χ(p)pk−nA(Ω′,Λ′; Φ(F )|T (n−1)j−1 (p2)),

pr+lA(Ω(3),Λ;F |T (n)j (p2)) = pn−jA(Ω′,Λ′; Φ(F )|T (n−1)

j−1 (p2)).

Substituting these in to (3.14) we obtain

B(Ω′,Λ′;F |T (n)j (p2))

= A(Ω′,Λ′; Φ(F )|T (n−1)j (p2))

+ (χ(p2)p2k−j−n + χ(p)pk−n + pn−j)A(Ω′,Λ′; Φ(F )|T (n−1)j−1 (p2))

+ pl∑u′

A(Ω(4)(u′),Λ;F |T (n)j (p2)).

(3.15)

42

Finally, from

Ej(Ω(4)(u′),Λ) = 2k − j − n− l − Ej−2(Ω′,Λ′)

χ(pj−n[Ω(4)(u′) : pΛ]) = χ(p2)χ(pj−n−1[Ω′ : pΛ′]),

and (3.12) we compute

pl∑u′

A(Ω(4)(u′),Λ;F |T (n)j (p2))

= χ(p2)p2k−j−n(p2k−2j+1 − p2k−j−n)A(Ω′,Λ′; Φ(F )|T (n−1)j−2 (p2))

Substituting this in to (3.15) we obtain

B(Ω′,Λ′; T(n)j (p2)) = A(Ω′,Λ′; T

(n−1)j (p2))

+ (χ(p2)p2k−j−n + χ(p)pk−n + pn−j)A(Ω′,Λ′; T(n−1)j−1 (p2))

+ χ(p2)(p2k−2j+1 − p2k−j−n)A(Ω′,Λ′; T(n−1)j−2 (p2))

(3.16)

This is (3.8), so the proof is complete.

From this it is straightforward to deduce Theorem 3.1:

Proof of Theorem 3.1 for T(n)j (p2). Applying Φ to the definition (3.2) we have

Φ(F |T (n)j (p2)) = p(n−j)(n−k+1)χ(pn−j)

j∑t=0

(n− tj − t

)p

Φ(F |T (n)t (p2)).

Now it is clear from Proposition 3.11 that

Φ(F |T (n)t (p2)) =

t∑s=0

c(n−1)t,s Φ(F )|T (n−1)

s (p2)

for some complex numbers c(n−1)t,s . Thus we can write

Φ(F |T (n)j (p2)) = p(n−j)(n−k+1)χ(pn−j)

j∑t=0

(n− tj − t

)p

t∑s=0

c(n−1)t,s Φ(F )|T (n−1)

s (p2). (3.17)

On the other hand

Φ(F |T (n)j (p2)) = Φ(F )|T (n−1)

j (p2) + c(n−1)j,j−1 Φ(F )|T (n−1)

j (p2)

+ c(n−1)j,j−2 Φ(F )|T (n−1)

j (p2)

43

which we can write as

Φ(F |T (n)j (p2))

= p(n−1−j)(n−k)χ(pn−1−j)

j∑t=0

(n− 1− tj − t

)p

Φ(F )|T (n−1)t (p2)

+ c(n−1)j,j−1 p

(n−j)(n−k)χ(pn−j)

j−1∑t=0

(n− 1− tj − 1− t

)p

Φ(F )|T (n−1)t (p2)

+ c(n−1)j,j−2 p

(n−j+1)(n−k)χ(pn+1−j)

j−2∑t=0

(n− 1− tj − 2− t

)p

Φ(F )|T (n−1)t (p2).

(3.18)

Comparing the coefficient of Φ(f)|T (n−1)j (p2) between (3.17) and (3.18) we have

p(n−j)(n−k+1)χ(pn−j)c(n−1)j,j = p(n−1−j)(n−k)χ(pn−1−j)

from which we get c(n−1)j,j = χ(p)pj+k−2n. Arguing similarly but with more tedious com-

putation we compute the remaining coefficients and deduce Theorem 3.1.

3.3 The intertwining relation for Φ and T (n)(p)

We now describe how one can use a similar (but much easier) argument to that of §3.2

to derive the intertwining relation for the operator T (n)(p) := T (n)(p, χ).1 As before let

F ∈M(n)k (N,χ) have Fourier expansion (3.1). Applying the Hecke operator T (n)(p) then

the Siegel lowering operator Φ to we obtain

Φ(F |T (n)(p))(Z ′) =∑Λ′

∑pΛ⊂Ω⊂ 1

pΛ

A(Ω,Λ;F |T (n)(p))eΛ′Z ′. (3.19)

If we apply Φ first then T (n−1)(p) we obtain

(Φ(F )|T (n−1)(p))(Z ′) =∑Λ′

∑pΛ′⊂Ω′⊂ 1

pΛ′

A(Ω′,Λ′; Φ(F )|T (n−1)(p))eΛ′Z ′, (3.20)

and we must compare the Fourier coefficients in (3.19) and (3.20). Fix an indexing lattice

Λ′. Let Ω′ be a rank n− 1 lattice, and define

B(Ω′,Λ′;F |T (n)(p)) =∑

Ω s.t.π(Ω)=Ω′

A(Ω,Λ;F |T (n)(p)).

1Like the previous section this notation clashes with our notation for Hecke operators on spaces withtrivial character. It will also be used only in this section.

44

As in the proof of Proposition 3.11 we find that it suffices to show that

B(Ω′,Λ′;F |T (n)(p)) = (1 + χ(p)pk−n)A(Ω′,Λ′; Φ(F )|T (n−1)(p2)) (3.21)

for each pΛ′ ⊂ Ω′ ⊂ 1pΛ′.

It is again useful to classify all the lattices Ω which project on to a given Ω′, and record

some of the properties of such Ω. This is provided by the following two lemmas:

Lemma 3.12. There is a one-to-one correspondence between:

1. lattices Ω such that pΛ ⊂ Ω ⊂ Λ with p-type (s, n− s),

2. s-dimensional subspaces ∆ of Λ/pΛ.

Lemma 3.13. Let Ω′ be a lattice with pΛ′ ⊂ Ω′ ⊂ Λ′ and p-type (r, n− r− 1). Under the

map π : Λ→ Λ′, the lattices that project on to Ω′ are classified as follows:

(A) one lattice with p-type (r + 1, n− r − 1), which we will denote Ω(1),

(B) pr lattices with p-type (r, n − r), which we will denote by Ω(2)((αi)1≤i≤s), where

αi ∈ Fp.

The proofs are similar to (but easier than) Lemmas 3.7 and 3.9. Then writing Ω(2) for

any Ω(2)(αi) we compute, using the notation of Theorem 3.4,

E(n)(Ω(1),Λ) = k − n+ E(n−1)(Ω′,Λ′),

E(n)(Ω(2),Λ) = −r + E(n−1)(Ω′,Λ′),

and

χ([Ω(1) : pΛ]) = χ(p)χ([Ω′ : pΛ′]),

χ([Ω(2) : pΛ]) = χ([Ω′ : pΛ′]),

so that

A(Ω(1),Λ;F |T (n)(p)) = χ(p)pk−nA(Ω′,Λ′; Φ(F )|T (n−1)(p))

A(Ω(2),Λ;F |T (n)(p) = p−rA(Ω′,Λ′; Φ(F )|T (n−1)(p)).(3.22)

But by Lemma 3.13 we have

B(Ω′,Λ′;F |T (n)(p)) = A(Ω(1),Λ;F |T (n)(p)) + prA(Ω(2),Λ;F |T (n)(p)).

Substituting (3.22) in to this we obtain (3.21). This proves the intertwining relation for

T (n)(p) stated in Theorem 3.1.

45

3.4 Review of the Satake compactification

Let Γ(n) be a congruence subgroup of Sp2n(Q), so that Γ(n) acts on Hn by (2.1), the

resulting quotient space Γ(n)\Hn is a complex analytic space of dimension n(n + 1)/2.1

There are various approaches to compactifying this space in the literature but the sim-

plest and most important for the classical theory of Siegel modular forms is the Satake

compactification. We will briefly review this construction; our account is based on [54], in

which a very explicit description of the cuspidal structure in the case of degree two and

paramodular level is also given. In the following section we will provide a similar explicit

description for level Γ(n)0 (N) when N is squarefree.

Let C2n×nrank n ⊂ C2n×n be the subset of rank n matrices. Let

GrC(2n, n) = C2n×nrank n/GLn(C)

be the Grassmannian of rank n subspaces of C2n, and consider the subspace of isotropic

subspaces

GrisoC (2n, n) =

MN

∈ GrC(2n, n);(tM tN

)0n −1n

1n 0n

MN

= 0

,

where [MN ] denotes the class of (MN ) ∈ C2n×nrank n in GrC(2n, n); the condition defining

GrisoC (2n, n) ⊂ GrC(2n, n) is immediately seen to be independent of the choice of repre-

sentative for the class. We shall endow GrisoC (2n, n) with the complex structure it naturally

inherits from these definitions. We let Sp2n(C) act on GrisoC (2n, n) via matrix multiplica-

tion from the left.

Consider the upper half space Hr for 0 ≤ r ≤ n, with the convention H0 = ∞. Let

jr,n : Hr → GrC(2n, n), for 0 < r ≤ n, be given by

jr,n(Z) =

1n(Z−1 0

0 0

) .

1Γ(n)\Hn is essentially a complex manifold, but, as in the case n = 1, the existence of elliptic pointsstops this from being strictly true. These subtleties, and the related issue of when we are actually allowedto say “variety” in the sequel, are irrelevant for our purposes.

46

For r = 0, set j0,n(Z) =[

1n0n

]. One easily sees that jr,n(Hr) ⊂ jn,n(Hn) (the closure taking

place inside GrisoC (2n, n)). For 0 ≤ r ≤ n consider the orbit Sp2n(Q)jr,n(Hr); note that

when r = n this is just Hn, but for r < n it is strictly larger than Hr. Now define a

subspace H∗n of GrisoC (2n, n) by

H∗n =n⊔r=0

Sp2n(Q)jr,n(Hr).

Then H∗n is naturally equipped with an action of Γ(n), and the Satake compactification of

Γ(n)\Hn is simply the quotient Γ(n)\H∗n. Now Γ(n)\H∗n, being a subquotient of GrC(2n, n),

comes equipped with a natural topology, under which it becomes a compact Hausdorff

space. We note that

Sp2n(Q)jr,n(Hr) =⊔i

Γ(n)γijr,n(Hr) (3.23)

where the γi are a system of representatives for

Γ(n)\ Sp2n(Q)/ StabSp2n(Q)(jr,n(Hr))

One can explicitly compute that this stabiliser is equal to Pn,r(Q), where Pn,r ⊂ Sp2n

is the parabolic subgroup defined in (2.4). Recall also the canonical surection ωn,r from

(2.5); this is split by the map ξr,n : Sp2r(Q)→ Pn,r(Q) defined by

ξr,n

A11 B11

C11 D11

=

A11 0 B11 0

0 1n−r 0 0n−r

C11 0 D11 0

0 0n−r 0 1n−r

.

Now consider an individual Γ(n)γjr,n(Hr) in (3.23). Let

Γ(r)γ := ωn,r(γ

−1Γ(n)γ ∩ Pn,r(Q)).

The map defined on Hr by

Z 7→ Γ(n)\Γ(n)γjr,n(Z)

induces an isomorphism

Γ(r)γ \Hr → Γ(n)\Γ(n)γjn,r(Hr) ⊂ Γ(n)\H∗n.

47

The space Γ(r)γ \Hr is therefore embedded inside Γ(n)\H∗n. This is a space of dimension

r(r+ 1)/2, and we shall (temporarily) refer to this as the r-cusp of Γ(n)\H∗n associated to

γ.

Now an r-cusp Γ(r)γ \Hr is the quotient of an upper half plane by a congruence subgroup,

and we could therefore form the compactification as above. Abstractly this would involve

forming the space H∗r =⊔rs=0 Sp2r(Q)js,r(Hs), and writing

Sp2r(Q)js,r(Hs) =⊔i

Γ(r)γ ρijs,r(Hs),

where the ρi are a system of representatives for

Γ(r)γ \ Sp2r(Q)/Pr,s(Q).

However we want to construct this compactification not with a new incarnation but rather

in the already-carnate space Γ(n)\H∗n. We therefore extend the embedding of Γ(r)γ \Hr to an

embedding of Γ(r)γ \H∗r as follows: given 0 ≤ s ≤ r, Z ∈ Hs, ρ ∈ Sp2r(Q), and γ ∈ Sp2n(Q)

consider the map

Γ(r)γ ρjs,r(Z) 7→ Γ(n)γξr,n(ρ)js,n(Z).

This induces a well-defined isomorphism

Γ(r)γ \Γ(r)

γ ρjs,r(Hs)→ Γ(n)\Γ(n)γξr,n(ρ)js,n(Hs).

Varying s and ρ we obtain an embedding Γ(r)γ \H∗r → Γ(n)\H∗n (in fact, the image of Γ

(r)γ \H∗r

under this embedding is simply the closure of Γ(r)γ \Hr in Γ(n)\H∗n). We shall replace our

earlier convention and now call Γ(r)γ \H∗r (viewed inside Γ(n)\H∗n) the r-cusp of Γ(n)\H∗N

associated to γ.

Remark 3.14. The arithmetic subgroup Γ(r)γ , and hence the structure of the cusp Γ

(r)γ \H∗r,

depends on the choice of representative γ for Γ(n)γPn,r(Q). More precisely, just as in

Remark 2.4, it is invariant under left multiplication by Γ(n), but changes by a conju-

gation if we right multiply by some element of Pn,r(Q). Similarly, one may work with

instead with the double coset space Γ(n)\GSp2n(Q)/P ∗n,r(Q) (which is in bijection with

Γ(n)\ Sp2n(Q)/Pn,r(Q)) where P ∗n,r is the parabolic subgroup of GSp2n which contains

48

Pn,r, and a similar statement holds. For the remainder of this section and for §3.5 we are

only interested in properties of the double cosets so this remark is unimportant. However,

this technicality will become important from §3.6 onwards.

We now record some observations regarding cusp crossings: let γ represent an r-cusp of

Γ(n)\H∗n and ρ represent an s-cusp of Γ(r)γ \H∗r. By the above embedding, the latter may

be thought of as an s-cusp of Γ(n)\H∗n; explicitly this is the s-cusp given by the double

coset Γ(n)γξr,n(ρ)Pn,s(Q). Then:

• if this same coset can be realised with two inequivalent γ and γ′ (i.e. the double

cosets Γ(n)γPn,r(Q) and Γ(n)γ′Pn,r(Q) are different), then the two distinct r-cusps

corresponding to γ and γ′ intersect at this s-cusp,

• if this same coset can be obtained with the same (or just equivalent) γ but inequiv-

alent ρ and ρ′ (i.e. Γ(r)γ ρPr,s(Q) and Γ

(r)γ ρ′Pr,s(Q) are different) then the r-cusp

corresponding to γ self-intersects at this s-cusp.

3.5 The Satake compactification of Γ(n)0 (N)\Hn

In this section we will provide an explicit description of the cuspidal configuration of

Γ(n)0 (N)\H∗n, where N is square-free. It is well-known that for n = 1 that the 0-cusps are

in bijection with positive divisors of N . For n = 2 one must consider not only 1- and

0-cusps, but also how the former may cross at the latter. An account of this is given in

[5]. Motivated by this we will proceed analogously for general n.

Recall from §3.4 that the r-cusps of Γ(n)0 (N)\H∗n correspond bijectively to

Γ(n)0 (N)\ Sp2n(Q)/Pn,r(Q).

We begin by describing representatives for this. For each 1 ≤ r ≤ n and each divisor l of

49

N fix a matrix γ(r)(l) ∈ Sp2r(Z) satisfying

γ(r)(l) ≡

0r −1r

1r 0r

mod l2,

1r 0r

0r 1r

mod (N/l)2.

(3.24)

This is possible since, for all M ∈ Z≥1, the reduction modulo M map Sp2r(Z) →

Sp2r(Z/MZ) is surjective. Next, given a sequence of positive integers l1, ..., ln−r, assumed

to be pairwise coprime and each a divisor of N , define

γ(n)r (ln−r, ..., l1) = γ(n)(ln−r)ξn−1,n(γ(n−1)(ln−r−1)) . . . ξr+1,n(γ(r+1)(l1)) (3.25)

Set l0 = N/(ln−r . . . l1). To explain the ordering of the indices, note that

γ(n)r (ln−r, ..., l1) ≡

0r+i −1r+i

1n−r−i 0n−r−i

1r+i 0r+i

0n−r−i 1n−r−i

mod l2i

for n − r ≥ i ≥ 1, and γ(n)r (ln−r, ..., l1) ≡ 12n mod l20. On the other hand, write any

element γ of Sp2n(Q) as ( A BC D ), and C in turn as

(C11 C12C21 C22

)with C22 size n − r. Then

the under the left action of Γ(n)0 (N) and the right action of Pn,r(Q) we see that rkp(C22),

the rank of C22 modulo p for p | N , is invariant. Going back to γ = γ(n)r (ln−r, ..., l1) with

ln−r, ..., l1 pairwise coprime divisors of N , we see that p | li = p; rkp(C22) = i. With

our definition this holds with i = 0 as well.

Lemma 3.15. Continue with the above notation. Then as (ln−r, ..., l1) varies over all

tuples of pairwise coprime positive divisors of N , the γ(n)r (ln−r, ..., l1) describe a complete

system of coset representatives for

Γ(n)0 (N)\ Sp2n(Q)/Pn,r(Q).

Proof. By the discussion preceding the statement of the lemma we see that the γ(n)r (ln−r, ..., l1)

are inequivalent for distinct tuples (ln−r, ..., l1). One can argue further form these rank ob-

servations to see that the γ(n)r (ln−r, ..., l1) actually form a complete set of representatives.

Alternatively this follows since they agree in number with those of [6] Lemma 8.1.

50

Henceforth we shall identify a cusp of Γ(n)0 (N)\H∗n with the corresponding tuple (ln−r, ..., l1)

of pairwise coprime positive divisors of N . With γ = γ(ln−r, ..., l1) one sees that

Γ(r)γ = Γ

(r)0 (l0, ln−r . . . l1),

where

Γ(r)0 (N1, N2) =

A B

C D

; C ≡ 0 mod N1; B ≡ 0 mod N2

.

Remark 3.16. The group Γ(r)0 (N1, N2) is conjugate to the group Γ

(r)0 (N1N2), so we see

that modular forms on the boundary components are isomorphic to modular forms on

Γ(r)0 (N). This is related to Remarks 2.4 and 3.14. In fact, it is not difficult to see that

one can also choose the representatives so that one manifestly has boundary components

of the form Γ(r)0 (N). The representatives we have chosen are convenient for the present

computations; we will work with a slight modification of them in §3.6 and §3.7 which will

be well-suited to studying modular forms on the boundary components.

We now describe the intersections between these boundary components. Of course, in

contrast to the issues raised in Remark 3.16, this is purely a question about the double

cosets and the result of this computation does not depend on the choice of representatives

we have made.

Theorem 3.17. Let n be a positive integer, N a square-free positive integer, and Γ(n)0 (N)\H∗n

the Satake compactification of Γ(n)0 (N)\Hn.

1. Let (ln−r, ..., l1) be an r-cusp represented by γ as above, and let 0 ≤ s ≤ r. Consider

two s-cusps on the Satake compactification Γ(r)γ \H∗r of the boundary component cor-

responding to (ln−r, ..., l1). If these two s-cusps are equal when viewed as s-cusps

of Γ(n)0 (N)\H∗n, then they are also equal when viewed as s-cusps of Γ

(r)0 (N)\H∗r. In

other words, no r-cusp can self-intersect at an s-cusp.

2. Let (ln−s, ..., l1) be an s-cusp, where 0 ≤ s < n− 1. Then the (s+ 1)-cusps on which

(ln−s, ..., l1) lies are precisely those of the form(ln−scn−s−1,

ln−s−1

cn−s−1

cn−s−2,ln−s−2

cn−s−2

cn−s−3, ...,l2c2

c1

),

where, for 1 ≤ i ≤ n− s− 1, ci | li.

51

Remark 3.18. Let 0 ≤ s < r < n. Part 2 of Theorem 3.17 can be applied by induction

to describe which r-cusps an arbitrary s-cusp lies on. Alternatively, enough ingredients

will be given in the proof of Theorem 3.17 to describe this in general, although we omit it

since it is notationally cumbersome.

Before proving Theorem 3.17 let us demonstrate the consistency of the numbers in it,

since it may not be immediately obvious that this is the case. We will count s-cusps

with the multiplicity, more precisely we shall count each s-cusps once for every (s + 1)-

cusps on which it appears. Let t denote the number of prime divisors of the squarefree

integer N . On the one hand the number of (s + 1)-cusps of Γ(n)0 (N)\H∗n is the num-

ber of tuples (ln−(s+1), ..., l1) of pairwise coprime positive divisors of N , of which there

are (n − (s + 1) + 1)t = (n − s)t; on each of these cusps the number of s-cusps is

((s+ 1)− s+ 1)t = 2t, so the number of s-cusps with multiplicity is 2t(n− s)t.

On the other hand, suppose we fix an s-cusp (l′n−s, ..., l′1). Let us write γi for the number

of prime divisors of li. Part 2 of Theorem 3.17 tells us that the number of (s + 1)-cusps

on which (l′n−s, ..., l′1) lies is 2γn−s−1 . . . 2γ22γ1 . Write also δi =

∑ij=1 γj for the number of

prime divisors of l′i . . . l′1, so that γi = δi − δi−1 for i > 1. Then the number of s-cusps

counted with multiplicity according to the number of (s+ 1)-cusps on which they appear

is

t∑δn−s=0

(t

δn−s

) δn−s∑δn−s−1=0

(δn−sδn−s−1

). . .

δ2∑δ1=0

(δ2

δ1

)2δn−s−1−δn−s−2 . . . 2δ2−δ12δ1 = 2t(n− s)t,

by repeatedly applying the binomial theorem.

Proof of Theorem 3.17. First note that if (ln−r, ..., l1) is an r-cusp of Γ(n)0 (N)\H∗n and

(mr−s, ...,m1) is an s-cusp on this r-cusp then viewed inside Γ(n)0 (N)\H∗n this s-cusp is

represented by the matrix

γ(n)r (ln−r, ..., l1)ξr,n(γ(r)

s (mr−s, ...,m1))

= γ(n)(ln−r)ξn−1,n(γ(n−1)(ln−r−1)) . . . ξr+1,n(γ(r+1)(l1))

× ξr,n(γ(r)(mr−s))ξr−1,r(γ(r−1)(mr−s−1)) . . . ξs+1,r(γ

(s+1)(m1)).

This is of course not one of our representatives. To determine this as an s-cusp of

Γ(n)0 (N)\H∗n is to determine which coset it is in in the space Γ

(n)0 (N)\ Sp2n(Q)/Pn,s(Q),

52

which is simply to determine the rank of the C22 block (of size n− s) of the above matrix

modulo p for each p | N . We write l0 = N/(ln−r . . . l1) and m0 = N/(ms−r . . .m1). By

multiplying out in the expression for γ(n)r (ln−r, ..., l1)ξr,n(γ

(r)s (mr−s, ...,m1)) one sees that

• if p | m0 and p | l0 then rkp(C22) = 0,

• if p | m0 and p | li where n− r ≥ i ≥ 1 then rkp(C22) = r − s+ i,

• if p | mj where r − s ≥ j ≥ 1 and p | l0 then rkp(C22) = j,

• if p | mj where r − s ≥ j ≥ 1 and p | li where n − r ≥ i ≥ 1 then rkp(C22) =

(r − s+ i)− j.

This is enough to deduce Part 1. Indeed, let (m′r−s, ...,m′1) be another s-cusp on (ln−r, ..., l1),

and let C ′22 be the corresponding block of size (n− s). We assume that this s-cusp when

viewed inside Γ(n)0 (N)\H∗n is the same as the one coming from (mr−s, ...,m1); equiva-

lently rkp(C22) = rkp(C′22) for all p | N . We claim that this implies (mr−s, ...,m1) =

(m′r−s, ...,m′1). Define m′0 = N/(m′r−s . . .m

′1). We will prove that p | mi and p | m′i

are the same; this is sufficient because everything is squarefree. Take a divisor p of N ,

and assume first that p | l0. From the above criteria we have under this assumption that,

for r − s ≥ j ≥ 0,

p | mj ⇐⇒ rkp(C22) = j ⇐⇒ rkp(C′22) = j ⇐⇒ p | m′j.

Now assume that p | li where n − r ≥ i ≥ 1. Again from the above criteria we have, for

r − s ≥ j ≥ 0,

p | mj ⇐⇒ rkp(C22) = r − s− j ⇐⇒ rkp(C′22) = r − s− j ⇐⇒ p | m′j.

Since every p | N divides some li, this proves Part 1. In fact, we see that if (ln−r, ..., l1)

is an r-cusp, and (mr−s, ...,m1) is an s-cusp on it, then the s-cusp when viewed inside

Γ(n)0 (N)\H∗n is (l′n−s, ..., l

′1) where, for r − s < i ≤ n− s,

l′i = (mr−s, li)(mr−s−1, li−1) . . . (m1, li−(r−s−1))(m0, li−(r−s)),

and for 1 ≤ i ≤ r − s,

l′i = (mr−s, li)(mr−s−1, li−1) . . . (mr−s−(i−1), l1)(mi, l0).

53

In the above formulas, if we refer to (la,mb) where either la or mb is not defined (e.g.

a ≤ −1 or a ≥ n − r + 1) then we understand that (la,mb) should be omitted from the

product.

In order to prove Part 2 we must start with an (s + 1)-cusp, say (dn−s−1, , ..., d1), and

exhibit and s-cusp m1 on this which is (ln−s, ..., l1), when viewed inside Γ(n)0 (N)\Hn.

Following the recipe above, where we are taking r = s + 1, we see that we must exhibit

(m1,m0) with m1m0 = N such that

ln−s = (m0, dn−s−1)

ln−s−1 = (m0, dn−s−2) (m1, dn−s−1)

ln−s−2 = (m0, dn−s−3) (m1, dn−s−2)

...

l3 = (m0, d2) (m1, d3)

l2 = (m0, d1) (m1, d2)

l1 = (m1, d0) (m1, d1)

l0 = (m0, d0).

If dn−s−1 = ln−scn−s−1 and di = (li+1/ci+i)ci for n− s− 2 ≥ i ≥ 1 as in the statement of

Part 2 then we take

m0 = ln−s ·ln−s−1

cn−s−1

· ln−s−2

cn−s−2

· · · l3c3

· l2c2

· 1 · l0,

m1 = 1 · cn−s−1 · cn−s−2 · · · c3 · c2 · l1 · 1;

this is written so as to emphasize which primes of li are in m0 and m1 respectively. To

finish it remains to show, given (ln−s, ..., l1), that if we have an (s+1)-cusp (dn−s−1, ..., d1)

which satisfies the above system equations (for some m1) then it must be of the form

stated in Part 2 of the Theorem. Now examining the equations for ln−s and ln−s−1 we see

that dn−s−1 must be a multiple of ln−s which divides ln−sln−s−1, so dn−s−1 = ln−scn−s−1

for some cn−s−1 | ln−s−1. Next examining the equations for ln−s−1 and ln−s−2 we see

that dn−s−2 must be a multiple of ln−s−1/cn−s−1 which divides (ln−s−1/cn−s−1)ln−s−2, so

dn−s−2 = (ln−s−1/cn−s−1)cn−s−2 for some cn−s−2 | ln−s−2. This pattern continues all the

way up to d1, and we see that it is necessary that (dn−s−1, ..., d1) has the form stated in

Part 2 of the Theorem. Since we’ve already seen that this is sufficient we are done.

54

3.6 Intertwining relations at arbitrary cusps for square-

free level

We continue with the imposition that N be squarefree. In this section we will obtain

versions of Theorem 3.1 where we restrict to any cusp. In the following section we will

show how the can be used to obtain information on the action of Hecke operators on

Klingen–Eisenstein series.

Write

κ(n)(l) =

1n 0n

0n l1n

γ(n)(l)

where γ(n)(l) is as in (3.24), so that

κ(n)(l) ≡

0n −1n

l1n 0n

mod l2

1n 0n

0n l1n

mod (N/l)2.

As l varies over all positive divisors of N the κ(l) represent the double coset space

Γ(n)0 (N)\GSp2n(Q)/P ∗n,n−1(Q),

where P ∗n,r is the parabolic subgroup of GSp2n which contains Pn,r (i.e. the similtudes

preserving the same flag). The inclusion induces a bijection

Γ(n)0 (N)\ Sp2n(Q)/Pn,r(Q) ' Γ

(n)0 (N)\GSp2n(Q)/P ∗n,r(Q),

so that the κ(n)(l) are in bijection with the (n − 1)-cusps of Γ(n)0 (N)\H∗n. An easy com-

putation shows that

κ(l)−1Γ(n)0 (N)κ(l) = Γ

(n)0 (N),

and that the map f 7→ f |kκ(l) defines an isomorphism M(n)k (N,χ)→M(n)

k (N,χlχN/l).

For a l positive divisor of N we write Φl for the operator defined by

Φl(F ) = Φ(F |kκ(l)),

so Φ1 = Φ.

55

Remark 3.19. Similarly to Remark 2.4 this definition depends on the choice of repre-

sentative. More precisely, if γ′ ∈ Γ(n) and δ ∈ P ∗n,n−1(Q),

Φ(F |kγ′κδ) = D−k22 χ(γ′)µn(δ)(n−r)k/2Φ(F |κ)|ωn,n−1(δ).

As a map of vector space we have Φl :M(n)k (N,χ)→M(n−1)

k (N,χlχN/l). In terms of the

Hecke module structure at primes not dividing the level we have the following:

Lemma 3.20. Let n and k be positive integers, N a squarefree positive integer, and

p a prime not dividing N . For l | N let κ(l) be as above. Then we have the following

commutative diagrams:

M(n)k (N,χ)

T (n)(p,χ)−−−−−→ M(n)k (N,χ)

|kκ(l)

y y|kκ(l)

M(n)k (N,χlχN/l) −−−−−−−−−−−−→

χl(pn)T (n)(p,χlχN/l)M(n)

k (N,χlχN/l),

and for 1 ≤ j ≤ n

M(n)k (N,χ)

T(n)j (p2,χ)−−−−−−→ M(n)

k (N,χ)

|kκ(l)

y y|kκ(l)

M(n)k (N,χlχN/l) −−−−−−−−−−−−−→

χl(p2n)T(n)j (p2,χlχN/l)

M(n)k (N,χlχN/l).

Proof. We shall show commutativity of the first diagram using an argument based on

[46] Theorem 4.5.5; the second will yield to similar reasoning.

Write α =(

1np1n

), so that T (p, χ) is given by the double coset Γ

(n)0 (N)αΓ

(n)0 (N). Note

that µ(α) = p. Write

Γ(n)0 (N)αΓ

(n)0 (N) =

⊔v

Γ(n)0 (N)αv. (3.26)

Since κ(l)−1Γ(n)0 (N)κ(l) = Γ

(n)0 (N) we also have

Γ(n)0 (N)αΓ

(n)0 (N) =

⊔v

Γ(n)0 (N)κ(l)−1αvκ(l). (3.27)

Now take F ∈M(n)k (N,χlχN/l), then

F |κ(l)−1|T (p, χ)|κ(l) = pnk2−n(n+1)

2

∑v

χ(αv)F |κ(l)−1αvκ(l)

56

where we have chosen the decomposition (3.26) for our definition of T (p, χ). Writing

αv =(Av BvCv Dv

)we have χ(αv) = χ(det(Av)), so

F |κ(l)−1|T (p, χ)|κ(l) = pnk2−n(n+1)

2

∑v

χ(det(Av))F |κ(l)−1αvκ(l). (3.28)

On the other hand,

F |T (p, χlχNl) = p

nk2−n(n+1)

2

∑v

χ(κ(l)−1αvκ(l))F |κ(l)−1αvκ(l)

where we have chosen the decomposition (3.27) for our definition of T (p, χlχN/l). This is

seen to be the same as

F |T (p, χlχNl) = p

nk2−n(n+1)

2

∑v

χl(det(Dv))χNl(det(Av))F |κ(l)−1αvκ(l).

Now det(A) det(D) ≡ det(α) ≡ pn mod N , so χl(det(Dv)) = χl(pn)χl(det(A)), hence

F |T (p, χlχNl) = χl(p

n)pnk2−n(n+1)

2

∑v

χ(det(Av))F |κ(l)−1αvκ(l) (3.29)

Comparing (3.28) and (3.29) we see that if we multiply the latter by χl(pn) then we

obtain the former; whence we obtain the stated commutative diagram.

Proof of Theorem 3.2. This follows immediately from Theorem 3.1 and Lemma 3.20.

3.7 Action of Hecke operators on Klingen–Eisenstein

series

We can now generalize the discussion of the Klingen–Eisenstein series from 2.3 to include

modular forms transforming with character. This is most conveniently done iteratively:

rather going directly from an r-cusp all the way to Γ(n)0 (N)\H∗n, we proceed via a sequence

of r-cusps. As usual, let N be a squarefree positive integer and χ a Dirichlet character

modulo N . Let l1 | N represent an (n − 1)-cusp, and set l0 = N/l1. We take F ∈

M(n−1)k (N,χl1χl0). Fix the representative κ(n)(l1) from §3.6 for l1. We define

E(n)l1

(Z;F ) = µn(κ(l1))−nk/2∑M

χ(κ(l1)M)j(M,Z)−kF (π(M〈Z〉)),

57

where M varies over a system of representatives of

(κ(l1)−1Γ(n)0 (N)κ(l1) ∩ Pn,n−1(Q))\κ(l1)−1Γ

(n)0 (N).

Since κ(l1)−1Γ(n)0 (N)κ(l1) = Γ

(n)0 (N) we can simply say that M varies over a system of

representatives of

Γ(n)0 (N) ∩ Pn,n−1(Q)\κ(l1)−1Γ

(n)0 (N).

Analogously to (2.12), one easily checks that El1(·;F ) is well-defined provided that

χ(κ(l1)δκ(l1)−1)D−k22 = 1, for all δ ∈ κ(l1)−1Γ(n)0 (N)κ(l1) ∩ Pn,n−1(Q),

which is equivalent to

χ(−1) = (−1)k.

Moreover the series converges absolutely provided that k > 2n, so under this assumptions

we have El1(·;F ) ∈M(n)k (N,χ).

Note that for l1 | N and F ∈M(n−1)k (N,χl1χl0) we have

Φl1(El1(·;F )) = F. (3.30)

Indeed, let Z ′ ∈ Hn−1 and put Zλ =(Z′

iλ

), then

Φl1(El1(·;F ))(Z ′) = limλ→∞

µn(κ(l1))−nk/2

(∑M

χ(κ(l1)M)j(M,κ(l1)〈Zλ〉)−k

×F (π(M〈κ(l1)〈Zλ〉〉)µn(κ(l1))nk/2j(κ(l1), Zλ)−k)

= limλ→∞

∑M

χ(κ(l1)M)j(Mκ(l1), Zλ)−kF (π(Mκ(l1)〈Zλ〉)

= limλ→∞

∑M ′

χ(κ(l1)M ′κ(l1)−1)j(M ′, Zλ)−kF (π(M ′〈Zλ〉)),

(3.31)

where M varies over (Γ(n)0 (N)∩Pn,n−1(Q))\κ(l1)−1Γ

(n)0 (N), and M ′ = Mκ(l1) varies over

(Γ(n)0 (N) ∩ Pn,n−1(Q))\Γ(n)

0 (N). Since the Klingen–Eisenstein series converges uniformly

and absolutely (under the assumption on k), an application of the dominated convergence

theorem allows us to interchange the summation and limit. Taking the limit of each sum-

mand individually one obtains zero unless M ′ ∈ Γ(n)0 (N) ∩ Pn,n−1(Q) (e.g. M ′ = 12n).

58

However, we do not obtain the full analogue of (2.12), since F ∈M(n−1)k (N,χl1χl0) need

not be an a cusp form. If F is not a cusp form then Φl′1(El1(F )) will be non-zero if l′1 and

l1 intersect on an (n− 2)-cusp at which F is non-vanishing. If F ∈ S(n−1)k (N,χl1χl0) is a

cusp form then the analogue of (2.12) holds and we easily prove the following:

Lemma 3.21. Let N ∈ Z≥1 be squarefree, l1 | N represent an (n − 1)-cusp of Γ(n)0 \H∗n,

and set l0 = N/l1. Let F ∈ S(n−1)k (N,χl1χl0) be an eigenfunction of T (n−1)(p, χl1χl0) with

eigenvalue λ(n−1)(p, χl1χl0). Then

El1(·;F )|T (n)(p, χ) = λ(n)(p, χ)El1(·;F ),

where

λ(n)(p, χ) =(χl1(p

n) + χl1(pn−1)χl0(p)p

k−n)λ(n−1)(p, χl1χl0).

Proof. First note that the Eisenstein subspace is invariant under the action of Hecke

operators outside the primes dividing the level. This is surely well-known, but can also

easily be proved using the (obvious) fact that the Hecke operators preserve the subspace

of cusp forms, and the fact that Hecke operators at p - N onM(n)k (N,χ) are normal with

respect to the Petersson inner product ([1] Lemma 4.6). Thus El1(·; f)|T (n)(p, χ) is an

Eisenstein series. Let l′1 be any divisor of N , and set l′0 = N/l′1. Then

Φl′1(El1(·;F )|T (n)(p, χ))

=(χl′1(p

n) + χl′1(pn−1)χl′0(p)p

k−n)Φl′1(El1(·;F ))|T (n−1)(p, χl′1χl′0).

If l′1 = l1 this becomes

Φl1(El1(·;F )|T (n)(p, χ))

=(χl1(p

n) + χl1(pn−1)χl0(p)p

k−n)λ(n−1)(p, χl1χl0)Φl1(El1(·;F )).(3.32)

If l′1 6= l1 we instead get

Φl′1(El1(·;F )|T (n)(p, χ)) = 0. (3.33)

Now consider the function

El1(·;F )|T (n)(p, χ)− (χl1(pn) + χl1(p

n−1)χl0(p)pk−n)El1(·;F ) ∈M(n)

k (N,χ).

By (3.32) and (3.33) this vanishes at all (n − 1)-cusps, so is a cusp form. On the other

hand, by the discussion at the beginning of the proof it is an Eisenstein series. Thus it

must be equal to zero.

59

The same argument also proves the following:

Lemma 3.22. Let F ∈ S(n−1)k (N,χl1χl0) be an eigenfunction all of T

(n−1)j (p2, χl1χl0),

0 ≤ j ≤ n− 1, with eigenvalues λ(n−1)j (p2, χl1χl0). Then

El1(·;F )|T (n)j (p2, χ) = λ(n)(p2, χ)El1(·;F ),

where

λ(n)(p2, χ) = χl1(p2n)[c

(n−1)j,j (χl1χl0)λ

(n−1)j (p2, χl1χl0)

+c(n−1)j,j−1 (χl1χl0)λ

(n−1)j−1 (p2, χl1χl0)

+c(n−1)j,j−2 (χl1χl0)λ

(n−1)j−2 (p2, χl1χl0)

],

where c(n−1)j,j , c

(n−1)j,j−1 , c

(n−2)j,j−2 are as in Theorem 3.1.

With a little more book-keeping we can generalise Lemmas 3.21 and 3.22 to all Klingen–

Eisenstein series. WriteM(n,n)k (N,χ) = S(n)

k (N,χ) ⊂M(n)k (N,χ) for the subspace of cusp

forms. It makes sense to define the orthogonal complement N (n)k (N,χ) of M(n,n)

k (N,χ)

with respect to (2.8). Define an operator

Φ : N (n)k (N,χ)→ ⊕l1|NM

(n−1)k (N,χl1χl0)

by

F 7→ (Φl1(F ))l1|N .

This map is not surjective, since the vectors in the image must agree on lower dimensional

intersections, as suggested above. In large enough weights, Φ is surjective on to the

subspace where this condition holds, as we shall see in a moment. First, we define some

subspaces M(n,i)k (N,χ) ⊂ N (n)

k (N,χ) for 0 ≤ i < n by induction on n. There is nothing

to do for n = 1: for any character ψ modulo N , M(1,0)k (N,ψ) = N (1)

k (N,ψ) is the usual

space of degree one Eisenstein series. For n > 1 and 0 ≤ i < n, we define

M(n,i)k (N,χ) = Φ−1

⊕l1|N

M(n−1,i)k (N,χl1χl0)

.

ThenM(n,i)k (N,χ) is a linear subspace, and Σn

i=0M(n,i)k (N,χ) is in fact direct. By double

induction (increasing on n, decreasing on r) one sees from the normality of the Hecke

operators with respect to the inner product (2.8) and Theorem 3.2 that the Hecke algebra

60

Hp (when p - N) preserves the decomposition⊕

iM(n,i)k (N,χ).

In order to produce some elements of M(n,i)k (N,χ) we will use Eisenstein series. At the

same time this will show Φ is surjective, i.e.

M(n)k (N,χ) =

n⊕i=0

M(n,i)k (N,χ). (3.34)

We work iteratively: let (ln−r, ..., l1) be a sequence of pairwise coprime divisors of N

corresponding to an r-cusp, and define

Φ(ln−r,...,l1) = Φl1 Φl2 · · · Φln−r .

In the other direction, let F ∈ S(r)k (N,χln−r...l1χl0), and define

E(ln−r,...,l1)(F ) = Eln−r Eln−r−1 · · · El1(F ).

The proof of (3.34) follows easily by induction once we know that E(ln−r,...,l1)(F ) ∈M(n,r)k .

This latter fact follows from a somewhat technical computation for which we refer to [59],

especially (2.12) and the discussion preceding it. See also [27] Corollary 2.4.6, which in-

cludes a detailed proof of this decomposition but is slightly removed from our context

since modular forms are identified with sections of automorphic vector bundles.

We can now iterate the idea of Lemmas 3.21 and 3.22 to handle lifts of cusp forms of any

degree 0 ≤ r < n:

Theorem 3.23. Let n be a positive integer, 0 ≤ r < n, N a squarefree positive integer,

and (ln−r, ..., l1) correspond to an r-cusp of Γ(n)0 (N)\H∗n. Let F ∈ S(r)

k (N,χln−r...l1χl0),

where k > n+ r + 1.

1. Assume that F is an eigenfunction of T (r)(p, χln−r...l1χl0), write λ(r)(p, χln−r...l1χl0)

for the eigenvalue. Then

E(ln−r,...,l1)(F )|T (n)(p, χ) = λ(n)(p, χ)E(ln−r,...,l1)(F ),

where

λ(n)(p, χ) = λ(r)(p, χln−r...l1χl0)n∏

t=r+1

χlt−r(pt)c(t)(χln−r...lt−rχlt−r−1...l0).

61

2. Assume that F is an eigenfunction of each T(r)j (p2, χln−r...l1χl0) with eigenvalues

λ(r)(p2, χln−r...l1χl0). Then

E(ln−r,...,l1)(F )|T (n)j (p2, χ) = λ

(n)j (p2, χ)E(ln−r,...,l1)(F ),

where λ(n)j (p2, χ) is given recursively by the following recursive formula

θ(n)(ln−r,...,l1)(λ

(n)j (p2, χ))

= χln−r(p2n)[c

(n−1)j,j (χln−rχln−r−1...l0)θ

(n−1)(ln−r−1,...,l1)(λ

(n−1)j (p2, χln−rχln−r−1...l0))

+c(n−1)j,j−1 (χln−rχln−r−1...l0)θ

(n−1)(ln−r−1,...,l1)(λ

(n−1)j−1 (p2, χln−rχln−r−1...l0))

+c(n−1)j,j−2 (χln−rχln−r−1...l0)θ

(n−1)(ln−r−1,...,l1)(λ

(n−1)j−2 (p2, χln−rχln−r−1...l0))

],

where at the final stage of the recursion the notation θ(r)() (λ

(r)k (p2, χln−r...l1χl0)) means

λ(r)k (p2, χln−r...l1χl0). Recall that c

(n−1)j,j (χ), c

(n−1)j,j−1 (χ), and c

(n−1)j,j−2 (χ) are given by The-

orem 3.1, and we have the convention that c(s)j,k = 0 if k < 0 or k > s (i.e. omit

terms that ask for Hecke eigenvalues for a Hecke operator that does not exist).

Proof. We prove Part 1 by induction on n, the proof of Part 2 follows by the same

argument. When n = r + 1 this is Lemma 3.21, so the base case is done. In general,

consider the function

E(ln−r,...,l1)(F )|T (n)(p, χ)− λ(n)(p, χ)E(ln−r,...,l1)(F ) ∈M(n,n−r)k (N,χ). (3.35)

Then

Φln−r(E(ln−r,...,l1)(F )|T (n)(p, χ)− λ(n)(p, χ)E(ln−r,...,l1)(F ))

= χln−r(pn)c(n)(χln−rχln−r−1...l0)T

(n−1)(p, χln−rχln−r−1...l1)E(ln−r−1,...,l1)(F )

− λ(n)(p, χ)E(ln−r−1,...,l1)(F )

= χln−r(pn)c(n)(χln−rχln−r−1...l0)

[T (n−1)(p, χln−rχln−r−1...l1)E(ln−r−1,...,l1)(F )

−λ(n−1)(p, χln−rχln−r−1...l0)E(ln−r−1,...,l1)

].

By induction hypothesis this is zero. On the other hand it is clear that

Φ(l′n−r,...,l′1)(E(ln−r,...,l1)(F )) = 0

if (l′n−r, ..., l′1) 6= (ln−r, ..., l1). Thus

Φ(l′n−r,...,l′1)(E(ln−r,...,l1)(F )|T (n)(p, χ)− λ(n)(p, χ)E(ln−r,...,l1)(F )) = 0,

62

for all (l′n−r, ..., l′1). But the containment (3.35) tells us that

E(ln−r,...,l1)(F )|T (n)(p, χ)− λ(n)(p, χ)E(ln−r,...,l1)(F )

is determined by its value on all r-cusps. Since we have shown it vanishes at all of these,

it must be zero, so we obtain the statement of the theorem.

63

Chapter 4

Fourier coefficients of level N

Siegel–Eisenstein series

Recall that Siegel’s Hauptsatz from §2.2 stated that the genus average theta series is a

linear combination of Siegel–Eisenstein series. From the viewpoint of quadratic forms it

is therefore essential to obtain formulas for the Fourier coefficients of Siegel–Eisenstein

series, but there are still some perhaps surprising gaps in our knowledge of these.

Various authors have worked on the case Siegel-Eisenstein series for the full symplec-

tic group Sp2n(Z) and we mention only a sample of the results here. Maass ([43], [44])

obtained a formula for the Fourier coefficients of degree 2 Siegel–Eisenstein series by

explicitly computing the local densities in Siegel’s formula. The same results were also

obtained by Eichler–Zagier ([17]) by realising the Siegel–Eisenstein series as the Maass lift

of the corresponding Jacobi–Eisenstein series. More recently, formulas for general degree

n have been obtained via different methods by Katsurada (first degree 3 [30], then any

degree [31]) and Kohnen (even degree [36]), Choie–Kohnen (odd degree [11]). Katsurada’s

approach has more in common with Maass’ approach, whereas Kohnen’s approach (for

even degree) is at least nominally closer to that of Eichler–Zagier with his linearised ver-

sion of the Ikeda lift playing a role simlar to the Maass lift.

In the case of modular forms transforming with character χ under the Hecke-type con-

gruence subgroup Γ(n)0 (N) the situation is less well-known. To the author’s knowledge,

64

the only explicit results in the literature pertain to a single Eisenstein series when the

degree is n = 2 and the character χ is primitive modulo N . First Mizuno ([47]) consid-

ered the case of squarefree N and obtained the Fourier coefficients by realising a level

N Eisenstein series as a Maass lift of a corresponding level N Jacobi–Eisenstein series.

The argument is more difficult than Eichler–Zagier’s at level 1 since the author requires

analytic machinery to prove the coincidence of the lift with the desired form. An ex-

tension of this result, dropping the assumption that N be squarefree, was obtained by

Takemori ([68]) by computing the local densities. A related result, again in the context

of n = 2 and arbitrary level N but now with no restrictions on the character, is due to

Yang, who explicitly computes the local densities in Siegel’s theorem. Yang’s methods are

rather different: following Weil and Kudla he interprets those local densities in terms of

the local Whittaker function coming from the representation attached to the Eisenstein

series. He explicitly computes this latter quantity ([74] for p 6= 2; [75] for p = 2), which

is essentially equivalent to computing the Fourier coefficients of the genus-average theta

series, hence the Fourier coefficients of a Siegel–Eisenstein series.

Since quadratic forms are rarely unimodular the corresponding theta series will usually

be modular forms of level N > 1. Additionally, it is important for the arithmetic theory of

these quadratic forms that we have explicit formulas for Fourier coefficients for a basis of

the space of Siegel-Eisenstein series. Although we do have some Fourier coefficients when

N > 1, we only have these for a single Eisenstein series, but (for large enough weight)

the dimension of the space of Siegel–Eisenstein series of level N is strictly larger than one.

In this chapter we consider the case n = 2, squarefree level N , the character χ being the

trivial character modulo N . Under these assumptions we can use the following very simple

method to obtain relations amongst the Fourier coefficients: let p be a prime not dividing

N , then an appropriate sum of level Np Eisenstein series produces a level N Eisenstein

series, and after acting on this relation with Hecke operators at p the explicit formulas

from [72] produce enough linear relations among the Fourier coefficients to write down a

formula for the coefficients of a level Np Eisenstein series in terms of those of the level N

one. Using a convenient level 1 formula, namely that of [17], one can then argue by induc-

65

tion to obtain Fourier coefficients for a full basis of the Eisenstein subspace in the case of

squarefree level and trivial character. This is carried out in Lemma 4.6: the main bulk of

the computation is then placing these in a more elucidating form as stated in Theorem 4.8.

Let us remark that the important feature that makes this work is that the level Np

Fourier coefficients add up to something known in the base case of the induction. So for

example in the case of primitive character this approach seems unlikely to succeed. In the

case of squarefree level the transformation character will always be a product of primi-

tive and trivial characters, and if one knows the Fourier coefficients for Eisenstein series

transforming with a given primitive character χ then one can argue as suggested above

to obtain Fourier coefficients of Eisenstein series of any squarefree level and character

which has χ as the underlying primitive character. However one would need to know the

Fourier coefficients for a full basis at the primitive stage in order to deduce the Fourier

coefficients for a full basis at later stages. As noted above no such formulas for a full basis

are currently available.

Finally in §4.3 we emphasise this point regarding the importance of having Fourier co-

efficients for a full basis of the Eisenstein subspace by showing how one can compute

the genus representation numbers of an integral quadratic form by combining knowledge

of the Fourier coefficients of a basis for the Eisenstein subspace with the well-known

formulas for the value a theta series takes at a 0-dimensional cusp of (the Satake com-

pactification of) Γ(2)0 (N)\H2. There is a finite number of integral quadratic forms which

have a single-class genus, and from the viewpoint of degree 2 representation numbers only

the 8-dimensional ones have dimension large enough to study via Siegel–Eisenstein series

(i.e. the Eisenstein series of degree 2 and weight 4 converges) of even weight.1 Amongst

these 8-dimensional integral quadratic forms, or equivalently even integral lattices, only

5 satisfy the condition that their level be squarefree and their character trivial. Of course

one of these is the unimodular lattice E8 for which degree 2 representation numbers (i.e.

explicit formulas for the number of times it represents a quadratic form in 2 variables)

1The necessary condition for the existence of Eisenstein series (c.f. Remark 2.5) says that the weightmust be even when the character is trivial.

66

follow (for example) from the formula of [17]. The remaining 4 have small prime level and

for these we will note in Corollary 4.10 how the methods of this chapter give new closed

formulas for their degree 2 representation numbers.

In this chapter we work mostly with Siegel modular forms of degree two. Consequently,

we omit the superscript (2) from the notation, so for example Mk(N) = M(2)k (N),

T (p) = T (2)(p), etc. This chapter is based on the paper [14].

4.1 The action of Hecke operators on Siegel–Eisenstein

series

We begin by recalling the notation and some results from [72]. First we must remark on

the difference between the ways we define Eisenstein series. We will define our Siegel–

Eisenstein using the coset representatives from §3, whereas [72] uses a different set of

representatives. However, these are also drawn from Sp2n(Z), and it follows that the ma-

trices δ in Remark 2.6 which quantify the difference between our representatives and those

of [72] must be integral. In the notation of Remark 2.6 this implies that det(D22) = ±1.

Since we must assume that k is even anyway, the Eisenstein series defined here and in

[72] are in fact the same. (Note that the action of ωn,r(δ) has no effect since r = 0.) As

we remarked at the end of §2 the definition of the Hecke operators are the same.

As we have seen in Theorem 3.17, the 0-cusps of Γ0(N)\H2 correspond bijectively to triples

of pairwise coprime divisors of N , say (N0, N1, N2) where N0N1N2 = N . Corresponding

to such a triple is a double coset Γ(n)0 (N)γPn,r(Q), where Ni is the product of the primes

P dividing N at which the rank of the C-block of γ ∈ Sp2n(Z) modulo p is i. Using, say,

the fixed choice of representatives γ = γ(N2,N1) from §3 we define

E(N0,N1,N2)(Z) = Eγ(Z). (4.1)

The change of notation is motivated by the fact that we will now be arguing with Eisen-

stein series of different levels, and it is important that we keep track of (N0, N1, N2) (and

not just (N2, N1) as we did in the previous notation) because the ambiguity in levels

67

might mean that recovering N0 is not immediate. There should be no confusion with the

notation of §3 because we use E in place of E.

Lemma 4.1. Let N be square-free, (N0, N1, N2) such that N0N1N2 = N , and let p be a

prime not dividing N . Then

E(N0,N1,N2) = E(pN0,N1,N2) + E(N0,pN1,N2) + E(N0,N1,pN2).

Proof. This follows by considering the rank of the C-block of a summand appearing on

the left hand side modulo p: it is either zero, one, or two, and that summand will therefore

appear in the corresponding Eisenstein series on the right hand side.

We now quote the results of [72] in the case of trivial character:

Proposition 4.2. The action of the Hecke operators U(p) on the level Np Eisenstein

series transforming with trivial character are as follows:

E(pN0,N1,N2)|U(p) = E(pN0,N1,N2) + (1− p−1)E(N0,pN1,N2) + (1− p−1)E(N0,N1,pN2),

E(N0,pN1,N2)|U(p) = pk−1E(N0,pN1,N2) + (pk−1 − pk−3)E(N0,N1,pN2),

E(N0,N1,pN2)|U(p) = p2k−3E(N0,N1,pN2).

Proof. These are special cases of [72] Propositions 3.5, 3.6, and 3.7.

Proposition 4.3. The action of the Hecke operators U1(p2) on the level Np Eisenstein

series transforming with trivial character are as follows:

E(pN0,N1,N2)|U1(p2) = (pk−2 + pk−3)E(pN0,N1,N2)

+ (pk−1 + 1)(pk−3 − pk−4)E(N0,pN1,N2)

+ (pk−3 − pk−5)E(N0,N1,pN2),

E(N0,pN1,N2)|U1(p2) = (p3k−5 + pk−2)E(N0,pN1,N2)

+ (pk−2 + 1)(pk−2 − pk−4)E(N0,N1,pN2),

E(N0,N1,pN2)|U1(p2) = (p3k−5 + p3k−6)E(N0,N1,pN2).

Proof. These are special cases of [72] Propositions 3.8, 3.9, and 3.10.

Proposition 4.4. The action of the Hecke operator T (p) on the level N (where p - N)

Eisenstein series transforming with trivial character is:

E(N0,N1,N2)|T (p) = (p2k−3 + pk−1 + pk−2 + 1)E(N0,N1,N2).

68

Proof. This is a special case of [72] Proposition 3.3 or of Theorem 3.23 of this thesis.

Proposition 4.5. Let F (Z) =∑

T a(T ;F )e(tr(TZ)) ∈ Mk(N). For M any matrix,

write T [M ] = tMTM . Then, for p | N ,

a(T ;F |U(p)) = a(pT ;F ),

a(T ;F |U1(p2)) = pk−3

∑α mod p

a

T1 0

α p

;F

+ a

Tp 0

0 1

;F

,and, for p - N ,

a(T ;F |T (p)) = a(pT ;F ) + pk−2

∑α mod p

a

1

pT

1 0

α p

;F

+a

1

pT

p 0

0 1

;F

+ p2k−3a

(1

pT ;F

).

Proof. Since the degree n is two this is well known. For a proof one could, for example,

consult [26], which does general degree n.

4.2 Calculation of the Fourier coefficients

A computation based on [72]. Fix a partition (N0, N1, N2) of the squarefree integer

N and let E(N0,N1,N2) be the associated Eisenstein series transforming with the trivial

character modulo N . Let us abbreviate a(T ) := a(T ;E(N0,N1,N2)), and define

a0(T ) = a(T ;E(pN0,N1,N2)),

a1(T ) = a(T ;E(N0,pN1,N2)),

a2(T ) = a(T ;E(N0,N1,pN2)).

Lemma 4.6. In the above notation

a0(T ) =1

(pk − 1)(p2k−2 − 1)

[(p3k−2 + p2k−1 − p2k−2 + pk+1 − pk − p+ 1)a(T )

−(p2k−1 + pk+1 + p2 − p)a(pT ) + p2a(p2T )],

a1(T ) =1

(pk − 1)(p2k−2 − 1)

[(−p2k−1 − pk+1 − p3 + p)a(T )

+(p2k−1 + pk+1 + p3 + p2 − p+ p−k+4)a(pT )− (p2 + p−k+4)a(p2T )]

a2(T ) =p3a(T )− (p3 + p−k+4)a(pT ) + p−k+4a(p2T )

(pk − 1)(p2k−2 − 1).

69

Proof. By Lemma 4.1 we have

E(N0,N1,N2) = E(pN0,N1,N2) + E(N0,pN1,N2) + E(N0,N1,pN2), (4.2)

and comparing the T th Fourier coefficient in this gives

a(T ) = a0(T ) + a1(T ) + a2(T ). (4.3)

Now apply U(p) to (4.2). By Proposition 4.2 we have

E(N0,N1,N2)|U(p) = E(pN0,N1,N2)

+ (pk−1 + 1− p−1)E(N0,pN1,N2)

+ (p2k−3 + pk−1 − pk−3 + 1− p−1)E(N0,N1,pN2).

Note that E(N0,N1,N2)|U(p) makes sense since E(N0,N1,N2), a priori a modular form of level

N , is also a modular form of level Np. Hence by Proposition 4.5 a(T ;E(N0,N1,N2)|U(p)) =

a(pT ;E(N0,N1,N2)) and we have

a(pT ) = a0(T ) + (pk−1 + 1− p−1)a1(T )

+ (p2k−3 + pk−1 − pk−3 + 1− p−1)a2(T ).(4.4)

Similarly, apply U1(p2) to (4.2) we obtain

a(T ;E(N0,N1,N2)|U1(p2))

= (pk−2 + pk−3)a0(T ) + (p3k−5 + p2k−4 − p2k−5 + pk−2 + pk−3 − pk−4)a1(T )

+ (p3k−5 + p3k−6 + p2k−4 − p2k−6 + pk−2 + pk−3 − pk−4 − pk−5)a2(T ).

(4.5)

Comparing Fourier expansions at pT in Proposition 4.4 we have

a(pT ;E(N0,N1,N2)|T (p)) = (p2k−3 + pk−1 + pk−2 + 1)a(pT )

On the other hand, by Proposition 4.5,

a(pT ;E(N0,N1,N2)|T (p))

= a(p2T ) + pk−2

∑α mod p

a

T1 0

α p

+ a

Tp 0

0 1

+ p2k−3a (T )

= a(p2T ) + pa(T ;E(N0,N1,N2)|U1(p2)) + p2k−3a(T ).

70

Hence

a(p2T ) = −pa(T ;E(N0,N1,N2)|U1(p2)) + (p2k−3 + pk−1 + pk−2 + 1)a(pT )− p2k−3a(T ).

Substituting (4.3), (4.4), (4.5) in to the right hand side we obtain

a(p2T ) = a0(T ) + (2p2k−4 + p2k−2 + pk−1 + pk−2 + 2pk−3 + 1 + p−1)a1(T )

+ (−p3k−5 + p2k−2 − p2k−3 + p2k−5 + pk−1 − pk−4 + 1− p−2)a2(T )

Solving (4.3), (4.4) and (4.5) simultaneously we obtain

a0(T ) =(p3k−2 − p2k−2 + pk+1 − pk − p+ 1)a(T ) + (pk + p)a(pT )

(pk − 1)(p2k−2 − 1)

−pka(T ;E(N0,N1,N2)|U1(p2))

(pk − 1)(p2k−2 − 1), (4.6a)

a1(T ) =(−p3 + p)a(T )− (pk+1 + pk + p2 + p)a(pT )

(pk − 1)(p2k−2 − 1)

+(pk + p2)a(T ;E(N0,N1,N2)|U1(p2))

(pk − 1)(p2k−2 − 1), (4.6b)

a2(T ) =(−pk+1 + p3)a(T ) + (pk+1 + p2)a(pT )− p2a(T ;E(N0,N1,N2)|U1(p2))

(pk − 1)(p2k−2 − 1). (4.6c)

Comparing Fourier expansions at pT in Proposition 4.4 we have

a(pT ;E(N0,N1,N2)|T (p)) = (p2k−3 + pk−1 + pk−2 + 1)a(pT )

On the other hand, by Proposition 4.5,

a(pT ;E(N0,N1,N2)|T (p)) = a(p2T ) + pk−2

∑α mod p

a

T1 0

α p

+a

Tp 0

0 1

+ p2k−3a (T )

= a(p2T ) + pk−2a(T ;E(N0,N1,N2)|U1(p2)) + p2k−3a(T ).

Hence

a(T ;E(N0,N1,N2)|U1(p2)) = −p−k+2a(p2T ) + (pk−1 + p+ 1 + p−k+2)a(pT )− pk−1a(T ),

and substituting this in to (4.6a), (4.6b), (4.6c) we obtain the lemma.

71

Formulas for the Fourier coefficients. Note that, given the Fourier coefficients a(T ) =

a(T ;E(N0,N1,N2)), Lemma 4.6 provides a formula for the Fourier coefficients a0(T ), a1(T ),

and a2(T ). As these are written these are, of course, unsatisfactory; we will now present

them in a more familiar form.

Before proceeding let us recall the formula from [17] for the Fourier coefficients of the

level 1 Siegel-Eisenstein series E of degree 2 at a positive definite matrix T . This formula

is

a(T ;E) =2

ζ(1− k)ζ(3− 2k)

∑d|e(T )

dk−1H

(∆(T )

d2

)(4.7)

where H denotes the function defined by Cohen in [12] (with first parameter in the

notation of [12] set equal to k − 1): writing a positive integer M with M ≡ 0,−1 mod 4

as M = −Df 2 where D < 0 is fundamental discriminant the function is

H(M) = L(2− k, χK)∑g|f

µ(g)χD(g)gk−2∑h|(f/g)

h2k−3,

where χD is the character associated with the extension Q(√D). Now let N be any

(squarefree) positive integer and let 1N denote the trivial character modulo N . For M =

−Df 2 as above we define

HN(M) = L(2− k, χK)∑g|f

1N(g)µ(g)χD(g)gk−2∑h|(f/g)

1N(h)h2k−3.

Note that H1 = H. Let us also remark that [17] provides a formula for the Fourier

coefficient a(T ;E) when T is singular, namely

a

n 0

0 0

; E

=

2

ζ(1−k)

∑d|n d

k−1 if n > 0,

1 if n = 0.

(4.8)

This is of course an illustration of how the Fourier coefficients of Eisenstein series of

degree n on singular matrices are given by those of Eisenstein series of degree n− 1.

Lemma 4.7. Let N be a squarefree positive integer, p a prime not dividing N , k a positive

integer, M a positive integer with M ≡ 0,−1 mod 4. Write M = −Df 2 as above, then

HNp(M)Cp,D(ordp(f)) = HN(M)

72

where

Cp,D(v) =v∑j=0

pj(2k−3) − χD(p)pk−2

v−1∑j=0

pj(2k−3).

Moreover, writing p2M = −D(pf)2, we also have

HNp(p2M) = HNp(M).

Proof. From the definition we have

HN(M) = L(2− k, χD)

∑g|f

ordp(g)=0

1N(g)µ(g)χD(g)gk−2∑h|(f/g)

1N(h)h2k−3

−χD(p)pk−2∑g|f

ordp(g)=0

1N(g)µ(g)χD(g)gk−2∑

h|(f/(pg))

1N(h)h2k−3

= L(2− k, χD)

∑g|f

1Np(g)µ(g)χD(g)gk−2∑h|(f/g)

ordp(h)=0

1N(h)h2k−3

×

ordp(f)∑j=0

pj(2k−3) − χD(p)pk−2

ordp(f)−1∑j=0

pj(2k−3)

= L(2− k, χD)

∑g|f

1Np(g)µ(g)gk−2∑h|(f/g)

1Np(h)h2k−3Cp,D(ordp(f)).

The second claimed equality follows immediately from the definition of HNp.

Theorem 4.8. Let T =(

m r/2r/2 n

)be positive semi-definite. Let ∆ = 4mn − r2 and

e = gcd(m,n, r). Write −∆ = Df 2 where D is a fundamental discriminant. Let χD

denote the character(D·

), and let 1N(·) be the trivial character modulo N . Then

1. at T = ( 0 00 0 ), the Fourier coefficients are as follows:

a(( 0 0

0 0 ) ;E(N0,N1,N2)

)=

1 if (N0, N1, N2) = (N, 1, 1),

0 otherwise.

2. for T 6= ( 0 00 0 ) but ∆ = 0, the Fourier coefficients are

a(T ;E(N0,N1,N2)) = Υ(T ;N0, N1, N2)2

ζ(1− k)

∑d|e

1N(d)dk−1,

73

where Υ(T ;N0, N1, N2) =∏

i

∏p|Ni υi(p, ordp(e)) with

υ0(p, up) =p(up+1)(k−1) − 1

pk−1 − 1− p(up+1)(k−1)p

pk − 1,

υ1(p, up) =p(up+1)(k−1)p

pk − 1,

υ2(p, up) = 0.

3. for T > 0, the Fourier coefficients are

a(T ;E(N0,N1,N2)) = Ψ(T ;N0, N1, N2)2

ζ(1− k)ζ(3− 2k)

∑d|e

1N(d)dk−1HN

(∆

d2

),

where Ψ(T ;N0, N1, N2) =∏

i

∏p|Ni ψi(p, ordp(e), ordp(f)) with

ψ0(p, up, vp) = (p2k−3 − χD(p)pk−2)

[pvp(2k−3)

(pk−2(p− 1)

(p2k−3 − 1)(p2k−2 − 1)(pk−2 − 1)

)−p(vp−up)(2k−3)pup(k−1)

(p− 1

(p2k−3 − 1)(pk − 1)(pk−2 − 1)

)]+ (χD(p)pk−2 − 1)

[pup(k−1)

(pk−1(p− 1)

(p2k−3 − 1)(pk − 1)(pk−1 − 1)

)− 1

(p2k−3 − 1)(pk−1 − 1)

],

ψ1(p, up, vp) = (p2k−3 − χD(p)pk−2)

[pvp(2k−3)

(pk−1(p2 − 1)

(p2k−2 − 1)(pk − 1)(pk−2 − 1)

)−p(vp−up)(2k−3)pup(k−1)

(p(pk−1 − 1)

(p2k−3 − 1)(pk − 1)(pk−2 − 1)

)]+ (χD(p)pk−2 − 1)pup(k−1) pk

(p2k−3 − 1)(pk − 1),

ψ2(p, up, vp) = (p2k−3 − χD(p)pk−2)pvp(2k−3) pk+1

(p2k−2 − 1)(pk − 1).

Proof. Arguing by induction on the number of divisors of N , using 4.6 and the base case

(4.8), one obtains 1. and 2. These Fourier coefficients could also be obtained by consid-

ering the cusp of support of Φ(E(N0,N1,N2) to identify this as a degree 1 Eisenstein series.

The more interesting case is that of 3. Here we will again proceed by induction on the

number of prime divisor of N , but now the calculations are more technical. The base case

is the formula (4.7) from [17] (we have the usual convention that any product indexed by

the empty set is equal to 1).

74

Now suppose we have a partition (N0, N1, N2) of square-free N , p is a prime not dividing

N , and the coefficients a(T ) = a(T ;E(N0,N1,N2)) are as stated in the theorem. To ease

notation we shall write up = ordp(e), vp = ordp(f). We can write

a(T ) = Ψ(T ;N0, N1, N2)2

ζ(1− k)ζ(3− 2k)

×up∑j=0

∑d|e

1Np(d)dk−1pj(k−1)HN

(∆

p2jd2

)= Ψ(T ;N0, N1, N2)

2

ζ(1− k)ζ(3− 2k)

×ordp(e)∑j=0

∑d|e

1Np(d)dk−1pj(k−1)HNp

(∆

p2jd2

)Cp,D (vp − j)

=2

ζ(1− k)ζ(3− 2k)

∑d|e

1Np(d)dk−1HNp

(∆(T )

d2

)

×

[Ψ(T ;N0, N1, N2)

up∑j=0

pj(k−1)Cp,D (vp − j)

].

(4.9)

Similarly,

a(pT ) =2

ζ(1− k)ζ(3− 2k)

∑d|e

1Np(d)dk−1HNp

(∆

d2

)

×

[Ψ(pT ;N0, N1, N2)

up+1∑j=0

pj(k−1)Cp,D (vp + 1− j)

],

a(p2T ) =2

ζ(1− k)ζ(3− 2k)

∑d|e

1Np(d)dk−1HNp

(∆

d2

)

×

[Ψ(p2T ;N0, N1, N2)

up+2∑j=0

pj(k−1)Cp,D (vp + 2− j)

].

75

By Lemma 4.6 we have

(pk − 1)(p2k−2 − 1)Ψ(T ; pN0, N1, N2) =

(p3k−2 + p2k−1 − p2k−2 + pk+1 − pk − p+ 1)

×

[Ψ(T ;N0, N1, N2)

up∑j=0

pj(k−1)Cp,D (vp − j)

]

+ (−p2k−1 − pk+1 − p2 + p)

×

Ψ(pT ;N0, N1, N2)

ordp(e(T ))+1∑j=0

pj(k−1)Cp,D (vp + 1− j)

+ p2

[Ψ(p2T ;N0, N1, N2)

up+2∑j=0

pj(k−1)Cp,D (vp + 2− j)

],

(pk − 1)(p2k−2 − 1)Ψ(T ;N0, pN1, N2) =

(−p2k−1 − pk+1 − p3 + p)

×

[Ψ(T ;N0, N1, N2)

up∑j=0

pj(k−1)Cp,D (vp − j)

]

+ (p2k−1 + pk+1 + p3 + p2 − p+ p−k+4)

×

[Ψ(pT ;N0, N1, N2)

up+1∑j=0

pj(k−1)Cp,D (vp + 1− j)

]

+ (−p2 − p−k+4)

[Ψ(p2T ;N0, N1, N2)

up+2∑j=0

pj(k−1)Cp,D (vp + 2− j)

],

(pk − 1)(p2k−2 − 1)Ψ(T ;N0, N1, pN2) =

p3

[Ψ(T ;N0, N1, N2)

up∑j=0

pj(k−1)Cp,D (vp − j)

]

+ (−p3 − p−k+4)

[Ψ(pT ;N0, N1, N2)

up+1∑j=0

pj(k−1)Cp,D (vp + 1− j)

]

+ p−k+4

[Ψ(p2T ;N0, N1, N2)

up+2∑j=0

pj(k−1)Cp,D (vp + 2− j)

].

We aim to find a solution to these equations subject to the initial condition Ψ(T ; 1, 1, 1) =

1. From this initial condition and the right and side of the above formulas we see that

Ψ(T ;N0, N1, N2) only depends on T via local quantities uq = ordq(e(T )) and vq =

ordq(f) at primes q | N0N1N2. In particular if p is a prime not dividing N0N1N2 then

76

Ψ(pT ;N0, N1, N2) = Ψ(T ;N0, N1, N2). Thus we can remove Ψ(T ;N0, N1, N2) as a com-

mon factor from all terms on the right hand side of the above system of equations, which

then simplify1 to

(pk − 1)(p2k−2 − 1)Ψ(T ; pN0, N1, N2) =[(p2k−1 − p2k−2 − p+ 1)

up∑j=0

pj(k−1)Cp,D(vp − j)

−(p2k−1 + p2 − p)Cp,D(vp + 1)

+p2Cp,D(vp + 2)]

Ψ(T ;N0, N1, N2),

(pk − 1)(p2k−2 − 1)Ψ(T ;N0, pN1, N2) =[(p3k−2 − p2k−1 − pk + p)

up∑j=0

pj(k−1)Cp,D(vp − j)

+(p2k−1 + p2 − p+ p−k+4)Cp,D(vp + 1)

−(p2 + p−k+4)Cp,D(vp + 2)]

Ψ(T ;N0, N1, N2),

(pk − 1)(p2k−2 − 1)Ψ(T ;N0, N1, pN2) =[−p−k+4Cp,D(vp + 1) + p−k+4Cp,D(vp + 2)

]×Ψ(T ;N0, N1, N2).

In the third of these we note that

Cp,D(vp + 2) = Cp,D(vp + 1) + p(vp+2)(2k−3) − χD(p)pk−2p(vp+1)(2k−3), (4.10)

thus

(pk − 1)(p2k−2 − 1)Ψ(T ;N0, N1, pN2)

= pk+1[p(vp+1)(2k−3) − χD(p)pk−2pvp(2k−3)

]Ψ(T ;N0, N1, N2)

= pvp(2k−3)[p3k−2 − χD(p)p2k−1

]Ψ(T ;N0, N1, N2).

This gives the formula for ψ2 stated in the theorem. For ψ1 we note that p3k−2 − p2k−1 −

1The usage is relative.

77

pk + p = (pk−1 − 1)(p2k−1 − p), so

(p3k−2 − p2k−1 − pk + p)

up∑j=0

pj(k−1)Cp,D(vp − j)

= (p2k−1 − p)

[up∑j=0

p(j+1)(k−1)Cp,D(vp − j)−up∑j=0

pj(k−1)Cp,D(v − j)

]

= (p2k−1 − p)[p(up+1)(k−1)Cp,D(vp − up)− Cp,D(vp)

+

up−1∑j=0

p(j+1)(k−1) [Cp,D(vp − j)− Cp,D(vp − j − 1)]

]

= (p2k−1 − p)[p(up+1)(k−1)Cp,D(vp − up)− Cp,D(vp)

+

up−1∑j=0

p(j+1)(k−1)[p(vp−j)(2k−3) − χD(p)pk−2p(vp−j−1)(2k−3)

]]Also, expanding as with (4.10) we have

(p2k−1 + p2 − p+ p−k+4)Cp,D(vp + 1)− (p2 + p−k+4)Cp,D(vp + 2)

= (p2k−1 − p)Cp,D(v)− (pk+1 + p)pvp(2k−3)(p2k−3 − χD(p)pk−2)

Combining these in to the formula for Ψ(T ;N0, pN1, N2) we obtain

(pk − 1)(p2k−2 − 1)Ψ(T ;N0, pN1, N2)

=

(p2k−1 − p)[p(up+1)(k−1)Cp,D(vp − up)

+

up−1∑j=0

p(j+1)(k−1)[p(vp−j)(2k−3) − χD(p)pk−2p(vp−j−1)(2k−3)

]]

−(pk+1 + p)pvp(2k−3)(p2k−3 − χD(p)pk−2

)Ψ(T ;N0, N1, N2)

=pup(k−1)pk−1(p2k−1 − p)

+

[(p2k−1 − p)

(pk−1p

(vp−up)(2k−3)pup(k−1) − pup(k−1)

p2k−3 − 1

+pvp(2k−3) − p(vp−up)(2k−3)pup(k−1)

pk−2 − 1

)−(pk+1 + p)pvp(2k−3) ] (p2k−3 − χD(p)pk−2)

Ψ(T ;N0, N1, N2)

=

(p2k−3 − χD(p)pk−2)

[pvp(2k−3)

(pk−1(p2 − 1)

pk−2 − 1

)−p(vp−up)(2k−3)pup(k−1)

(p(p2k−2 − 1)(pk−1 − 1)

(p2k−3 − 1)(pk−2 − 1)

)]+(χD(p)pk−2 − 1)pup(k−1)p

k(p2k−2 − 1)

p2k−3 − 1

Ψ(T ;N0, N1, N2).

78

This gives the formula for ψ1 stated in the theorem. One can argue in a similar fashion

to derive the formula for ψ0, but given that we have found these formulas for ψ1 and ψ2

it is less painful to instead argue from the observation that by (4.9) we have

Ψ(T ; pN0, N1, N2) + Ψ(T ;N0, pN1, N2) + Ψ(T ;N0, N1, pN2)

= Ψ(T ;N0, N1, N2)

up∑j=0

pj(k−1)Cp,D(vp − j)

so we can obtain ψ0 by evaluating the sum on the right hand side. But this is easily done,

namely

up∑j=0

pj(k−1)Cp,D(vp − j)

=

up∑j=0

pj(k−1)

[1 + (p2k−3 − χD(p)pk−2)

vp−j−1∑i=0

pi(2k−3)

]

=

(p(up+1)(k−1) − 1

pk−1 − 1

)+ (p2k−3 − χD(p)pk−2)

up∑j=0

pj(k−1)

(p(vp−j)(2k−3) − 1

p2k−3 − 1

)= (p2k−3 − χD(p)pk−2)

[pvp(2k−3)

(pk−2

(p2k−3 − 1)(pk−2 − 1)

)−p(vp−up)(2k−3)pup(k−1)

(1

(p2k−3 − 1)(pk−2 − 1)

)]+ (χD(p)pk−2 − 1)

[pup(k−1)

(pk−1

(p2k−3 − 1)(pk−1 − 1)

)− 1

(p2k−3 − 1)(pk−1 − 1)

].

Subtracting ψ1 + ψ2 from this we obtain the formula for ψ0 stated in the theorem.

4.3 Applications to representation numbers of quadratic

forms

Recall from §3.5 and §3.6 that the (n − 1)-cusps of Γ0(N)(n)\Hn are in bijection with

positive divisors d of N , and we defined Φd to be the Siegel lowing operator to the cusp.

The following proposition describes how this operator acts on theta series:

79

Proposition 4.9. Let L be an even lattice and S a Gram matrix of L. Suppose that the

level N is squarefree and det(S) is a square, so that θL ∈Mk(N). For a prime divisor p

of N , let sp(L) be the Hasse invariant of Q on L ⊗Z Zp, normalized as in [60]. For any

divisor d of N , let dp be the highest power of p dividing det(S). Then

Φd(θ(n)L ) =

∏p|d

(d−n/2p sp(L)n

)θ

(n−1)

L#,d

where

L#,d = L# ∩ Z

[1

p; p | d

]denotes the lattice dualized at all primes p | d.

Proof. Let p be any prime divisor of d. Applying [6] Lemma 8.2(a) with l = n (noting

that the factor γp(dp) = 1 since dp is a square) we obtain

θ(n)L |kγ

(n)(p) = d−n/2p sp(L)nθ(n)

L#,p

where

L#,p = L# ∩ Z

[1

p

]L

denotes the lattice dualized only at p, and γ(n)(p) is as in (3.24). Since dp and sp are local

to p and N is squarefree we can apply this result now at other primes dividing d to see

that

θ(n)L |γ

(n)(d) =∏p|d

(d−n/2p sp(L)n

)θ

(n)

L#,d

where L#,d is as in the statement of the proposition. Applying the Siegel lowering operator

we obtain the result.

Corollary 4.10. Let L be an even integral lattice of rank 2k, k ≥ 4. Assume that the

level N is squarefree and the transformation character of θL trivial, so that θL ∈Mk(N).

Let E(N0,N1,N2) be the Eisenstein series in the natural basis. Then

θ(n)gen(L) =

∑c(N0, N1, N2)E(N0,N1,N2)

where the sum is over all tuples (N0, N1, N2) of positive integers such that N0N1N2 = N ,

and the coefficients are given by

c(N0, N1, N2) =

∏p|N1

d−1/2p sp(L)

∏p|N2

d−1p

.

80

In particular if T ∈ Q2×2sym is positive semi-definite and semi-integral then the average

representation number rgen(S)(T ) is given by

rgen(S)(T ) =∑

c(N0, N1, N2)a(T ;E(N0,N1,N2))

where the sum and c(N0, N1, N2) are as above, and a(T ;E(N0,N1,N2)) is given by Theorem

4.8.

Proof. By Siegel’s Hauptsatz we know that θgen(L) is a linear combination of Eisenstein

series. Now the Eisenstein series comprising our basis are characterised by E(N0,N1,N2)

being the unique weight k and level N = N0N1N2 Eisenstein series which takes the value

1 at the cusp corresponding to (N0, N1, N2) and the value 0 at all others.1 Characterising

a degree two Siegel–Eisenstein series by its value at 0-cusp corresponds to characterising

its image under two applications of Siegel lowering operators; by unwinding our defini-

tions one easily checks that the condition of taking value at the cusp corresponding to

(N0, N1, N2) is that

ΦN1

(ΦN2

(E(N0,N1,N2)

))= 1.

Thus to express θ(n)L as a linear combination of Eisenstein series it suffices to compute the

value of θ(n)L at the 0-cusp (N0, N1, N2). Applying Proposition 4.9 with n = 2 we have

ΦN2(θ(n)L ) =

∏p|N2

d−1p θ

(n−1)

L#,d ,

since the Hasse invariant is either 1 or −1. Next applying ΦN1 and using Proposition 4.9

with n = 1 we have

ΦN1

(ΦN2

(θ

(n)L

))=

∏p|N1

d−1/2p sp(L)

∏p|N2

d−1p

,

using the fact that N1 and N2 are coprime.

We emphasise that everything in Corollary 4.10 is completely explicit. To illustrate this

we consider the case when the genus of the quadratic form corresponding to S contains

only one isometry class. Then the average and exact representation numbers rgen(S)(T )

1To be precise, the term “value at cusp” depends on how one defines the Siegel lowering operator,which in turns depends on a choice of coset representative. The representatives we take are those givenby Lemma 3.15, which are the same ones used to define our Eisenstein series. The claimed property ofE(N0,N1,N2) then follows from (2.13).

81

and rS(T ) are the same object and Corollary 4.10 gives us an exact formula for these.

Now if S is of size 2k then it describes a modular form of weight k; in order to analyse

this with Eisenstien series we require k to be even and at least 4. According to the Nebe–

Sloane database there are 36 8-dimensional lattices which form a single-class genus (and

none in higher dimensions divisible by 4); of these there are 5 which satisfy the condition

that the level be squarefree and the transformation character trivial. As noted in the

introduction one of these (which has matrix S1 in the following) is E8, the others are

not unimodular but have small prime level. Explicitly these lattices are the following: we

regard a symmetric matrix S = (sij) of size 8 as being determined by a tuple

v(S) = (s11, s21, s22, s31, s32, s33, ..., s81, s82, s83, s84, s85, s86, s87, s88).

Then the Gram matrices Si for the 8-dimensional single-genus even lattices of squarefree

level and trivial character are determined by

v(S1) = (2, 1, 2, 1, 1, 2, 1, 0, 0, 2, 1, 1, 0, 0, 2, 1, 1, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 2,

1, 1, 0, 1, 1, 1, 0, 2),

v(S2) = (2,−1, 2, 0,−1, 2, 0, 0,−1, 2, 0, 0, 0,−1, 2, 0, 0,−1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2,

0, 0, 0, 0, 0, 0, 1, 2),

v(S3) = (2, 1, 2,−1,−1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2,

1, 1, 0, 1, 1, 1, 1, 2),

v(S4) = (2, 0, 2, 0, 0, 2, 1, 1, 1, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2,

0, 0, 0, 0, 1, 1, 1, 2),

v(S5) = (2, 0, 2, 0, 0, 2, 1,−1, 1, 4, 0, 0, 0, 1, 2, 0, 0, 0,−1, 0, 2, 0, 0, 0,−1, 0, 0, 2,

0, 0, 0, 1, 0, 0, 0, 2).

Computing the level of each lattices and applying Proposition 4.9 we obtain the following:

Matrix Level Number of representations of T

S1 1 a(T ;E4,(1,1,1))

S2 3 a(T ;E4,(3,1,1)) + (1/3) a(T ;E4,(1,3,1)) + (1/9) a(T ;E4,(1,1,3))

S3 2 a(T ;E4,(2,1,1)) + (1/2) a(T ;E4,(1,2,1)) + (1/4) a(T ;E4,(1,1,2))

S4 2 a(T ;E4,(2,1,1)) + (1/4) a(T ;E4,(1,2,1)) + (1/16) a(T ;E4,(1,1,2))

S5 2 a(T ;E4,(2,1,1)) + (1/8) a(T ;E4,(1,2,1)) + (1/64) a(T ;E4,(1,1,2))

82

With the easily computable formulas of Theorem 4.8 one can now compute representa-

tion numbers of these quadratic forms very quickly on a computer. Of course the same

reasoning applies to allow quick computation of representation numbers of genus-averages

of quadratic forms, provided that the level is squarefree and the transformation character

is trivial.

83

Chapter 5

Equidistribution of Satake

parameters attached to Siegel cusp

forms

For the remainder of the thesis we will mostly be concerned with the space S(2)k (N), as

the parameters N and k vary. We will therefore drop the subscript (2) from the notation,

and there will no longer be any transformation characters in our notation (e.g. for Hecke

operators) either. The questions we ask about the elements of Sk(N) are mostly concerned

about Hecke eigenvalues or, equivalently, Satake parameters. It would be interesting to

have versions of the equidistribution results described herein for Fourier coefficients, as

one could interpret this as having results on the distribution of the error term in degree

two representation numbers for (certain) quadratic forms. However, as described in the

introduction, Fourier coefficients are generally more difficult to get ones hands on, so we

must make do with Satake parameters.

We will think of the elements of Sk(N) as vectors inside cuspidal automorphic repre-

sentations of GSp4. As one increases the weight and level one might expect that these

representations vary in a family. A recent survey article of Kowalski ([37]) proposes that a

reasonable family of automorphic representations should satisfy the following condition:

after defining an appropriate notion of a conductor and way to count (i.e., an appropriate

84

measure on) cuspidal automorphic representations up to a given conductor, the local com-

ponents of these representations should, as we increase the conductor, be equidistributed

amongst all possibilities. Not only this, but the distributions of the local components at

different places should be independent. In this chapter we show, building on the paper

[39], that this property holds for the representations generated by holomorphic Siegel

cusp forms of degree 2.

A motivating example of this behaviour is due to Serre ([64], Theoreme 1) and, indepen-

dently, Conrey–Duke–Farmer ([13], Theorem 1). Fix a finite set S of primes, and consider

the set of all holomorphic cusp forms on SL2(Z) of weight k, level N , and trivial neben-

typus, with k varying over even positive integers and N varying over integers with none

of their prime factors in S. At the places p ∈ S the local representation attached to a

holomorphic cuspidal eigenform is unramified and is determined by the Hecke eigenvalue

λ(p; f) of f under T (p). The above papers prove that, as k and N vary as above, normal-

ized eigenvalues λ′(p; f) = λ(p; f)/2p(k−1)/2 are equidistributed with respect to the p-adic

Plancherel measure on [−1, 1].1 This implies, without invoking the machinery of Deligne,

that “most” cusp forms satisfy the Ramanujan–Petersson conjecture λ′(p; f) ∈ [−1, 1];

but it also ties down a precise measure with respect to which the points λ′(p; f) equidis-

tribute. Moreover, the joint asymptotic distribution for different primes p are independent

(see §5.2 of [64]). Similarly, an earlier result of Sarnak ([58]) states that if one considers

the pth Fourier coefficient of Maass forms averaged over the Laplacian eigenvalues, one

finds that they are equidistributed with respect to this same measure. Another example,

pertinent for us, is a weighted version of the quoted result of [64] and [13] which is implicit

in the work of Bruggeman ([8]).

The above results can be understood as solutions to equidistribution problems, which

can be set up very generally as follows: Let X be a topological space, V a finite dimen-

sional complex vector space endowed with an inner product 〈, 〉, Q any fixed non-negative

quadratic form on V , and H a finitely generated commutative algebra of hermitian oper-

1This is related to, but easier than, the Sato–Tate problem, where one fixes the modular form butvaries the prime.

85

ators acting on V . Suppose that whenever v ∈ V is an eigenvector for H it has associated

to it a point a(v) ∈ X such that if v1 and v2 lie in the same eigenspace then a(v1) = a(v2).

For each v ∈ V let ω(v) = Q(v)/〈v, v〉. For each orthogonal basis B of V consisting of

eigenforms for H, consider the measure on X given by νV,ω =∑

v∈B ω(v)δa(v) (where δ is

the Dirac mass).

Now suppose we keep X fixed but vary V over a sequence of finite dimensional vector

spaces: then it makes sense to ask whether the sequence of measure νV,ω converges weakly

to some canonical measure µ on X. In other words, we ask whether the points a(v) with

v ∈ B counted with the “weighting” ω equidistribute. We are specifically interested in the

case when V is the space of automorphic forms on some group of some fixed conductor

and infinity type (as we allow the conductor to increase), H is the local Hecke algebra at

p for some fixed prime p (or more generally the algebra generated by finitely many Hecke

operators), and for an eigenform F of H the point a(F ) is the local Satake parameter

at the prime p. The problem is then one of spectral equidistribution, and the results of

Serre and Conrey–Duke–Farmer is spectral equidistribution of holomorphic cusp forms

when the weighting ω is constant.

We are concerned not with S(1)k (N) but rather S(2)

k (N). In order to state a version of the

main result we assume a certain amount of familiarity with automorphic representations;

the relevant parts of the theory will be explained in §5.1 and §5.2. As before fix a finite

set of primes S. Let F ∈ Sk(N), and suppose that F is an eigenform for the local Hecke

algebras at all primes in S. Then for each prime p ∈ S there is a spherical principal series

representation πF,p of GSp4(Qp) generated by F (see Remark 5.3). The isomorphism class

of πF,p is determined by the eigenvalues of F for the Hecke operators T (2)(p) and T(2)1 (p2).

Equivalently, the isomorphism class of πF,p is determined by the orbit of the Satake

parameters (ap(F ), bp(F )) of πF,p under a certain Weyl group which we denote by W . Now

it is known that the Satake parameters satisfy 0 < |ap(F )| , |bp(F )| ≤ √p, and the very

deep generalized Ramanujan conjecture for GSp4 (a proof of which has recently appeared

in [73]) implies that if F is not a Saito-Kurokawa lift then |ap(F )| = |bp(F )| = 1. (If F is

a Saito–Kurokawa lift then it easily follows that for an appropriate representative in the

86

Weyl group orbit we have |ap(F )| = 1 and |bp(F )| =√p.) We can therefore regard the

local components as points on the space Yp = (a, b) ∈ C× ×C×; 0 < |a| , |b| ≤ √p/W ,

and the local components of the representations attached to non-Saito–Kurokawa lifts

actually lie inside Ip the product of the two unit circles in Yp (or rather its image under

the quotient by W ; this is the subspace of tempered representations). The main theorem

can then be stated as follows:

Theorem 5.1 (Local equidistribution and independence, prototypical version1). Let S

be a finite set of primes. Let k ≥ 6 be even and let N ≥ 1 have none of its prime factors

in S. Let YS =∏

p∈S Yp, and define a measure νS,N,k on YS by

νS,N,k =∑

f∈Sk(N)∗

ωF,N,kδπS(F ),

where

ωF,N,k =

√π(4π)3−2kΓ

(k − 3

2

)Γ(k − 2)

vol(Γ(2)0 (N)\H2)

|a(12;F )|2

4〈F, F 〉,

Sk(N)∗ is any orthogonal basis for Sk(N) consisting of eigenforms for the local Hecke

algebra at all p ∈ S, πS(F ) =∏

p∈S(ap(F ), bp(F )) ∈ YS, and δ denotes Dirac mass.

Then, as k + N → ∞ with k ≥ 6 varying over even integers and N ≥ 1 varying over

integers with none of their prime factors in S, the measure νS,N,k converges weak-∗ to a

certain product measure µS =∏

p µp on YS, which is the measure2 referred to in [24] as

the Plancherel measure for the local Bessel model associated to (4,1). That is, for any

continuous function ϕ on YS,

limk+N→∞

∑F∈Sk(N)∗

ωF,N,k ϕ((ap(F ), bp(F ))p∈S) =

∫YS

ϕ dµS.

In particular if ϕ =∏

p∈S ϕp is a product function then

limk+N→∞

∑F∈Sk(N)∗

ωF,N,kϕ((ap(F ), bp(F ))p∈S) =∏p∈S

∫Yp

ϕp dµp.

1Theorem 5.5 is a slightly more general version of this theorem, which in fact contains an infinitefamily of local equidistribution and independence statement indexed by fundamental discriminants −d(d > 0) and characters Λ of the ideal class group of Q(

√−d). The above is d = 4, Λ = 1. The measure

νS,N,k does not depend on the choice of basis (see Lemma 5.4) and in Theorem 5.5 a slight relaxationon the basis is allowed. Theorem 5.17 is a quantitative version of Theorem 5.5.

2The measures µp and their product µS will be constructed in detail in §5.2. In particular we will seethat µS is actually supported on Ip.

87

Theorem 5.1 is a generalization of Theorem 1.6 of [39], which deals with the case when

N is fixed equal to 1.1 Note that the cusp forms are only required to be eigenfunctions

at p ∈ S, a point which does not seem to have been emphasised in previous work in

equidistribution. Our methods of proof follow those of [39], with some small changes

when arguing with Bessel models in §5.3 and the main modifications coming from the

need to track the dependency on both the weight k and level N in certain estimates of

Fourier coefficients of cusp forms, carried out in §5.4. As well as the case considered in

[39] (N = 1, k varying) this result generalizes one which recently appeared in [10] (k

treated constant, N varying); in treating the mixed case we also obtain a better decay

with respect to N than that in [10]. Allowing both the weight and level allows for the

most general notion of conductor in this context, so Theorem 5.1 settles the question of

local equidistribution and independence (at unramified primes) for representations at-

tached to classical Siegel modular forms. We remark now that we will actually prove

a quantitative version of this (Theorem 5.17), which will be useful for applications, for

example in Chapter 6.

In recent work ([65], [66]) Shin and Shin–Templier have proved a very general local

equidistribution and independence statement. For any cuspidal automorphic represen-

tation of a reductive group G with a discrete series representation at the archimedean

place2, they are able to count cuspidal automorphic representations with their natural

weight 1 (in contrast to the weight ωF,N,k appearing in ours) and prove a local equidis-

tribution, with the limit measure a suitable normalization of Plancherel measure, on the

unitary dual of G(Qp); and moreover they prove the expected independence as well. The

limits are taken in either the increasing weight or level aspect, and it is expected an

appropriate combination of their arguments would deal with the mixed case. The groups

1Setting N = 1 it appears we have an extra factor of vol(Γ(2)0 (1)\H2). However, our Petersson norms

are normalised whereas those of [39] are not, so the weights are in fact the same.2We have stated a weak version of their result relevant to our setup, but their theorem is much more

general. In particular Q can be replaced by any totally real field and what follows is true verbatim. Thecondition that the archimedean component admits a discrete series representation is, however, important.The proof assumes that transfer to GLn is known for G. For G = GSp4, N = 1, this is a theorem ofPitale–Saha–Schmidt (see [52]). More general results should follow from the work of Arthur. At the timeof writing, to the best of the author’s knowledge, a satisfactory version of transfer is known for G = Sp2n,but not for G = GSp2n.

88

satisfying these hypotheses include GSp4 and also higher rank symplectic groups, as well

as GL1 and GL2 (but not GLn for n ≥ 3).

Our Theorem 5.1 (even for either fixed level or weight aspect) is not contained in the

work of Shin–Templier, due to our different weights in counting. In fact, we see that the

presence of the arithmetic factor in our weight affects the limiting measure (that is, our

limit measure is not Plancherel measure – for more details on our limit measure see §5.2).

Such behaviour has been observed in local equidistribution problems of cuspidal automor-

phic representations on general linear groups in various families when the representations

are counted weighted by special values of associated L-functions. Our arithmetic factor

|a(12;F )|2 /〈F, F 〉 prima facie does not appear to be so significant, but, at least when F is

an eigenform, a deep conjecture of Bocherer relates |a(12;F )|2 to L(1/2, F )L(1/2, F×χ−4)

(the L-functions are normalized to have functional equation relating s with 1 − s). We

will discuss this feature more fully in Chapter 6.

In this chapter we work with trivial character and (mostly) degree two, so we again omit

the superscript (2) from the notation. We also write G = GSp4. In contrast to §4 we do

not assume squarefree level. This chapter is based on part of the paper [15].

5.1 The representation attached to a Siegel modular

form

Adelization. We begin by recalling the construction of the adelization of a Siegel cusp

form. Our setup follows that of [55], which we refer to for proofs. Let F ∈ Sk, so there is

a positive integer N such that F ∈ Sk(Γ(N)). Define, for each prime p, a compact open

subgroup KNp of G(Zp) by

KNp =

g ∈ G(Zp); g ≡

12

a12

mod NZp, a ∈ Z×p

. (5.1)

89

Note that KNp = G(Zp) for all primes p - N , and that the multiplier map µ2 : G(Zp)→ Z×p

is surjective for every prime p. The strong approximation theorem for G then says that

G(A) = G(Q)G+(R)∏p<∞

KNp .

Define the adelization ΦF of F by

ΦF (gQg∞h) = µ2(g∞)kj(g∞, i12)−kF (g∞〈i12〉)

where gQ ∈ G(Q), g∞ ∈ G+(R), h ∈∏

p<∞KNp .1 Since G(Q) ∩ G+(R)

∏pK

Np = Γ(N),

the modularity of F implies that ΦF is well-defined. Moreover, one can easily check that

ΦF is independent of the choice of N made in its construction.

The map F 7→ ΦF injectively assigns to each degree 2 Siegel modular form a function

on G(A). Immediately from the definition it is clear that ΦF (gQg) = ΦF (g) for all gQ ∈

G(Q), g ∈ G(A). Also |ΦF (g)|2 is invariant under the centre ZG(A), and so we can form

the following integral, which will in fact be finite by the moderate growth of F :∫ZG(A)G(Q)\G(A)

|ΦF (g)|2 dg <∞. (5.2)

We follow the standard abuse of notation and write L2(G(Q)\G(A)) for the space of func-

tions on G(Q)\G(A) whose absolute value descends to a function on G(Q)ZG(A)\G(A)

which is square-integrable as in (5.2). The above then shows that F 7→ ΦF defines a linear

injection Sk → L2(G(Q)\G(A)). One can list a few more conditions on that ΦF must

satisfy by virtue of coming from a cusp form (e.g. cuspidality is given by the vanishing

of a sort of adelic Fourier coefficient, holomorphicity is reflected in the action of the Lie

algebra of GSp4(R), etc.) We refer to [55] for the precise characterisation of the image

Vk ⊂ L2(G(Q)\G(A)); and in particular Theorem 1 which shows that the map Sk → Vk

is an isometry of vector spaces, the inner product on Sk being (2.8).

Hecke operators. We now introduce the adelic counterpart the the classical Hecke

algebra. Fix a prime p, and define Hp to be the set of locally constant compactly supported

1Unfortunately there is a slight clash between this and our notation for the Siegel lowering operator.However, the Siegel lowering operator will not appear in the remainder of this thesis, so we henceforthuse Φ to denote functions on G(A).

90

functions on G(Qp) which are both left and right invariant by G(Zp). This is equipped

with convolution product, namely for f1, f2 ∈ Hp define

(f1 ∗ f2)(g) =1

vol(G(Zp))

∫G(Qp)

f1(gh)f2(h−1)dh.

There is a canonical map Hp → Hp defined to be the Z-linear extension of the map

Γ(1)MΓ(1) 7→ 1G(Zp)MG(Zp). Denoting the image of an arbitrary element T ∈ H by T ,

the map T 7→ T induces an isomorphism of rings Hp ⊗Z C → Hp. Now the classical

Hecke algebra Hp acts on p-spherical modular forms as explained in §2.4. On the adelic

side we also have a notion of Φ ∈ Vk being p-spherical, namely that Φ(gk) = Φ(g) for all

k ∈ G(Zp) (so by definition any given Φ is p-spherical for all but finitely many p). An

element f in the adelic Hecke algebra Hp acts on a p-spherical Φ by the rule

(fΦ)(g) =1

vol(G(Zp))

∫G(Qp)

f(h)Φ(gh)dh.

If we restrict the map F 7→ ΦF to a map between p-spherical elements, then it is Hecke

equivariant in the sense that

ΦF |T = TΦF

for any p-spherical F ∈ Sk and T ∈ Hp. Again we refer to [55] for a proof of these facts.

The automorphic representation attached to a cusp form. Let F ∈ Sk, and

let ΦF ∈ Vk be its adelization. Letting G(A) act on ΦF by the right regular action

ΦF (g) 7→ ΦF (gh) for h ∈ G(A) we generate a cuspidal automorphic representation πF of

G(A).1 This decomposes as a direct sum of finitely many irreducible cuspidal automorphic

representations of G(A), say

πF =m⊕i=1

π(i)F (5.3)

with each (π(i)F , V

(i)) irreducible.2 Let π = π(i)F be any irreducible constituent of πF . By

the tensor product theorem there exist irreducible, unitary, admissible representations πv1As usual, we actually only allow the action of a (g,K)-module at the infinite place, but suppress this

technicality.2Such a decomposition need not be unique, since an irreducible constituent appearing in the sum

could do so with multiplicity greater than one. However, it is expected that weak multiplicity one holds(i.e. that any (isomorphism class of) irreducible cuspidal automorphic representation of G(A) occurringin L2(G(Q)\G(A)) does so with multiplicity one), which would rule this possibility out. We do notneed to assume anything about the uniqueness of this decomposition, since the local components we areinterested in will always turn out to be isomorphic, regardless of which constituent we have chosen.

91

of G(Qv) (one for each place v of Q) such that

π ' ⊗′vπv, (5.4)

where the prime denotes a restricted tensor product, and for almost all v the local repre-

sentation πv is spherical. Since F ∈ Sk the archimedean component π∞ is a certain lowest

weight representation as described in [4]. Similarly, the following proposition describes πp

when F is an eigenfunction for Hp:

Proposition 5.2. Let p be a prime and suppose F ∈ Sk is p-spherical. Assume moreover

that F is an eigenfunction for the Hecke operators T (p) and T1(p2) (and hence, by Lemma

2.7, an eigenfunction for Hp), with corresponding eigenvalues λ(p) and λ1(p2). Then, for

any irreducible constituent π of πF , the local component πp in any isomorphism of the

form (5.4) is a spherical principal series representation1 of G(Qp) whose isomorphism

class is determined uniquely by λ(p) and λ1(p2).

Proof. See [55] Proposition 3.9.

Remark 5.3. It follows that there is a well-defined isomorphism class of local represen-

tations at p (which are necessarily spherical principal series) attached to a p-spherical

element F ∈ Sk under the assumption that F is an eigenfunction of T (p) and T1(p2).

This is well-defined in the sense that it is independent of the (possible) choice of decom-

position in (5.3), the choice of irreducible constituent π = π(i)F from this decomposition,

and the choice of isomorphism in (5.4).

5.2 The equidistribution problem

We now describe in detail the equidistribution problem addressed in this chapter. Fix

a finite set of primes S, let k be any even integer ≥ 6, and let N be a positive integer

with gcd(N,S) = 1. Let Sk(N)∗ denote any2 orthogonal basis of Sk(N) consisting of

forms that are eigenfunctions of T (p) and T1(p2) whenever p ∈ S (there is no ambigu-

ity in our notation for the Hecke operators – see the discussion following Lemma 2.7).

1We will recall the construction of these representations in §5.2.2The definitions we make in the following appear to depend on the choice of basis. However we will

show in Lemma 5.4 that this is not the case.

92

Let F ∈ Sk(N)∗. By Remark 5.3 we can attach to F an isomorphism class of spherical

principal series representations of G(Qp) for each p ∈ S. Since F ∈ Sk(N)∗ has trivial

character, the central character of the corresponding representation will be trivial.

Spherical principal series representations. We now recall the construction of the

spherical principal series representations of G(Qp) with trivial central character. Let

χ1, χ2, σ be unramified quasi-characters of Q×p , and define a character of the Borel sub-

group a1 ∗ ∗ ∗

a2 ∗ ∗

λa−11

∗ λa−12

7→ χ1(a1)χ2(a2)σ(λ).

We require the central character to be trivial, so χ1χ2σ2 = 1. Via normalized induction

we obtain a representation of G(Qp), and this has a unique spherical constituent, denoted

χ1 × χ2 o σ as in the notation of [57]. Since the quasi-characters χ1, χ2, σ are unramified

they are completely determined by their values on p ∈ Q×p . Since the central character

is trivial, χ1 × χ2 o σ is therefore determined by (a, b) = (σ(p), σ(p)χ1(p)) ∈ C× ×C×.

We refer to (a, b) as the Satake parameters of χ1 × χ2 o σ. By the classification in [53],

it follows that 0 < |a| , |b| ≤ √p. The form of the generalized Ramanujan conjecture for

GSp4 proved by Weissauer (see [73]) states that if the global representation π is not CAP

then in fact |a| = |b| = 1. We will discuss this further in §6.1 and §6.2.

Any spherical principal series representation of G(Qp) with trivial central character is

isomorphic to some χ1×χ2oσ. Moreover, the representations χ1×χ2oσ and χ′1×χ′2oσ′,

with associated (a, b) and (a′, b′) respectively, are isomorphic if and only if (a, b) and (a′, b′)

lie in the same orbit under the action of the Weyl group W of order 8 generated by the

transformations

(a, b) 7→ (b, a), (a, b) 7→ (a−1, b), (a, b) 7→ (a, b−1). (5.5)

Let Xp be the set of isomorphism classes of spherical principal representations of G(Qp).

Let Yp = (a, b) ∈ C××C×; 0 < |a| , |b| ≤ √p/W . Then we have a well-defined injection

93

Xp → Yp. The space Yp therefore provides a natural choice of co-ordinates on Xp. Fix a

finite set of primes S. We also form the product spaces

XS =∏p∈S

Xp, YS =∏p∈S

Yp. (5.6)

We form the natural injection XS → YS, which allows us to view XS as a subspace of

YS. We will formulate our equidistribution problem on YS, doing so by defining two mea-

sures νS,N,k and µS on YS and showing that these agree in an appropriate weak-∗ limit.

The measure µS is a certain natural measure on YS. The measure νS,N,k reflects the dis-

tribution of the spherical principal series representations attached to eigenforms in Sk(N).

The measure νS,N,k. As mentioned in the introduction, our distribution will be weighted

by a certain “arithmetic factor”; our first task is to define this. Let k ≥ 6 be even

and N a positive integer with gcd(N,S) = 1. Let d ∈ Z≥1 be such that −d is the

discriminant of Q(√−d). Let w(−d) denote the number of roots of unity in Q(

√−d). Let

Cld denote the ideal class group of Q(√−d), and let Λ be any character of Cld. Recall the

isomorphism between Cld and the set of SL2(Z) equivalence classes of primitive, semi-

integral, positive definite matrices with determinant d/4. We write this map from Cld to

the set of (equivalence classes of) such matrices as c 7→ Sc.1 Define

cd,Λk =√π(4π)3−2kΓ

(k − 3

2

)Γ(k − 2)

(d

4

)−k+ 32 dΛ

w(−d) |Cld|,

where

dΛ =

1 if Λ2 = 1,

2 otherwise.

Define also

a(d,Λ;F ) =∑c∈Cld

Λ(c)a(Sc;F ), (5.7)

which is well-defined since the Fourier coefficients a(T ;F ) depend only on the equivalence

class of T modulo SL2(Z)-conjugation (the same is even true for GL2(Z)-conjugation,

1We apologize for the various incarnations of the letter S at this point. Eventually the S = Sc willbe used to define a Bessel model, and following the standards in the literature we should use S = Sc forthis. In order to to minimize confusion with our finite set of primes (and the spaces of cusp forms), weuse the standard letter but in sanserif font.

94

since k is even).

The weight1 we use is

ωd,ΛF,N,k =cd,Λk

vol(Γ0(N)\H2)

|a(d,Λ;F )|2

〈F, F 〉. (5.8)

Recall that the Petersson inner product, defined by (2.8), is independent of the choice of

congruence subgroup. The dependence on N is therefore solely via vol(Γ0(N)\H2), in the

sense that if F ∈ Sk(N) ⊂ Sk(NN1), then

ωd,ΛF,NN1,k=

vol(Γ0(N)\H2)

vol(Γ0(NN1)\H2)ωd,ΛF,N,k.

The asymptotics as a function of N is therefore determined by the index of Γ0(N) inside

Sp4(Z), and one can easily check vol(Γ0(N)\H2) [Γ0(N) : Sp4(Z)] N3. The depen-

dency on k is already explicit from the form of cd,Λk .

A more subtle point is the dependency of this weight on F . In the parlance of general

equidistribution problems from the introduction we have chosen the quadratic form Q to

be

F 7→ cd,Λk |a(d,Λ;F )|2

vol(Γ0(N)\H2).

It is believed that the term |a(d,Λ;F )|2 carries deep arithmetic information: when F

is an eigenform, a conjecture of Bocherer relates this quantity to the central value

L(1/2, πF × χ−d) of the Langlands L-function L(s, πF × χ−d), where χ−d is the char-

acter corresponding to the imaginary quadratic extension Q(√−d). This deep conjecture

can be viewed as an analogue of Waldspurger’s famous theorem in the case of elliptic

modular forms. To the best of the author’s knowledge this has only been proved for cer-

tain “lifts” (e.g. Saito–Kurokawa and Yoshida lifts).

In our investigation of the asymptotics of this measure we will work with a fixed but

arbitrary choice of d and Λ. Consequently we will often abbreviate ωd,ΛF,N,k to ωF,N,k. The

limiting distribution, µS defined below, will also depend on the choice of d,Λ. In order to

1When N = 1 this is the weight used in [39], though one must recall that we normalize our Peterssoninner products differently.

95

simplify notation one may wish to focus on the simplest case, which is d = 4 and Λ = 1,

giving the weight used in Theorem 5.1. We will also restrict to this weight in §6.1-§6.3.

With d and Λ fixed, now fix S and form Sk(N)∗ as we did at the beginning of this section.

To each F ∈ Sk(N)∗ we have associated a tuple πS(F ) = (πp(F ))p∈S, where each πp(F ) is

an isomorphism class of spherical principal series representations of G(Qp). We also write

πS(F ) ∈ YS for the image of this tuple under the map XS → YS. The measure νS,N,k on

YS, which is supported on (the image of) XS, is defined by

νS,N,k =∑

F∈Sk(N)∗

ωF,N,kδπS(F ), (5.9)

where δ denotes Dirac mass.

In a moment we will compare this with the general equidistribution set up in the intro-

duction. First we prove, in that generality, that the measure is independent of the choice

of basis:

Lemma 5.4. Let X be a topological space, V a finite dimensional complex inner product

space, Q a fixed non-negative hermitian form on V , and H a finitely generated commu-

tative algebra of hermitian operators acting on V . Suppose that whenever v ∈ V is an

eigenvector for H is has associated to it a point a(v) ∈ X such that if v1, v2 lie in the same

eigenspace then a(v1) = a(v2). For each v ∈ V let ω(v) = Q(v)/〈v, v〉. For each orthogonal

basis B of V consisting of eigenforms of H define a measure X by νB =∑

v∈B ω(v)δa(v).

Then νB is independent of the choice of B.

Proof. V can be written as a direct sum ofH-eigenspaces, different eigenspaces necessarily

being orthogonal, and hence we reduce to the case when all v ∈ V have the same a(v).

Let A denote the linear operator such that Q(v) = 〈Av, v〉. Take a function F : X → C,

then ∫FdνB = F (a)

∑v∈B

Q(v)

〈v, v〉= F (a)

∑v∈B

〈Av, v〉〈v, v〉

= F (a) tr(A),

which is independent of B. Thus νB is independent of B.1

1This proof actually shows how we can define νB without picking a basis: namely νB :=∑E tr(AE)δa(E) where E varies over the distinct H-eigenspaces, AE is the operator representing Q

restricted to E, and a(E) = a(v) for any v ∈ E.

96

For our present situation in the notation of the lemma we have:

• the topological space X is YS,

• the finite dimensional vector space V is Sk(N), equipped with the Petersson inner

product,

• the algebra of operators H consists of the local Hecke algebras for p ∈ S,

• the point a(F ) ∈ X for F ∈ V is the Satake parameters of the local representation,

πS(F ),

• the quadratic form Q is F 7→ cd,Λk |a(d,Λ;F )|2 / vol(Γ0(N)\H2).

The measure µS. Our limiting measure is the measure referred to in [24] as the Plancherel

measure for the local Bessel model associated to (d,Λ). In [39] is appears as the limiting

measure for νS,1,k as k → ∞ over even integers. We follow this paper for our definition

now. Let

Ip = (a, b) ∈ C× ×C×; |a| = |b| = 1/W ⊂ Yp,

where W is the Weyl group generated by (5.5). We write (a representative of) the point

(a, b) ∈ Ip using the co-ordinates (a, b) = (eiθ1 , eiθ2). We define a measure dµp on Ip by

dµp(θ1, θ2) =4

π2(cos(θ1)− cos(θ2))2 sin2(θ1) sin2(θ2) dθ1dθ2.

Note that this is independent of the choice of representative (eiθ1 , eiθ2). This can be

obtained as a pushforward of the probability Haar measure on USp4 (the compact form

of Sp4) to Ip, in analogy with the construction of the classical Sato–Tate measure. We

extend µp to a measure on Yp, also denoted µp, by extending by zero. The measure

µp = µp,d,Λ is now defined by

dµp =

(1−

(−dp

)1

p

)∆−1p,d,Λdµp.

This measure is also supported on Ip ⊂ Yp. The function ∆p,d,Λ is given by

∆p,d,Λ(θ1, θ2) = δ1δ2

97

where

δi =

((1 + 1

p

)2

− 4 cos2(θi)p

)p inert in Q(

√−d),((

1− 1p

)2

+(

2 cos(θi)√p− λp

p

)(2 cos(θi)√

p− λp

))p splits in Q(

√−d),(

1− 2λp cos(θi)√p

+ 1p

)p ramifies in Q(

√−d),

and where λp =∑

N(p)=p Λ(p) (a sum over the one or two prime ideals in Q(√−d) of norm

p). Note that ∆p,d,Λ is again independent of the choice of Weyl group orbit representative.

Finally, we define the measure µS = µS,d,Λ on XS by

dµS =∏p∈S

dµp. (5.10)

Although the definition is rather complicated this measure is at least completely explicit.

Along with the fact that the measure is supported on Ip, it is perhaps also worth noting

that dµp tends towards the Sato–Tate measure as p→∞.

Theorem 5.5 (Local equidistribution and independence, qualitative version). Fix any

d > 0 such that −d is the discriminant of Q(√−d), and let Λ be any character of Cld.

For any finite set of primes S, the measure νS,k,N converges weak-∗ to µS as k+N →∞

with k ≥ 6 varying over positive even integers and N ≥ 1 varying over positive integers

with gcd(N,S) = 1. That is, for any continuous function ϕ on YS,

limk+N→∞

∑F∈Sk(N)∗

ωF,N,k ϕ((ap(F ), bp(F ))p∈S) =

∫YS

ϕ dµS.

In particular if ϕ =∏

p∈S ϕp is a product function then

limk+N→∞

∑F∈Sk(N)∗

ωF,N,kϕ((ap(F ), bp(F ))p∈S) =∏p∈S

∫Yp

ϕp dµp.

The proof of (a quantitative version of) this theorem is the goal of the next three sections.

Before proceeding, let us remark on the cases of low (even) weight which are not covered

by Theorem 5.5 (i.e. k = 2, 4). As we shall see, the condition k ≥ 6 is necessary for

absolute convergence of a certain Poincare series (required also in related calculations

in [39] and [10]) and is an artefact of our method. In [39] this condition is not an issue

98

as they work in the limit k → ∞. However, in our context, the level aspect for fixed

small weight is an interesting case which is not addressed by our results. Note that the

weight k = 4 (for which the ∞-type is cohomological) in the level aspect is included in

the work of [66]. It would be particularly interesting to study this problem in the case

of k = 2 and paramodular newforms, since rational paramodular newforms conjecturally

correspond to abelian surfaces over Q (with a certain genericity assumption), and so the

problem is related to a vertical Sato–Tate problem for abelian surfaces. The arithmetic

weighting then, under the Bocherer and Birch–Swinnerton-Dyer conjectures, carries deep

information about the arithmetic of the abelian surface.

5.3 Bessel models

Global Bessel models. We begin by recalling the definition of the global Bessel model

for a cuspidal representation of G(A) in the fashion of [20], [39]. Let S ∈ Q2×2sym be positive

definite. Let disc(S) = −4 det(S) < 0 and d = 4 det(S) > 0. If we write S =(

a b/2b/2 c

),

then we define ξ = ξS by

ξ =

b/2 c

−a −b/2

.

Let L = Q(√−d). We have an isomorphism

Q(ξ)→ L

defined by

a+ bξ 7→ a+ b

√−d2

.

Now define the algebraic group

T = g ∈ GL2; tgSg = det(g)S.

A straightforward computation shows that Q(ξ)× = T (Q), and hence we can identity

T (Q) with L×. We embed T as a subgroup of G via

g 7→

g 0

0 det(g)tg−1

. (5.11)

99

Define another subgroup of G by

U =

u(X) =

12 X

02 12

; tX = X

,

and let R = TU .

Let ψ =∏

v ψv be a character of A such that the conductor of ψp is Zp for all finite

primes p, ψ∞(x) = e(x) for x ∈ R, and ψ|Q = 1. Define a character θ of U(A) by

θ(u(X)) = ψ(tr(SX)).

Let Λ be a character of T (A)/T (Q) such that Λ|A× = 1. Using the above isomorphism

we see that this can be thought of as a character of AL×/L× such that Λ|A× = 1. Define

a character Λ⊗ θ of R(A) by (Λ⊗ θ)(tu) = Λ(t)θ(u) for t ∈ T (A), u ∈ U(Q).

Now let π be a cuspidal representation of G(A) with trivial central character, and let Vπ

be its space of automorphic forms. For Φ ∈ Vπ, we define a function BΦ on G(A) by

BΦ(g) =

∫R(Q)ZG(A)\R(A)

(Λ⊗ θ)(r)Φ(rg)dr. (5.12)

Note that the complex vector space C〈BΦ; Φ ∈ Vπ〉 is preserved under the right regular

action of G(A), since Φ ∈ Vπ is.

Consider the case that π =⊗

v πv is an irreducible cuspidal representation with trivial

central character, with space of automorphic forms Vπ. If C〈BΦ; Φ ∈ Vπ〉 is non-zero then

the representation afforded by the right regular action of G(A) on this space is isomor-

phic to π. We call the resulting representation a global Bessel model of type (S, θ,Λ) for π.

Local Bessel models. Let π be an irreducible cuspidal representation of G(A) with

trivial central character. Fix an isomorphism π ' ⊗′vπv, where the πv are irreducible,

unitary, admissible representations of G(Qv). Let Ω be a finite set of places, containing

∞, such that if p /∈ Ω then πp is a spherical principal series representation. We now

describe the local Bessel function on G(Qp) associated to πp for p /∈ Ω. From the character

100

data Λ, θ for the global Bessel model we have induced characters Λp, θp of T (Qp), U(Qp)

respectively. Let B be the space of locally constant functions ϕ on G(Qp) such that

ϕ(tug) = Λp(t)θp(u)ϕ(g), for t ∈ T (Qp), u ∈ U(Qp), g ∈ G(Qp).

From the results of [49] we know that there is a unique subspace B(πp) of B such that

the right regular action of G(Qp) on B(πp) is isomorphic to πp. Let Bp be the unique

G(Zp)-fixed vector in B(πp) such that Bp(14) = 1. As explained in [20], Bp is completely

determined by the values Bp(hp(l,m)) where

hp(l,m) = diag(pl+2m, pl+m, 1, pm) (5.13)

for l,m ≥ 0. The following theorem of Sugano gives a formula for these values:

Theorem 5.6 (Sugano, [67] p544; see also [20] (3.6)). Let πp be a spherical principal

series representation of G(Qp) with associated parameters (a, b) = (σ(p), σ(p)χ1(p)) as

described in §5.2. Let Bp be the normalized spherical vector in the local Bessel model. Let

l,m ≥ 0 be integers, and hp(l,m) ∈ G(Qp) be defined by (5.13). Then

Bp(hp(l,m)) = p−2m− 3l2 U l,m

p (a, b),

for U l,mp given by the coefficients of the power series in [20] (3.6). The set of functions

U l,mp ; l,m ≥ 0 linearly generate a dense subspace of the space C(Yp) of continuous

functions on Yp.

The point of Theorem 5.6 is, of course, that we have an explicit formula for Bp(hp(l,m)).

The formula is fairly involved (for an exposition in a situation similar to our own, see

[20] (3.6)) and so we do not recall it here. For the proof of our local equidistribution

statement we only require two properties, namely the (already stated) fact that that U l,mp

generate (a dense subspace of) C(YS) (see [39] Proposition 2.7), and our Proposition 5.15

(for which we will refer to [24] or [39]). For our application to low-lying zeros we will also

use the formulas for the first few U l,mp , given as follows: as in the definition of µS write

λp =∑

N(p)=p Λ(p) where Λ is our fixed character of Cld and p is prime in Q(√−d). Let(

d·

)be the character of the extension Q(

√−d)/Q, which takes the value 1, 0,−1 on a

rational prime p according to whether p is split, ramified, or inert in Q(√−d). Set

σ(a, b) = a+ b+ a−1 + b−1,

τ(a, b) = 1 + ab+ a−1b+ ab−1 + a−1b−1.

101

Then

U0,0p (a, b) = 1,

U1,0p (a, b) = σ(a, b)− p−1/2λp,

U2,0p (a, b) = a2 + b2 + a−2 + b−2 + 2τ(a, b) + 2− p−1/2λpσ(a, b) + p−1

(d

p

),

U0,1p (a, b) = τ(a, b)−

(p−

(d

p

))−1(p1/2λpσ(a, b)−

(d

p

)(τ(a, b)− 1)− λ2

p

).

(5.14)

Recall that π '⊗′

v πv is an irreducible cuspidal representation with trivial central char-

acter. Suppose further that Φ = ⊗vΦv is a pure tensor in Vπ. Let Ω be as above, and for

g = (gv) ∈ G(A) let gΩ =∏

v∈Ω gv. Then by uniqueness of local Bessel models

BΦ(g) = BΦ(gΩ)∏p/∈Ω

Bp(gp). (5.15)

Note that (5.15) makes sense even if both sides are zero.

Bessel models for Siegel cusp forms. We now consider the implications of (5.15) for

the class of Siegel modular forms we are interested in. Let S be a finite set of primes

and let F ∈ Sk(N) where gcd(N,S) = 1. Assume that F is an eigenform for the local

Hecke algebras at p ∈ S. Recall the representation πF attached to F decomposes1 as

πF =⊕m

i=1 π(i)F , where each π

(i)F is an irreducible cuspidal representation of G(A). Thus

each vector Φ ∈ πF is a sum of vectors Φi in the irreducible cuspidal representations π(i)F .

Also, for each 1 ≤ i ≤ m, by the tensor product theorem, we have π(i)F ' ⊗′vπ

(i)F,v where

the π(i)F,v are irreducible, unitary, admissible representations of G(Qv). Thus each vector

Φi ∈ π(i) is in turn a sum of pure tensors ⊗vΦ(j)i,v ∈ ⊗′vπ

(i)F,v. In particular, suppressing the

subscript i, we can write

ΦF =n∑j=1

⊗vΦ(j)F,v (5.16)

where each ⊗vΦ(j)F,v is a pure tensor in some irreducible cuspidal representation π

(i)F with

1 ≤ i ≤ m.

Let S, θ,Λ be given. For the representation π = πF we can define, for any vector Φ ∈ Vπ,

the Bessel functional BΦ by (5.12). We ease notation by temporarily writing BΦ(·) =

1Still not necessarily uniquely, and this is still not a problem.

102

B(·; Φ). From the definition and (5.16) it is clear that

B(·; ΦF ) =n∑j=1

B(·; ⊗Φ(j)F,v). (5.17)

Fix some 1 ≤ j ≤ n and consider B(·; ⊗vΦ(j)F,v). Let Ω = ∞ ∪ p | N. All of the

local components π(i)F,p at p /∈ Ω are spherical principal series so Ω satisfies the hypotheses

necessary for (5.15). Thus we have, for any g ∈ G(A),

B(g; ⊗vΦ(j)F,v) = B(gΩ; ⊗vΦ(j)

F,v)∏p/∈Ω

B(i)p (gp), (5.18)

where B(i)p is the spherical vector in the Bessel model for the spherical principal series

representation π(i)F,p (recall ⊗vΦ(j)

v ∈ ⊗′vπ(i)F,v ' π

(i)F ). As i varies, the local representations

π(i)F,p for p ∈ S lie in the same isomorphism class. In particular, as i varies, the associated

Bessel models to π(i)F,p is the same space of functions on G(Qp), and each B

(i)p is the same

vector Bp. So (5.18) becomes

B(g; ⊗vΦ(j)F,v) = B(gΩ; ⊗vΦ(j)

F,v)∏p∈S

Bp(gp)∏

p/∈(Ω∪S)

B(i)p (gp), (5.19)

and putting these in to (5.17) we obtain

B(g; ΦF ) =∏p∈S

Bp(gp)

n∑j=1

B(gΩ; ⊗Φ(j)F,v)

∏p/∈(Ω∪S)

B(i)p (gp)

(5.20)

where i = i(j) is such that ⊗vΦ(j)v ∈ ⊗′vπ

(i)F,v. In particular, if g has the form

gv =

14 v /∈ S

gp v ∈ S

then, by our normalisation of the B(i)p , (5.20) reads

B(g; ΦF ) =∏p∈S

Bp(gp)

(n∑j=1

B(14; ⊗Φ(j)F,v)

). (5.21)

We will use (5.21) by explicitly computing the left hand side for certain g ∈ G(A). Namely,

let L,M be integers with all their prime factors in S, and define H(L,M) ∈ G(A) by

H(L,M)v =

diag(LM2, LM, 1,M) v ∈ S,

14 v /∈ S.

103

In particular, H(1, 1) = 14. The first step is to reduce the computation of B(H(L,M); ΦF )

to the computation for H(1, 1) with a possibly different modular form:

Lemma 5.7. Let S be a finite set of primes, N be a positive integer with gcd(N,S) = 1,

and L,M positive integers with all their prime factors in S. Let F ∈ Sk(N). Then there

exists F ′ ∈ Sk such that

B(H(L,M); ΦF ) = B(H(1, 1); ΦF ′)

Proof. Define ΦL,MF (g) = ΦF (gH(L,M)). Then clearlyB(H(L,M); ΦF ) = B(H(1, 1); ΦL,M

F ).

Now let H∞ = diag(LM2, LM, 1,M) ∈ G+(R) and define

F ′(Z) = (LM)−kF (H−1∞ 〈Z〉).

One easily checks that F ′(γ〈Z〉) = j(γ, Z)kF (Z) for

γ ∈ H∞Γ0(N)H−1∞ =

∗ M∗ LM2∗ LM∗

M−1∗ ∗ LM∗ L∗

L−1M−2N∗ L−1M−1N∗ ∗ M−1∗

L−1M−1N∗ L−1N∗ M∗ ∗

∈ Sp4(Q); ∗ ∈ Z

.

(5.22)

This contains Γ(NLM2) as a subgroup of finite index, so is a congruence subgroup and

F ′ ∈ Sk. Recall the choice of open compact subgroups (5.1). The adelization ΦF ′ of F ′ is

left invariant under G(Q) and right invariant under∏

p<∞KNLM2

p .

We claim that ΦF ′ = ΦL,MF . Now one easily checks that ΦL,M

F is also left-invariant under

G(Q) and right-invariant under∏

p<∞KNLM2

p , so it suffices to show that ΦF ′ and ΦL,MF

agree as functions on G+(R). For g∞ ∈ G+(R)

ΦL,MF (g∞) = ΦF (g∞H(L,M)) = ΦF

((L−1M−2

L−1M−1

1M−1

)g∞H(L,M)

)(5.23)

where the first equality is the definition and the second follows from left-invariance of ΦF

under G(Q). Similarly using the right-invariance by∏

p<∞KNLM2

p we can right-multiply

the variable by the adele which is diag(LM2, LM, 1,M) when v /∈ S ∪ ∞ and is 14

otherwise (note that we are using the restriction on the prime factors of L and M here)

to obtain

ΦL,MF (g∞) = ΦF (H−1

∞ g∞).

104

Then from this we then simply compute

ΦL,MF (g∞) = µ2(H−1

∞ g∞)j(H∞g∞, i12)−kF (H−1∞ g∞〈i12〉)

= (LM)−kµ2(g∞)j(g∞, i12)−kF (H−1∞ g∞〈i12〉)

= ΦF ′(g∞).

Lemma 5.7 reduces the computation ofB(H(L,M); ΦF ) to the computation ofB(H(1, 1); ΦF ′),

which is precisely the approach taken in [39] (although note the slight change in our def-

inition of H(L,M)). In order to quote the result of the latter computation, we introduce

their notation. Given M , define

Cld(M) = T (A)/T (Q)T (R)∏p<∞

(T (Qp) ∩K(0)p (M)),

where K(0)p (M) = g ∈ GL2(Zp); g ≡ ( ∗ 0

∗ ∗ ) mod M. Cld(1) = Cld is isomorphic to the

ideal class group of Q(√−d); in general Cld(M) can be interpreted as a ray class group.

Pick coset representatives tc ∈ T (A) (indexed by c ∈ Cld(M)) for this quotient, and write

(by strong approximation for T )

tc = γcmcκc

with γc ∈ GL2(Q), mc ∈ GL+2 (R), κc ∈

∏p<∞K

(0)p (M). Let

Sc :=1

det(γc)tγcSγc, (5.24)

where S is the matrix for our choice of Bessel model. We also define, for any symmetric

matrix Q, the matrix

QL,M :=

LL

M1

Q

M1

. (5.25)

Proposition 5.8. Suppose we have the same hypotheses as Lemma 5.7. Then

B(H(L,M); ΦF ) =re−2π tr(S)(LM)−k

|Cld(M)|∑

c∈Cld(M)

Λ(c)a(SL,Mc ;F )

where r is a nonzero constant depending only on the normalization of Haar measure on

the Bessel subgroup R.

105

Proof. Lemma 5.7 reduces this to the case in [39] Proposition 2.1. Note that this compu-

tation uses the fact that ΦF ′ is right invariant under

g ∈ GL2(Zp); g ≡ ( ∗ 0∗ ∗ ) mod M ,

embedded as a subgroup of G(Zp) via (5.11). That this still holds in our case is clear

from (5.22).

Let L,M be integers with all their prime factors in S and H(L,M) be as above. By (5.21)

we have

B(H(L,M); ΦF ) =∏p∈S

Bp(hp(lp,mp))

(n∑j=1

B(14;⊗vΦ(j)F,v)

)

where lp = ordp(L), mp = ordp(M) and hp(lp,mp) = diag(plp+2mp , plp+mp , 1, pmp). Also

from (5.21) we have

B(H(1, 1); ΦF ) =

(n∑j=1

B(14; ⊗vΦ(j)F,v)

),

so

B(H(L,M); ΦF ) = B(H(1, 1); ΦF )∏p∈S

Bp(hp(lp,mp)).

Using Proposition 5.8 twice we obtain

(LM)−k

|Cld(M)|∑

c∈Cld(M)

Λ(c)a(SL,Mc ;F ) =∏p|LM

Bp(hp(lp,mp))×1

|Cld|∑c∈Cld

Λ(c)a(Sc;F ), (5.26)

and hence using Theorem 5.6

|Cld||Cld(M)|

∑c∈Cld(M)

Λ(c)a(SL,Mc ;F )

= Lk−32Mk−2

∑c∈Cld

Λ(c)a(Sc;F )∏p|LM

U lp,mpp (ap(F ), bp(F )).

(5.27)

Equation (5.27) is crucial to our argument. It allows us to reduce the study of certain

continuous functions Ulp,mpp on the space Xp ⊂ Yp at the parameters corresponding to F

to the study of certain sums of the Fourier coefficients of F . In the next section we will

prove a result that allows us to do the latter.

106

5.4 Estimates for sums of Fourier coefficients of cusp

forms

They key to estimating (5.27) is the following proposition:

Proposition 5.9. Let k ≥ 6 be even, N ≥ 1, and let Sk(N)∗ be any orthogonal basis of

Sk(N). Let d < 0 be a fundamental discriminant, L and M positive integers. Recall the

definition of Cld(M); for c′ ∈ Cld(M) and c ∈ Cld recall also the matrices Sc′ and SL,Mc

defined by (5.24) and (5.25). Then

2√π(4π)3−2k

vol(Γ0(N)\H2)Γ

(k − 3

2

)Γ(k − 2)

(d

4

)−k+ 32 ∑F∈Sk(N)∗

a(Sc′ ;F )a(SL,Mc ;F )

〈F, F 〉

= δ(c, c′, L,M) + E(N, k; c, c′, L,M),

where

δ(c, c′, L,M) = #U ∈ GL2(Z);USc′tU = SL,Mc

(which may equal zero), and the error term satisfies

E(N, k; c, c′, L,M)ε N−1k−

23 (LM)k−

12

+ε.

We will prove this using estimates for Fourier coefficients of Poincare series. Given Q ∈

Q2×2sym positive definite and semi-integral and a positive even integer k, we define the

associated Poincare series of weight k and level N by

GQ,N,k(Z) =∑

M∈∆\Γ0(N)

j(M,Z)−ke(tr(Q ·M〈Z〉)), (5.28)

where ∆ =(

12 U02 12

)∈ Sp4(Z)

. This series converges uniformly and absolutely on com-

pact subsets of H2 provided k ≥ 6.

The following property of Poincare series is well-known, and can be found for the case

N = 1 in for example [34]. We include the argument for any level N here since the value

of the constant of proportionality will be important for our application.

107

Lemma 5.10. Let Q ∈ Q2×2sym be positive definite symmetric, k ≥ 6 be even, and N be a

positive integer. Let GQ,N,k be defined by (5.28), and let F =∑

T>0 a(T ;F )e(tr(TZ)) ∈

S(2)k (N). Then

〈GQ,N,k, F 〉 =2

vol(Γ0(N)\H2)

√π(4π)3−2kΓ

(k − 3

2

)Γ(k − 2) det(Q)−k+ 3

2a(Q;F ),

where 〈, 〉 is the Petersson inner product defined by (2.8).

Proof. Proceeding formally we have

〈GQ,N,k, F 〉 =1

vol(Γ0(N)\H2)

∫Γ0(N)\H2

GQ,N,k(Z)F (Z) det(Y )kdXdY

det(Y )3

=1

vol(Γ0(N)\H2)

∫Γ0(N)\H2

∑M∈∆\Γ0(N)

j(M,Z)−ke(tr(Q ·M〈Z〉))

× F (Z) det(Y )kdXdY

det(Y )3

where Z = X + iY . Now for M ∈ Γ0(N) we have det(Im(Z))kF (Z)j(M,Z)−k =

F (MZ) det(Im(MZ))k, so we can write

〈GQ,N,k, F 〉 =1

vol(Γ0(N)\H2)

∫Γ0(N)\H2

∑M∈∆\Γ0(N)

e(tr(Q ·M〈Z〉))

× F (MZ) det(Im(MZ))kdXdY

det(Y )3

=2

vol(Γ0(N)\H2)

∫∆\H2

e(tr(QZ))F (Z) det(Y )kdXdY

det(Y )3

=2

vol(Γ0(N)\H2)

∫Y >0

∫X mod 1

e(tr(QZ))F (Z) det(Y )kdXdY

det(Y )3.

Here the integral with respect to X is over Rn×nsym with entries taken modulo 1, and the

integral with respect to Y is over all positive definite matrices in R2×2sym. The factor of 2

appears in the second line because −14 /∈ ∆ but it acts trivially on H2. Substituting in

the Fourier expansion F (Z) =∑

T>0 a(T ;F )e(tr(TZ)) and integrating with respect to X

we see that only the T = Q term survives, giving

〈GQ,N,k, F 〉 =2a(Q;F )

vol(Γ0(N)\H2)

∫Y >0

e−4π tr(QY ) det(Y )k−3dY.

It remains to compute this integral. However, this is well-known (or easily computable

by induction), for example by ([42], (40)) we have∫Y >0

e−4π tr(QY ) det(Y )k−3dY =√π(4π)3−2kΓ

(k − 3

2

)Γ(k − 2) det(Q)−k+ 3

2 .

108

An immediate corollary of Lemma 5.10 is that the GQ,N,k generate Sk(N) as Q varies.

Thus one can obtain results on the growth of Fourier coefficients of Siegel cusp forms

by studying the growth rate Fourier coefficients of Poincare series. Such studies were

initiated in [33], who considered the dependency on det(T ) only. In [39] a saving with

respect to the weight k was obtained, and similarly in [10] for the level N . We need a

version which saves with both k and N . Our estimations are based on those of [39] and

in fact obtain a better decay than the N−1/2 of [10]. We shall prove:

Theorem 5.11. Let k ≥ 6 be even and let N be a positive integer. Let Q ∈ Q2×2 be

positive definite and semi-integral. Then for any positive definite semi-integral matrix T

a(T ;GQ,N,k) = δ(T,Q) + E(N, k, T ), (5.29)

where

δ(T,Q) = #U ∈ GL2(Z);UQtU = T

(which may equal zero), and the error term satisfies

E(N, k, T )ε,Q N−1k−

23 det(T )k/2−1/4+ε.

It is easy to prove Proposition 5.9 from Theorem 5.11:

Proof of Proposition 5.9. Since Sk(N)∗ is an orthogonal basis

GQ,N,k =∑

F∈Sk(N)∗

〈GQ,N,k, F 〉〈F, F 〉

F

and hence for any positive definite semi-integral T ∈ Q2×2sym

a(T ;GQ,N,k) =∑

F∈Sk(N)∗

〈GQ,N,k, F 〉〈F, F 〉

a(T ;F ). (5.30)

We take c ∈ Cld(M), c′ ∈ Cld, and T = SL,Mc , Q = Sc′ . Using det(Sc′) = d/4 we then have

that the right hand side of (5.30) is

2√π(4π)3−2k

vol(Γ0(N)\H2)Γ

(k − 3

2

)Γ(k − 2)

(d

4

)−k+ 32 ∑F∈Sk(N)∗

a(Sc′ ;F )a(SL,Mc ;F )

〈F, F 〉.

But the left hand side of (5.30) is estimated by Theorem 5.11, with error term

E(k,N, SL,Mc )ε N−1k−2/3 det(SL,Mc )

k2− 1

4+ε.

109

One easily computes that |T | = L2M2d/4, and since d is treated as constant we obtain

the statement of the corollary.

Before embarking on the proof of Theorem 5.11 let us consider whether it is possible

to give a simpler qualitative proof of this “asymptotic orthogonality” of Poincare series

Fourier coefficients. The motivation for this question is the argument in [38], which does

precisely this in the k-aspect (the proofs there and here also works more generally for

Siegel modular forms of degree g not necessarily equal to 2). A sketch of the proof is as

follows: one uses the fact that the Siegel fundamental domain can be characterised by a

finite list of conditions along with

limy→∞|det(Cyi12 +D)| = +∞ (5.31)

(valid for (C,D) the bottom block-rows of any real symplectic matrix, where C 6= 0) to

produce a positive real number y0 and a set Ug(y0) = X + iy012; X ∈ R2×2sym; |xij| ≤ 1

2

such that, for M = ( A BC D ) ∈ Sp4(Z) with C 6= 0 and Z ∈ Ug(y0), we have |j(M,Z)| > 1.

Since |e(tr(Q ·M〈Z〉))| ≤ 1 (for any M,Z), an application of dominated congergence for

series shows that for Z ∈ Ug(y0) the Poincare series

GQ,1,k(Z) =∑

M s.t.C=0

j(M,Z)−ke(tr(Q ·M〈Z〉))

+∑

M s.t. C 6=0

j(M,Z)−ke(tr(Q ·M〈Z〉))

converges to the sub-series defined by the first sum, as k →∞. On the other hand, using

dominated convergence again, one sees that the Fourier coefficients in the limit can be

computed by integrating this limiting sub-series over Ug(y0), and a simple computation

(c.f. Lemma 5.12) therefore gives the qualitative version of Theorem 5.11.

In order to extend this argument to deal with the case N > 1 one can argue as in the

proof of Proposition 2 of [38]. Let Ug(Y0) be as above and write

GQ,N,k(Z) =∑

M s.t.C=0

j(M,Z)−ke(tr(Q ·M〈Z〉))

+∑

M s.t.C 6=0

∆N(C)j(M,Z)−ke(tr(Q ·M〈Z〉))

110

where ∆N(C) = 1 if C ≡ 02 mod N , and is zero otherwise. The limit of each term in

the second series is 0 as N + k → ∞: indeed the large k limit was treated above, and

∆N(C) = 0 for N sufficiently large. Thus by dominated convergence for series we again

reduce to the first sum, and can then argue as above.

The remainder of this section is occupied with the (somewhat technical) proof of Theo-

rem 5.11. We will treat Q as being fixed, and will therefore suppress the dependency of

implied constants on Q. To ease notation we will also write |·| for det(·).

Now let hN be a complete set of representatives for ∆\Γ0(N)/∆. For M ∈ Γ0(N), let

θ(M) =

S ∈ Z2×2sym; M

12 S

02 12

M−1 ∈ ∆

.

Note that Z2×2sym/θ(M) is in bijection with ∆/(∆ ∩M−1∆M); we will identify these. We

then clearly have

∆M∆ =⊔

S∈Z2×2sym/θ(M)

∆M

12 S

02 12

. (5.32)

Define H(M, ·) = HQ,N,k(M, ·) by

H(M,Z) =∑

S∈Z2×2sym/θ(M)

j(M,Z + S)−ke(tr(Q ·M〈Z + S〉)) (5.33)

so that by (5.32)

G(Z) =∑M∈hN

H(M,Z) (5.34)

where G = GQ,N,k. Let h(M,T ) = hQ,N,k(M,T ) be given by h(M,T ) = a(T ;H(M, ·)).

Then by (5.34) we have

a(T ;G) =∑M∈hN

h(M,T ). (5.35)

In order to estimate (5.35) we split the sum over the subsets

h(i)N =

M =

A B

C D

∈ hN ; rk(C) = i

by defining, for i = 0, 1, 2,

Ri =∑M∈h(i)N

h(M,T ). (5.36)

111

In Lemmas 5.12, 5.13, and 5.14 we shall treat the cases i = 0, i = 1, and i = 2 respectively.

Lemma 5.12. In the notation of (5.36),

R0 = #U ∈ GL2(Z); UQtU = T

Proof. Straightfoward computation (see [10] Proposition 3.2).

Lemma 5.13. Let ε > 0. In the notation of (5.36),

|R1| ε N−1k−

56 |T |

k2− 1

4+ε .

Proof. We choose our representatives in h(1)N to be of the form

M =

∗ ∗

U−1 (Nc 00 0 ) tV U−1

(d1 d20 d4

)V −1

where

U ∈

∗ ∗

0 ∗

∈ GL2(Z)

\GL2(Z),

V ∈ GL2(Z)/

1 ∗

0 ∗

∈ GL2(Z)

,

c ≥ 1, d4 = ±1, (Nc, d1) = 1 and d1, d2 vary modulo Nc. For such an M , we have

θ(M) =

S ∈ Z2×2sym; tV SV =

0 0

0 ∗

.

When N = 1 the set of such M are the representatives for h(1)1 used by Kitaoka. It easily

follows that we have a complete set of representatives when N > 1 as well. These are also

the representatives used in [10].1

Consider now a fixed M as above. Let P =(

p1 p2/2p2/2 p4

)= UQtU , S =

(s1 s2/2s2/2 s4

)=

V −1T tV −1, and let a1 be any integer such that a1d1 ≡ 1 mod Nc. By the discussion in

1Note that there is a typo in the statement of Lemma 4.1 of [10]: the conditions on d1, d2 should beas we have stated them (modulo Nc1 in their notation), and following this their a1 should also be aninverse modulo Nc1. However, during the subsequent computations the variables are taken in the correctranges, so this does not affect the results of their computation. In the statement of [10] Lemma 4.2 thelast term in the exponential should have −d4 in place of d4, but this again has no effect.

112

the previous paragraph, we can apply [33] §3 Lemma 1 to our representatives (a subset

of Kitaoka’s), which gives

h(M,T ) = δp4,s4(−1)k2

√2π

(|T ||Q|

) k2− 3

4

s− 1

24 (Nc)−

32Jk− 3

2

(4π

√|T | |Q|Ncs4

)

× e(a1s4d

22 − (a1d4p2 − s2)d2

Nc+a1p1 + d1s1

Nc− d4p2s2

2Ncs4

),

(5.37)

where δ is the Kronecker delta, and J is the ordinary Bessel function. As in [33] and [10]

we sum (5.37) over d2 mod Nc, using the well-known bound on quadratic Gauss sums∑x mod c

e

(ax2 + bx

c

) (a, c)

12 c

12

(see [10] for a proof) for the first term in the exponential sum, and bounding the other

two in absolute value by 1 to get∣∣∣∣∣ ∑d2 mod Nc

h(M,T )

∣∣∣∣∣ δp4,s4

(|T ||Q|

) k2− 3

4

s− 1

24 (s4, Nc)

12 (Nc)−1

∣∣∣∣∣Jk− 32

(4π√|T | |Q|

Ncs4

)∣∣∣∣∣ .Now sum this over d1 mod Nc such that (d1, Nc) = 1, and d4 = ±1. Since the sum over

d1 has length O(Nc) we have

∑d1,d4

∣∣∣∣∣∑d2

h(M,T )

∣∣∣∣∣ δp4,s4

(|T ||Q|

) k2− 3

4

s− 1

24 (s4, Nc)

12

∣∣∣∣∣Jk− 32

(4π√|T | |Q|

Ncs4

)∣∣∣∣∣ .We next sum this over all possible U and V for a fixed c. Let us write U = ( ∗ ∗u3 u4 ). By

definition of our choice of M , the choice of (u3, u4) determines U up to sign. Note also

that, writing u = ( u3u4 ), p4 = Q[u]. So

∑U

∑d1,d4,

∣∣∣∣∣∑d2

h(M,T )

∣∣∣∣∣ r(s4;Q)

(|T ||Q|

) k2− 3

4

s− 1

24 (s4, Nc)

12

∣∣∣∣∣Jk− 32

(4π√|T | |Q|

Ns4c

)∣∣∣∣∣where r(s4;Q) = |( u3u4 ) ∈ Z2×2; (u3, u4) = 1; Q[u] = s4|. Similarly, write V = ( v1 ∗v2 ∗ ).

The choice of (v1, v2) determines V , by the definition of our representatives M . Summing

over all (v1, v2) such that gcd(v1, v2) = 1 we get

∑U,V

∑d1,d4

∣∣∣∣∣∑d2

h(M,T )

∣∣∣∣∣∑m≥1

r(m;T )r(m;Q)

(|T ||Q|

) k2− 3

4

m−12 (m,Nc)

12

∣∣∣∣∣Jk− 32

(4π√|T | |Q|

Nmc

)∣∣∣∣∣ .113

Now it is well known that the number of proper representations of m by a primitive

positive definite quadratic form isη mη, for any η > 0. Applying this with η/2 we have∑

U

∑d1,d4,

∣∣∣∣∣∑d2

h(M,T )

∣∣∣∣∣η

∑m≥1

(|T ||Q|

) k2− 3

4

m−12

+η(m,Nc)12

∣∣∣∣∣Jk− 32

(4π√|T | |Q|

Nmc

)∣∣∣∣∣ .Finally we sum over c ≥ 1 to get, for any η > 0,

|R1| η

(|T ||Q|

) k2− 3

4 ∑c,m≥1

m−12

+η(m,Nc)12

∣∣∣∣∣Jk− 32

(4π√|T | |Q|

Nmc

)∣∣∣∣∣ . (5.38)

As in [39] we split the sum on the right hand side of (5.38) up in to R11 +R12 +R13, but

where R1i now corresponds to

4π√|T ||Q|N

≤ mc if i = 1,

4π√|T ||Q|√kN

≤ mc ≤ 4π√|T ||Q|N

if i = 2,

mc ≤ 4π√|T ||Q|√kN

if i = 3.

So by definition we have

|R1| η

(|T ||Q|

) k2− 3

4

(R11 +R12 +R13), (5.39)

and we proceed to estimate each R1i individually.

Case R11: We are estimating

R11 =∑c,m≥1

mc≥ 4π√|T ||Q|N

m−12

+η(m,Nc)12

∣∣∣∣∣Jk− 32

(4π√|T | |Q|

Nmc

)∣∣∣∣∣ .In this range we use the estimate

Jk(x) xk

Γ(k), if k ≥ 1, 0 ≤ x

√k + 1, (5.40)

(i.e. [39] (3.1.3)), to get

Jk− 32

(4π√|T | |Q|

Nmc

) 1

Γ(k − 32)

(4π√|T | |Q|

Nmc

)k− 32

.

Substituting this in to R11 gives

R11 1

Γ(k − 3

2

) ∑m,c≥1

mc≥ 4π√|T ||Q|N

m−12

+η(m,Nc)12

(4π√|T | |Q|

Nmc

)k− 32

.

114

Since (4π√|T | |Q|)/(Nmc) ≤ 1 and k ≥ 6 we can replace the exponent k− 3

2with 1 + δ,

where 0 < δ ≤ 1. Doing this, and putting π and d in to the implied constant, we get

R11 N−1−δ |T |

12

+ δ2

Γ(k − 32)

∑m,c≥1

mc≥ 4π√|T ||Q|N

m−12

+η(m,Nc)12

(1

mc

)1+δ

Taking δ = 2η (assuming η is sufficiently small) and using (m,Nc)12 ≤ m

12 in the double

sum gives ∑m,c≥1

mc≥ 4π√|T ||Q|N

m−12

+η(m,Nc)12

(1

mc

)1+2η

≤∑m,c≥1

mc≥ 4π√|T ||Q|N

m−1−ηc−1−2η

which is manifestly convergent. Thus we have

R11 N−1−2η |T |

12

+η

Γ(k − 3

2

) .

Since the gamma function grows superexponentially we have Γ(k − 32) kE for any

E ≥ 1, so for any such E

R11 η N−1k−E |T |

12

+η . (5.41)

Case R12: We are now estimating

R12 =∑m,c≥1

4π√|T ||Q|

N√k≤mc≤ 4π

√|T ||Q|N

m−12

+η(m,Nc)12

∣∣∣∣∣Jk− 32

(4π√|T | |Q|

Nmc

)∣∣∣∣∣ . (5.42)

In this range we can still use the estimate (5.40). This, together with4π√|T ||Q|

Nmc≤√k (in

this range), gives

Jk− 32

(4π√|T | |Q|

Nmc

) 1

Γ(k − 32)

(4π√|T | |Q|

Nmc

)k− 32

kk2− 3

4

Γ(k − 32).

Substituting this in to (5.42) we have

R12 kk2− 3

4

Γ(k − 32)

∑m,c≥1

4π√|T ||Q|

N√k≤mc≤ 4π

√|T ||Q|N

m−12

+η(m,Nc)12 . (5.43)

115

Now we can easily see that, for any δ > 0,

∑m,c≥1mc≤X

m−12

+η(m,Nc)12 =

∑r≤X

∑e|r

e−12

+η

(e,Nr

e

) 12

δ X1+η+δ. (5.44)

Taking X =4π√|T ||Q|N

we can bound the sum in (5.43), with δ = η this gives

R12 ηkk2− 3

4

Γ(k − 32)

(√|T |N

)1+2η

.

Using Stirling’s formula we see that, for any E ≥ 1, kk2−

34

Γ(k− 32

) kE, so for any E ≥ 1, η > 0

we have

R12 η N−1k−E |T |

12

+η . (5.45)

Case R13: Finally we consider

R13 =∑m,c≥1

mc≤ 4π√|T ||Q|

N√k

m−12

+η(m,Nc)12

∣∣∣∣∣Jk− 32

(4π√|T | |Q|

Nmc

)∣∣∣∣∣ . (5.46)

Here we use

Jk(x) min(1, xk−1)k−13 , if k ≥ 1, x ≥ 1 (5.47)

(i.e. [39] (3.1.4)). By definition of this range4π√|T ||Q|

Nmc≥√k ≥ 1, so (5.47) is applicable

and gives

Jk− 32

(4π√|T | |Q|

Nmc

)(k − 3

2

)− 13

k−13 .

Substituting this in to (5.46) we get

R13 k−13

∑c,m≥1

cm≤ 4π√|T ||Q|

N√k

m−12

+η(m,Nc)12

Using (5.44) with X =4π√|T ||Q|

N√k

and δ = η gives

R13 η k− 1

3

(√|T |

N√k

)1+2η

η N−1k−

56 |T |

12

+η .

(5.48)

116

Combining (5.41), (5.45), and (5.48) in (5.39) we have

R1 η

(|T ||Q|

) k2− 3

4 (N−1k−E |T |

12

+η +N−1k−E |T |12

+η +N−1k−56 |T |

12

+η)

η N−1k−

56 |T |

k2− 1

4+η .

Taking η = ε this is precisely the statement of the lemma.

Before proceeding let us remark that it is in the proof of this lemma that we obtain the

improvement on [10]. The relevant quantities to compare are our estimates of R11, R12,

R13 and [10] Lemma 4.4 (which is the result used for estimates which are |T |k/2−1/4).

The bottleneck in their estimate is [10](4.9), corresponding to our (R12 and) R13. For

small x ([10](4.10)) they use the same estimate for the Bessel function as we do and one

can check that their exponent on N can be made to improve by assuming larger k as

we have. However, they estimate Jk(x) for large x ([10](4.9)) by x−1/2 which ultimately

introduces a factor of N1/2; we estimate Jk(x) for large x by k−1/3 which avoids this, as

well as giving the saving we require with respect to k.

Finally we estimate the remaining term R2:

Lemma 5.14. Let ε > 0. In the notation of (5.36),

|R2| ε N−2k−

23 |T |

k2− 1

4+ε .

Proof. We choose

h(2)N =

M =

∗ ∗

NC D

∈ Sp4(Z); |C| 6= 0; D mod NC

,

for such M we have θ(M) = 0. When N = 1 our h(2)1 is the set of representative of [33]

§2 Lemma 5. Again it easily follows that we have a complete set of representatives when

N > 1 as well, and also that Kitaoka’s computations are applicable. Note that these are

once again the same as the representatives used in [10]. We can then write

R2 =∑

C∈Z2×2

|C|6=0

∑D mod NC

h(M,T ) (5.49)

117

with M = ( ∗ ∗NC D ) as above. Fix a matrix C and consider the sum over all M = ( ∗ ∗

NC D ) ∈

h(2)N . Using the arguments of [33] §4 following Lemma 1 up to the second equation on p166,

we obtain the following: let

• P (NC) := T t(NC)−1Q(NC)−1,

• ||NC|| be the absolute value of |NC| = N2 det (C),

• K(Q, T ;NC) be the matrix Kloosterman sum defined (and bounded) by Kitaoka

([33], §1)

• 0 < s1 ≤ s2 be such that s21, s

22 are the eigenvalues of the positive definite matrix

P (NC), and write

Jk(P (NC)) =

∫ π/2

0

Jk− 32(4πs1 sin(θ))Jk− 3

2(4πs2 sin(θ)) sin(θ)dθ.

Then ∑D mod NC

h(M,T ) =1

2π4

(|T ||Q|

) k2− 3

4

||NC||−32K(Q, T ;NC)Jk(P (NC)).

Using principal divisors we can write NC ∈ Z2×2 with |C| 6= 0 uniquely (see [33] §4

Lemma 1) as

NC = U−1

Nc1 0

0 Nc2

V −1

where 1 ≤ c1, c1 | c2, U ∈ GL2(Z) and V ∈ SL2(Z)/Γ0(c2/c1). Here

Γ0(m) = ( a bc d ) ∈ SL2(Z); b ≡ 0 mod m .

We will thus consider our matrix NC to be parameterised by (U,Nc1, Nc2, V ). To handle

the sum over the NC, first suppose that (Nc1, Nc2, V ) is fixed. Pick U1 ∈ GL2(Z) such

that

A = A(Nc1, Nc2, V ) := t(V(Nc1 0

0 Nc2

)−1U1

)T(V(Nc1 0

0 Nc2

)−1U1

)(5.50)

is Minkowski-reduced. Clearly we have that the matricesNC with parameters (Nc1, Nc2, V )

are precisely the matrices

NC = U−1U−11

Nc1 0

0 Nc2

V −1

118

as U varies over GL2(Z). Hence we can write, for any NC with parameters (Nc1, Nc2, V ),

P (NC) = T t(NC)−1Q(NC)−1

= T t(V(Nc1 0

0 Nc2

)−1U1U

)Q(V(Nc1 0

0 Nc2

)−1U1U

) (5.51)

From (5.50) and (5.51) we immediately see |P (NC)| = |Q| |A|. On the other hand,

|P (NC)| = s21s

22, by definition of s1, s2. Now A, being positive definite symmetric, is

diagonizable, say to

H = H(Nc1, Nc2, V ) :=

a 0

0 c

,

where 0 < a ≤ c. Hence we have, recalling that |Q| is treated constant,

|H| = ac s21s

22 = |P (NC)| . (5.52)

By computing the determinant in (5.50) we have

s21s

22

|T |N4c2

1c22

. (5.53)

Since A is Minkowski-reduced, we also have

tr(P (NC)) tr(A[U ]) = tr(H[U ]). (5.54)

Continuing to work with any NC having parameters (Nc1, Nc2, V ), [33] §1 Prop. 1 gives

us

K(Q, T ;NC)ε N52

+εc21c

12

+ε

2 (Nc2,tvTv)

12

for any ε > 0, where v is the second column of V . Thus

∑D mod NC

h(M,T )ε

(|T ||Q|

) k2− 3

4

N−12

+εc121 c−1+ε2 (Nc2,

tvTv)12 |Jk(P (NC))| . (5.55)

We handle the different NC according to the properties of P (NC) by partitioning in to

the following sets:

C1 =NC ∈ Z2×2; |C| 6= 0; tr(P (NC)) < 1

,

C2 =NC ∈ Z2×2; |C| 6= 0; tr(P (NC)) ≥ max(2 |P (NC)| , 1)

,

C3 =NC ∈ Z2×2; |C| 6= 0; 1 ≤ tr(P (NC)) < 2 |P (NC)|

.

119

Recall we had 0 < s1 ≤ s2 such that s21, s

22 were the eigenvalues of P (NC). So tr(P (NC)) =

s21 + s2

2 and |P (NC))| = s21s

22. For C1, tr(P (NC)) < 1 implies s2

1, s22 ≤ 1. For C2,

s21 + s2

2 ≥ 2s21s

22 and s2

2 ≥ s21 imply s2

1 ≤ 1; in addition s21 + s2

2 ≥ 1 then gives s22 ≥

max(1− s21, s

21) ≥ 1/2. For C3, we have 2s2

1s22 ≥ s2

1 + s22 which, together with the AM-GM

inequality s21 + s2

2 ≥ 2s1s2 gives s1s2 ≥ 1, so s2 ≥ 1; and 2s21s

22 ≥ s2

1 + s22 also gives

s21 ≥ s2

2/(2s22 − 1), hence s2

1 ≥ 1/2. It then follows that

C1 ⊂NC ∈ Z2×2; 0 < s1 ≤ s2 ≤ 1

,

C2 ⊂NC ∈ Z2×2; 0 < s1 ≤ 1; s2 ≥ 1/

√2,

C3 ⊂NC ∈ Z2×2; s1 ≥ 1/

√2; s2 ≥ 1

.

This characterization of the Ci based on the values of the si will be important in the

following case analysis. For now we also define

Ci(Nc1, Nc2, V ) = NC ∈ Ci; NC has final three parameters (Nc1, Nc2, V ),

so⋃

(Nc1,Nc2,V ) Ci(Nc1, Nc2, V ) = Ci. We recall the (weighted) sizes of these sets as proved

in [33] §4 Lemma 2 and stated in Lemma 3.4 of [39]: for any ε, δ > 0,

|C1(Nc1, Nc2, V )| ε (ac)−12−ε (5.56)

∑NC∈C2(Nc1,Nc2,V )

|A|1+δ tr(tUAU)−54−δ δ,ε

(ac)12

+δ−ε if ac < 1

(ac)14

+ε if ac ≥ 1

(5.57)

|C3(Nc1, Nc2, V )| ε (ac)12

+ε (5.58)

Note that again our (Nc1, Nc2, V ) are simply a subset of the (c1, c2, V ) considered in [39].

Finally, write

R2 = R21 +R22 +R33,

where

R2i =∑NC∈Ci

∑D mod NC

h(M,T ).

Then by (5.55) we have

R2i ε

(|T ||Q|

) k2− 3

4

N−12

+ε∑

(Nc1,Nc2,V )

c121 c−1+ε2 (Nc2,

tvTv)12R2i(Nc1, Nc2, V ) (5.59)

120

where

R2i(Nc1, Nc2, V ) =∑

NC∈Ci(Nc1,Nc2,V )

|Jk(P (NC))| .

We will again bound each of these terms individually.

Case R21: Here we have s1, s2 ≤ 1. Using the esimate (5.40) we have

|Jk(P (NC))| 1

Γ(k − 1

2

)2 (4πs1)k−32 (4πs2)k−

32

(s1s2)2+2δ

kE,

where the final line holds for any reasonably small δ > 0, E ≥ 1, by using the fact that

k ≥ 6 and the superexponential growth of the gamma function. Also, by (5.56),

|C1(Nc1, Nc2, V )| δ (ac)−12− δ

2 (s1s2)−1−δ.

So

R21(Nc1, Nc2, V )δ(s1s2)1+δ

kE |T |

12

+ δ2

kE(N2c1c2)1+δ,

using (5.53). Thus, with c1, c2 fixed,∑V

c121 c−1+ε2 (Nc2,

tvTv)12R21(Nc1, Nc2, V )

δ|T |

12

+ δ2

kEN2+2δ

∑V

c− 1

2−δ

1 c−2−δ+ε2 (Nc2,

tvTv)12 .

(5.60)

By [33] §1 Proposition 2 with n = c2/c1 we have, for any η > 0,∑V

(c2

c1

, tvTv

) 12

η

(c2

c1

)1+η (cont(T ),

c2

c1

) 12

where, writing T =(

t1 t2/2t2/2 t3

), cont(T ) = gcd(t1, t2, t3). Using (Nc2,

tvTv)12 ≤ N

12 c

121

(c2c1, tvTv

) 12

in (5.60) then gives∑V

c121 c−1+ε2 (Nc2,

tvTv)12R21(Nc1, Nc2, V )

δ,η|T |

12

+ δ2

kEN−

32−2δc−1−δ−η

1 c−1−δ+ε+η2

(c2

c1

, cont(T )

) 12

.

Substituting this in to (5.59), and writing c2 = nc1,

R21 δ,η,ε

(|T ||Q|

) k2− 3

4

|T |12

+ δ2 N−2−2δk−E

∑c1,n≥1

c−2−2δ+ε1 n−1−δ+ε+η (n, cont(T ))

12 .

121

Take η = ε, δ = 3ε. Clearly the sum over c1 is convergent. For the sum over n we note that∑n≥1 n

−1(n, cont(T ))12 may be written as

∑e|cont(T ) e

− 12

∑m≥1m

−1 ∑

e|cont(T ) e− 1

2 ε

cont(T )ε. Then using the inequality cont(T )2 ≤ 4 det(T ) we see that the sum over n is

thus ε |T |ε. Thus, after redefining ε, we have for any E ≥ 1

R21 ε N−2k−E |T |

k2− 1

4+ε .

Case R22: This is the case s1 ≤ 1, s2 1. Now we have

|Jk(P (NC))| sk− 3

21

Γ(k − 32)

2k

s122

,

where we have used (5.40) to bound the Bessel function involving s1, and the estimate

Jk(x) 2kx−1/2 in the range k ≥ 1, x > 0 (i.e. [39] (3.1.5)) for the one involving s2. Let

NC have parameters (U,Nc1, Nc2, V ), and recall A = A(Nc1, Nc2, V ) defined by (5.50).

We have |A| |P (NC)| = s21s

22, so

|Jk(P (NC))| 2k

Γ(k − 32)

|A|k2− 3

4

sk−12

.

Also, by (5.54), tr(tUAU) tr(P (NC)) = s21 + s2

2 s22, since s1 ≤ 1. So

|Jk(P (NC))| 2k

Γ(k − 32)

|A|k2− 3

4

tr(tUAU)k−12

.

For any δ > 0 we may write

|A|k2− 3

4 tr(tUAU)1−k2 = |A|1+δ tr(tUAU)

54−δ(

|A|tr(tUAU)

) k2− 7

4−δ

.

But |A|tr(tUAU)

s21s22

s22= s2

1 ≤ 1. Now k ≥ 6 and we can assume δ is small, so

|Jk(P (NC))| 2k

Γ(k − 32)|A|1+δ tr(tUAU)

k2− 5

4−δ.

Using (5.57) (with ε = δ/2) and the superexponential growth of the gamma function

gives

R22(Nc1, Nc2, V ) k−E ×

(ac)12

+ δ2 if ac < 1,

(ac)14

+ δ2 if ac ≥ 1.

122

for any E ≥ 1. Recalling from (5.52) that ac |T | /(N4c21c

22) we can now write bound

the sum for R22 by

R22 ε

(|T ||Q|

) k2− 3

4

N−12

+εk−E

×

∑c1c2>

√|T |N2

( √|T |

N2c1c2

)1+δ∑V

c121 c−1+ε2 (Nc2,

tvTv)12

+∑

c1c2≤√|T |N2

( √|T |

N2c1c2

) 12

+δ∑V

c121 c−1+ε2 (Nc2,

tvTv)12

.

Now in the second sum the base with exponent 12

+ δ is larger than 1, so we can certainly

increase the exponent to 1 + δ. This then reduces to

R22 ε

(|T ||Q|

) k2− 3

4

N−52−2δ+εk−E

∑c1,c2≥1c1|c2

(√|T |

c1c2

)1+δ∑V

c121 c−1+ε2 (Nc2,

tvTv)12

ε

(|T ||Q|

) k2− 3

4

N−52−2δ+εk−E |T |

12

+ δ2

∑c1,c2,V

c− 1

2−δ

1 c−2−δ+ε2 (Nc2,

tvTv)12

But the sum over c1, c2, V is now exactly the same as the sum appearing in (5.60) (more

precisely summed over c1, c2, as we proceeded to do there). Thence we conclude that this

sum over c1, c2, V is δ N12 |T |

δ2 , so taking δ = ε we obtain

R22 ε |T |k2− 1

4+εN−2k−E

for any E ≥ 1 as before.

Case R23: In this case 1 s1 ≤ s2. Let M1 = θ ∈ [0, 2π); 4πs2 sin θ ≤ 1 (note that if

θ ∈M1 then 4πs1 sin θ ≤ 1 as well), and let M2 = θ ∈ [0, 2π); 4πs1 sin θ ≥ 1 (and note

that if θ ∈M2 then 4πs2 sin θ ≥ 1 as well). Then

|Jk(P (NC))| (∫

M1

+

∫M2

) ∣∣∣Jk− 32(4πs1 sin θ)Jk− 3

2(4πs2 sin θ) sin θ

∣∣∣ dθ.We estimate using (5.40) and (5.47) on M1 and M2 respectively. Since the argument of

the Bessel functions is ≤ 1 on M1, we may replace the exponent k− 32

by δ for any δ > 0.

123

Since the gamma functions grow superexponentially we may replace these by 2−k, giving

|Jk(P (NC))| δ(s1s2)δ

2k+ k−

23 ,

hence

|Jk(P (NC))| δ k− 2

3 (s1s2)δ.

Also, from (5.58) and (5.52), |C3(Nc1, Nc2, V )| ε (ac)12

+ε (s1s2)1+2ε, so taking ε = δ

we have

R23(Nc1, Nc2, V )δ k− 2

3 (s1s2)1+3δ.

Replacing δ by δ/3 and recalling (5.53) gives

R23(Nc1, Nc2, V )δ k− 2

3N−2−2δ|T |12

+ δ2 (c1c2)−1−δ,

hence

R23 δ

(|T ||Q|

) k2− 3

4

|T |12

+ δ2k−

23N−

52−2δ+ε

∑c1,c2,V

c− 1

2−δ

1 c−2−δ+ε2 (Nc2,

tvTv)12 .

The sum over c1, c2, V is once again the sum we dealt with for R21, so again taking δ = ε

we have

R23 ε |T |k2− 1

4+εN−2k−

23 .

Putting these three cases in to (5.49) we obtain the result.

5.5 The main theorem

Fix d,Λ and a finite set of primes S. Recall the definitions of the spaces XS and YS from

(5.6). Recall also the measures dνS,N,k and dµS defined by (5.9) and (5.10) respectively.

Our aim was to prove Theorem 5.5; that is, for any choice of d and Λ, the measure νS,N

converges weak-∗ to the measure µS as k and N vary admissibly.

Proposition 5.15. Let S be a finite set of primes, and let l = (lp)p∈S, m = (mp)p∈S be

tuples of non-negative integers. Define L =∏

p∈S plp, M =

∏p∈S p

mp. Let Sk(N)∗ be an

orthogonal basis of Sk(N) consisting of eigenforms for Hp when p ∈ S. Then∑F∈Sk(N)∗

ωF,N,k∏p∈S

U lp,mpp (ap(F ), bp(F )) = δ(l,m) +Od,ε

(N−1k−

23L1+εM

32

+ε),

124

where

δ(l,m) =

1 if lp = mp = 0 for all p ∈ S,

0 otherwise,

and the functions Ulp,mpp ∈ C(YS) are as in Theorem 5.6.

Proof. Recall the definition of a(d,Λ;F ) given by (5.7). Computing, using this definition

for the first and third line and the crucial formula (5.27) for the second,

|a(d,Λ;F )|2

〈F, F 〉∏p∈S

U lp,mpp (ap(F ), bp(F ))

=a(d,Λ;F )

〈F, F 〉∑c∈Cld

Λ(c)a(Sc;F )∏p∈S

U lp,mpp (ap(F ), bp(F ))

=a(d,Λ;F )

〈F, F 〉L

32−kM2−k |Cld||Cld(M)|

∑c∈Cld(M)

Λ(c)a(SL,Mc ;F )

=L

32−kM2−k |Cld||Cld(M)|

∑c′∈Cld

c∈Cld(M)

Λ(c′)Λ(c)a(Sc′ ;F )a(SL,Mc ;F )

〈F, F 〉.

Including the full weight ωF,N,k given by (5.8) and summing over our basis Sk(N)∗ we

obtain∑F∈Sk(N)∗

ωF,N,k∏p∈S

U lp,mpp (ap(F ), bp(F ))

=|Cld|L

32−kM2−k

|Cld(M)| vol(Γ0(N)\H2)

∑c′∈Cld

c∈Cld(M)

Λ(c′)Λ(c)cd,Λk∑

F∈Sk(N)∗

a(Sc′ ;F )a(SL,Mc ;F )

〈F, F 〉.

Using Corollary 5.9,∑F∈Sk(N)∗

ωF,N,k∏p∈S

U lp,mpp (ap(F ), bp(F ))

=|Cld|M2−kL

32−k

|Cld(M)|dΛ

2w(−d) |Cld|

×∑c′∈Cld

c∈Cld(M)

Λ(c′)Λ(c)[δ(c, c′, L,M) + E(N, k; c, c′, L,M)].

(5.61)

If LM = 1 then the right hand side of (5.61) is

dΛ

2w(−d) |Cld|∑

c,c′∈Cld

Λ(c′)Λ(c)[δ(c, c′, 1, 1) + E(k,N ; c, c′, 1, 1)].

125

Using [39] Lemma 3.7 (note that our δ includes the number of the GL2(Z)-automorphisms

in its definition) we evaluate this as

1 +dΛ

2w(−d) |Cld|∑

c,c′∈Cld

Λ(c′)Λ(c)E(N, k; c, c′, 1, 1) = 1 +Od,ε(N−1k−

23 ).

If LM > 1 then det(SL,Mc ) = det(Sc′)(LM)2 and it is clear that δ(c, c′, L,M) = 0. So

using Corollary 5.9 again the right hand side of (5.61) is simply

|Cld|M2−kL32−k

|Cld(M)|dΛ

2w(−d) |Cld|∑c′∈Cld

c∈Cld(M)

Λ(c′)Λ(c)E(N, k; c, c′, L,M)

= Od,ε(N−1k−

23L1+εM

32

+ε).

Proposition 5.16. Let S be a finite set of primes, and let l = (lp)p∈S, m = (mp)p∈S be

tuples of non-negative integers. Let µS be the measure on YS. Then∫YS

∏p∈S

U lp,mpp (ap, bp)dµS = δ(l,m),

where δ(l,m) is as in Proposition 5.15.

Proof. This is [39] Proposition 4.2.

It is now simple to obtain the quantitative version of our local equidistribution statement:

Proof of Theorem 5.5. By Weyl’s criterion ([28] §21.1) it suffices to show that the claimed

convergence holds for all ϕ in a set of continuous functions whose linear combinations

span C(YS). As (lp)p∈S and (mp)p∈S vary over all tuples of non-negative integers, Ulp,mpp

describes such a family. The result then follows immediately from Propositions 5.15 and

5.16.

Theorem 5.17 (Local equidistribution and independence, quantitative version). Fix any

d and Λ, and finite set of primes S. Let ϕ =∏

p ϕp be a product function on YS such

that ϕp is a Laurent polynomial in (a, b, a−1, b−1) invariant under the action of the Weyl

group generated by (5.5) and of total degree dp as a polynomial in (a+a−1, b+b−1). Write

D =∏

p∈S pdp. Then, for all ε > 0,∑

F∈Sk(N)∗

ωF,k,Nϕ((ap(F ), bp(F ))p∈S) =

∫YS

ϕ dµS +Od,ε(N−1k−

23D1+ε||ϕ||∞),

where ||ϕ||∞ = maxXS |ϕ|.

126

Proof. We may assume (by working with a smaller S if necessary) that each ϕp is non-

constant (i.e. dp ≥ 1). Since the functions Ulp,mpp linearly generate C(Yp)

ϕp =∑

0≤lp≤ep

∑0≤mp≤fp

ϕp(lp,mp)Ulp,mpp ,

where at least one of ep, fp is ≥ 1. Note that by Proposition 5.16∫YS

ϕ dµS =∏p∈S

ϕp(0, 0). (5.62)

Moreover, ∑F∈Sk(N)∗

ωF,N,kϕ((ap(F ), bp(F ))p∈S)

=∏p∈S

∑0≤lp≤ep

0≤mp≤fp

ϕp(lp,mp)∑

F∈Sk(N)∗

ωF,N,kUlp,mpp (ap(F ), bp(F ))

=∏p∈S

ϕp(0, 0) +N−1k−2/3R,

where, using Proposition 5.15, we have the following bounds on R: write Lϕ =∏

p∈S pep ,

Mϕ =∏

p∈S pmp , then

Rε

∑L|Lϕ

∑M |Mϕ

L1+εM3/2+ε∏p∈S

|ϕp(vp(L), vp(M))| .

Comparing with (5.62) it suffices to show from this that R D1+ε||ϕ||∞. This is carried

out in the proof of Theorem 1.6 of [39] and we do not repeat the details.

127

Chapter 6

L-functions, low-lying zeros, and

Bocherer’s conjecture

In this final chapter we discuss L-functions attached to eigenforms in Sk(N) := S(2)k (N).

Although there are many L-functions attached to F will study one of the most classical:

the “spin” L-function L(s, πF ), where πF is (an irreducible constituent of) the cuspidal

automorphic representation generated by F . Using Theorem 5.17 we will describe the

(weighted) distribution of the “low-lying zeros” for the L-functions attached to Siegel

cusp forms of increasing weight and level. More precisely, for any even Schwartz function

Φ whose Fourier transform has compact support we consider

D(πF ; Φ) =∑ρ

Φ( γ

2πlogCk,N

)where ρ = 1/2+iγ varies over all zeros of L(s, πF ) inside the critical strip with multiplicity,

and Ck,N is a certain analytic conductor as defined in §6.3. We assume the Riemann

hypothesis: namely all γ ∈ R. D(πF ; Φ) reflects the distribution of the low-lying zeros of

the single L-function L(s, πF ). We study an averaged version of this: let

D(N, k; Φ) =1∑

F∈Sk(N)# ωF,N,k

∑F∈Sk(N)#

ωF,N,kD(πF ; Φ)

where ωF,N,k is the weight from Theorem 5.11 and Sk(N)# consists of eigenfunctions of

all Hecke operators at all p - N (in contrast to Sk(N)∗ above). The distribution of the

low-lying zeros is then described as follows:

1Theorem 5.1 has a version for more general weights. In our treatment of low-lying zeros we stick tothis special case for simplicity.

128

Theorem 6.1. Let Φ : R → R be an even Schwartz function such that the Fourier

transform Φ(t) =∫R Φ(x)e−2πixtdx has compact supported contained in [−α, α] where

α < 2/9. Then

limk+N→∞

D(N, k; Φ) =

∫R

Φ(x)W (Sp)(x)dx

as k varies over even integers and N varies over squarefree1 positive integers, and where

W (Sp) is the kernel for symplectic symmetry

W (Sp)(x) = 1− sin 2πx

2πx.

The proof of Theorem 6.1 is a fairly standard exercise, combining Theorem 5.1 and ex-

plicit formulas for L-functions. The first thing to notice about the result is that there is no

restriction to newforms, so representations are counted with multiplicity, as in [66] (this

means that we must take our conductor Ck,N to be a log-average one). Once again we see

the effect of the weight ωF,N,k, as [66] (Theorem 1.5/11.5) shows that these low lying ze-

ros with constant weight exhibit even orthogonal symmetry (in the weight or level aspect).

Another noteworthy feature of Theorem 6.1 is the contribution of Saito–Kurokawa lifts

at ramified primes, which does not appear in the work of [39] (where there are no ram-

ified primes) or [66] (where transfer to GL4 is assumed, and thus the Saito–Kurokawa

forms are not present because their transfer to GL4 is not cuspidal). The point is that

Saito–Kurokawa lifts do not satisfy the Ramanujan conjecture. At unramified primes their

contribution is handled already in Theorem 5.1, but at ramified primes we must show that

their contribution in the explicit formula calculation can be neglected. In order to get a

handle on these exceptional cases we restrict to square-free level. After doing so we prove

that a cusp form which violates the Ramanujan conjecture at a single ramified prime gives

rise to a vector in the same representation as that of a classical Saito–Kurokawa lift. It

is well-known that Saito–Kurokawa lifts are few amongst all Siegel cusp forms, but we

require a quantitative estimate of how few they are when counted with the weight ωF,N,k.

To achieve this we combine classical and representation theoretic methods to show that

the Saito–Kurokawa contribution can be neglected as desired. We remark that, as sug-

gested by the previous paragraph, our treatment of Saito–Kurokawa lifts is not restricted

1Note that this was not assumed when considering the local equidistribution and independence.

129

to newforms (as is often the case in the literature).

In the final section we discuss the arithmetic weight ωF,N,k in the context of Bocherer’s

conjecture, and how this should explain the discrepancy between Theorem 6.1 and The-

orem 1.5/11.5 of [66]. We also explain how this conjecture is related to the “information

gap” between Hecke eigenvalues and Fourier coefficients for degree two Siegel cusp forms.

For this chapter we restrict to modular forms of squarefree level N , and take the weight

ωF,N,k to be defined with d = 4 and Λ = 1. This chapter is an elaboration of part of the

paper [15].

6.1 Background on L-functions

Given an irreducible automorphic representation π of GSp2n, one can form the Langlands

L-function L(s, π, r) for any representation r of the dual group GSp2n = GSpin(2n+1,C).

There are two L-functions which commonly appear in the theory of Siegel modular forms;

firstly, one has the standard representation

r : GSpin(2n+ 1,C)→ GL2n+1(C),

giving the “standard” L-function. On the other hand, one has the spinor representation

r : GSpin(2n+ 1,C)→ GL2n(C),

giving the “spin” L-function. These yield L-functions of degree 2n + 1 and degree 2n

respectively. Specializing to n = 2 the spin L-function has smaller degree and we will

therefore focus on this one.1 We therefore write L(s, π) for the degree four L-function

attached to a cuspidal automorphic representation π of GSp4(Q).2

1For an alternative perspective on this choice, note that since our representations of GSp4(A) havetrivial central character they descend to PGSp4(A). Now there is an exceptional isomorphism PGSp4 'SO(3, 2), coming from the symmetry in the root datum; the same symmetry is of course also present in thedual root datum. Thus if we view our representations as SO(3, 2)-representations, then the “standard”L-function coming from the tautological representation is in fact the “spin” L-function we are considering.

2Similar arguments could be used to deal with the standard and more general L-functions.

130

In this discussion of L-functions we restrict our attention to representations π which are

self-dual, since the representations generated by modular forms with trivial character

are self-dual. We also assume π is irreducible, since we can always choose an irreducible

constituent of the representation generated by a modular form. Since a different choice

changes the Euler factors at at most finitely many places the choice is unimportant for

the analytic considerations to follow. For p a finite prime the write the local Euler factor

as

Lp(s, π) =4∏i=1

(1− αi(p)p−s)

so that the (finite part of) the L-function is

L(s, π) =∏p

Lp(s, π)−1.

The αi(p) are the local factors, defined via the local Langlands correspondence for GSp4.

At the unramified primes (those where πp is spherical) these are the Satake parameters.

Using the notation of §5.2 these are (ap(π), bp(π)) = (σ(p), σ(p)χ1(p)). Thus labelling

appropriately we have

α1(p) = α2(p)−1 = ap(π),

α3(p) = α4(p)−1 = bp(π).

At the ramified primes (those where πp is not spherical) the αi(p) can be zero; it is a

delicate question to say precisely what the local factor are in these case. For our consider-

ation of low-lying zeros attached to these L-functions in §6.3 we will require some bounds

on these quantities. Whilst the Ramanujan conjecture, proved by Weissauer, provides the

optimal bound for the local parameters at unramified places (certainly the most impor-

tant case in general) for non-CAP representations, we are not aware of such results for

ramified places explcitly mentioned in the literature. We make the following assumption:

if π is non-CAP then there exists 0 ≤ θ < 1/2 such that

|αi(p)| ≤ pθ. (6.1)

We suspect this might be known, expecially given that we are assuming squarefree level.

It certainly follows if we assume transfer of π to GL4 (which has been proven for N = 1 in

[52]; to the best of the author’s knowledge this is a bona fide theorem in light of Arthur’s

131

latest book), as non-CAP representations will have cuspidal transfer so one can use [48]

Proposition 3.3 (the ramified analogue of [41] Theorem 2) to take θ = 12− 1

42+1. Alterna-

tively, it should be possible to argue similarly to [48] Proposition 3.3 without having to

move on to GL4 at all.

On the other hand we must also take in to account some CAP representations, since

the representations attached to Saito–Kurokawa lifts are so. These are certain cuspidal

automorphic representations of PGSp4 whose local factors do not satisfy the Ramanujan

conjecture: at almost all places some of the local factors are as large as p1/2. For these

representations the expected transfer to GL4 is no longer cuspidal (and in particular

(6.1) will not hold). It turns out that the ramified local factors for these representations

are large enough that we have to handle these representations exceptionally. Just from

dimension formulas one would expect the Saito–Kurokawa contribution to be negligible,

but it is more complicated in our situation since we require a weighted version of this

statement. We will explain our resolution of this issue in §6.2. Although we restrict to

squarefree level to deal with this, we expect this issue should really be minor in any case.

If this issue was resolved for general N then the rest of the results of this section would

also apply in that generality. At the unramified places the Saito–Kurokawa contribution

is already handled in Theorem 5.17.

We now continue with the definition of the L-function. For the infinite place we have a

gamma factor determined by the representation type of π∞. When π is an irreducible

constituent of the representation generated by a Siegel cusp form F of weight k the

gamma factor is

γ(s, π) = (2π)−2sΓ

(s+

1

2

)Γ

(s+ k − 3

2

)(6.2)

We shall assume the existence of a “nice L-function theory”: there exists an integer q(π),

divisible only by ramified primes of π, such that the completed L-function

Λ(s, π) = q(π)s/2γ(s, π)L(s, π)

extends to a meromorphic function satisfying the functional equation

Λ(s, π) = ε(π)Λ(1− s, π).

132

Here ε(π) ∈ ±1 is determined by the local ε-factors, in turn defined by the local Lang-

lands correspondence. A “nice L-function theory” would follow from ([51]), once it has

been verified in all cases that the local factors defined there agree with those of defined

by the local Langlands correspondence. Given such an L-function we define the analytic

conductor to be C(π) = q(π)q∞(π), where q(π) is the factor appearing in the functional

equation, and for π the representation generated by a weight k Siegel modular form

q∞(π) := k2.

Background on low-lying zeros. We are interested in the low-lying zeros of the L(s, π)

on the critical line s = 1/2. The explicit formula is key to this: for example from [28]

Theorem 5.12 we have, for h : R→ R an even Schwartz function with Fourier transform

h, ∑ρ

h( γ

2π

)= h(0) log q(π)

+1

2π

∫R

(γ′

γ

(1

2+ it, π

)+γ′

γ

(1

2− it, π

))h

(t

2π

)dt

− 2∑p

log p∑m≥1

c(π, pm)p−m/2h (m log p) ,

(6.3)

where the sum on the left hand side is over zeros ρ = 12

+ iγ, and the double sum on the

right involves moments of the local factors of the representation:

c(π, pm) =4∑i=1

αi(p)m. (6.4)

However, in order to have enough zeros to do a meaningful statistical study we will av-

erage over a suitable family of representations π as above, which we now describe: let

Sk(N)# be an orthogonal basis of Sk(N) consisting of eigenfunctions of all T (p) and

T1(p2) when p - N . Then for any F ∈ Sk(N)# we have an associated cuspidal automor-

phic representation of GSp4(A). Let πF be any irreducible consitutent of this, and write

C(πF ) be the analytic conductor as above.

We will consider the representations we obtain as we vary F ∈ S#k (N), in particular

there is no restriction to “newforms”. It may be possible to set up the problem in terms

133

of newforms using the description in [61], but this presents difficulties because the no-

tion of newforms is only really well-behaved on paramodular congruence subgroups. We

choose not to restrict to newforms for this reason, and also so that we can apply Theorem

5.17 directly. This means that as we vary over F ∈ Sk(N)#, the (isomorphism class of)

a representation may be repeated.

In any case when working with forms that are not necessarily “new” the q(πF ) is by no

means the same for each element in our family. It is therefore prudent to introduce a

log-average conductor, defined by

logCk,N =1∑

F∈Sk(N)# ωF,k,N

∑F∈Sk(N)#

ωF,k,N logC(πF ).

Recall that N is squarefree. From Table 3 of [61], particularly the fact that the conductors

of representations which have invariant vectors for P1 (the local version of Γ0(N)) have

conductor ≤ 2, it easily follows that Ck,N N2. By using the fact that representations

containing newforms for P1 have conductor ≥ 1 one can argue by induction to obtain a

lower bound and deduce that

logCk,N logN. (6.5)

Finally, let Φ be an even Schwartz function (the Fourier transform of which we will

eventually assume to have sufficiently small compact support), and let

D(k,N ; Φ) =1∑

F∈Sk(N)# ωF,k,N

∑F∈Sk(N)#

ωF,k,ND(πF ; Φ),

where

D(πF ; Φ) =∑ρ

Φ( γ

2πlogCk,N

).

The quantity D(k,N ; Φ) measures the low-lying zeros of the L-functions associated to

the representations in our family.

6.2 Saito–Kurokawa lifts

Recall that we stated in the preceding section that certain representations, namely CAP

representations, require special treatment. To this end we begin by recalling the descrip-

tion of Saito–Kurokawa lifts from [62]; at the end of this section we will show that these

134

essentially exhaust all problem cases in our context. First take an irreducible cuspidal

automorphic representation π of PGL2, and assume that π corresponds to a holomorphic

cusp form of weight 2k − 2, so that π∞ is the discrete series representation with lowest

weight 2k − 2. Let Σ be the set of places at which π is a discrete series. We pick a set

S with ∞ ∈ S ⊂ Σ such that (−1)|S| = ε(π), with the usual ε-factor of the cuspidal

automorphic representation π. Define a representation πS of GL2 by

πS =

1v if v /∈ S,

Stv if v ∈ S,

where Stv denotes the Steinberg representation. At the infinite place this is taken to mean

the lowest discrete series representation. πS is in fact a constituent of a globally induced

representation, so it is automorphic. For any choice of S as above a lift Π(π× πS) can be

defined; it is an irreducible cuspidal automorphic representation of PGSp4.

Most importantly for us is a case when π corresponds to a newform g ∈ S(1)2k−2(M) of

squarefree level M considered in detail in [63], where S is chosen to be the set of primes

p | M for which the newform g has Atkin–Lehner eigenvalue −1. The lift SK(π) =

Π(π × πS) is then an irreducible cuspidal automorphic representation of PGSp4. The

local component SK(π)∞ is the holomorphic discrete series representation of PGSp4(R)

with scalar minimal K-type of weight (k, k). This is the∞-type of the representation at-

tached to a holomorphic Siegel modular form; in fact it follows from Theorem 5.2 of [63]

that there is a unique (up to scalars) modular form F ∈ Sk(M) such that ΦF generates

the representation Π(π× πS). Indeed this function F is the classical Saito–Kurokawa lift

SK(g) of the newform g as defined in [45]. Using results from [50], it is also described in

the proof of Theorem 5.2 of [63] how the representation SK(π) occurs with multiplicity

one in the space of automorphic forms on PGSp4.

Our L-functions however are formed from Sk(N)# and therefore we must take in to

account that whilst there is a unique modular form F of level M | N whose representation

is πF , there will be more forms of level N describing the same representation. We shall

now count how many vectors in the representation SK(π) give rise to modular forms of

135

level N :

Lemma 6.2. Let π be the cuspidal automorphic representation of PGL2 associated to a

classical newform g of level M | N (N squarefree), and SK(π) its Saito–Kurokawa lift.

Then the vector space consisting of modular forms F ∈ Sk(N) such that ΦF ∈ SK(π) has

dimension 3r, where r is the number of prime divisors of N/M .

Proof. Set

P1(p) =

A B

C D

∈ GSp4(Zp); C ≡ 0 mod NZp

.

To count the number of vectors in SK(π) which come from level N modular forms it suf-

fices to count the number of vectors invariant under∏

p P1(p) =∏

p|N P1(p)∏

p-N GSp4(Zp).

It is shown in [63] that Π(π×πS)p has an essentially unique (i.e. up to scalars) vector un-

der the right-action of P1(p) for each p |M . For p - N there is an essentially unique vector

for the action of GSp4(Zp). The case p | N/M is not written down in the work of Schmidt

but follows easily from it: we know when p | N/M that πp = π(χ, χ−1) is a spherical prin-

cipal series representation of PGL2(Qp), and by [62] §7 we have Π(π×πS)p ' χ1GL2oχ−1

(in the notation of [57]). By Table 3 in [61] this has three linearly independent vectors

invariant under P1(p). Piecing this together for each prime dividing N/M we obtain the

statement of the lemma.

Lemma 6.2 does not give us the modular forms f ∈ Sk(N) explicitly, but we can easily

provide a basis for the vector space it considers via classical means:

Lemma 6.3. Let π be the cuspidal automorphic representation of PGL2 associated to

a classical newform g of level M , and SK(π) its Saito–Kurokawa lift. Let SK(g) be the

classical Saito–Kurokawa lift of g. Define1 the following maps on Fourier coefficients:

F (Z) =∑T>0

a(T ;F )e(tr(TZ)) 7→

∑

T>0 a(T ;F )e(tr(pTZ)) =: T1(p, F ),∑T>0 a(pT ;F )e(tr(pTZ)) =: T3(p, F ).

Define a set for squarefree multiples of M inductively as follows: BM = SK(g), and if N ′

is a squarefree multiple of M and p - N ′ is a prime set BN ′p = F, T1(p, F ), T3(p, F ); F ∈1The subscripts are thus to be consistent with the notation of [61].

136

BN ′. Then, for any squarefree multiple N of M , BN is a basis for the space of modular

forms F ∈ Sk(N) such that ΦF ∈ SK(π).

Proof. It suffices to prove that BN is a linearly independent set since if so it has the

dimension required by Lemma 6.2 by construction. By writing out a dependence relation

and picking off leading Fourier coefficients we see that proving linear independence boils

down to showing that there are no nontrivial dependence relations of the form

∑e|d

cea(eT ; SK(g)) = 0, for all T > 0 (6.6)

where d is a fixed divisor of N/M . Suppose we have such a nontrivial relation involving a

minimal number of divisors e. Now for any p -M we have that SK(g) is an eigenfunction

of T (p), hence there is λ ∈ C such that

λa(T ; SK(g)) = a(pT ; SK(g)) + pk−1a(T ; SK(g)) + p2k−3a(T ; SK(g)).

This follows from using the formula for the action of T (p) on Fourier expansions and the

fact that the Fourier coefficients of a Saito–Kurokawa lift depend only on the determinant

of the indexing matrix. Repeatedly using this allows us to derive from (6.6) a dependence

relation involving fewer e, and thence a contradiction.

Now we use a result of Brown and the structure of the basis in 6.3 to show that the

weights ωF,k,N are small for any F this basis:

Theorem 6.4. [Brown, [7] Theorem 1.1] Let M be a squarefree positive integer, say

M =∏m

i=1 pi, let g ∈ S(1)2k−2(M) be a newform, and let SK(g) ∈ Sk(M) be the classical

Saito–Kurokawa lift of g. Write Sh(g) for the Shimura lift of g, and a(n; Sh(g)) for its

Fourier coefficients. Let D < 0 be a fundamental discriminant such that gcd(M,D) = 1

and a(|D| , Sh(g)) 6= 0. Then

〈SK(g), SK(g)〉 = Bk,M|a(|D| ; Sh(g))|2 L(1, πg)

π |D|k−32 L(1

2, πg × χD)

〈g, g〉, (6.7)

where

Bk,M =Mk(k − 1)

∏mi=1(p4

i + 1)

2m+33[Sp4(Z) : Γ0(M)][Γ0(M) : Γ0(4M)].

137

Corollary 6.5. Let M be a squarefree positive integer and g ∈ S(1)k (M) be a newform,

and let SK(g) ∈ Sk(M) be the classical Saito–Kurokawa lift of g. Let S(1)k (M)#

new denote

an orthogonal basis for the space of newforms. Then, for any δ > 0,∑g∈S(1)k (M)#new

ωSK(g),M,k δ1

M5−δk2−δ .

Proof. Let g ∈ S(1)k (M)#

new, and assume for now a(12; SK(g)) 6= 0. By the construction of

the classical Saito–Kurokawa lifting we have a(4; Sh(g)) = a(12; SK(g)), so we can apply

Theorem 6.4 with D = −4. Substituting this in to the formula for ωSK(g),M,k = ω4,1SK(g),M,k

we have

ωSK(g),M,k =π2

2 vol(Γ(2)0 (M)\H2)Bk,M(k − 2)

Γ(2k − 3)

(4π)2k−3〈g, g〉L(1

2, πg × χD)

L(1, πg).

If a(12; SK(g)) = 0 then clearly the weight is zero. In any case the sum we are trying to

bound is majorized by a constant (depending on k and M) multiplied by∑g∈S(1)k (M)#new

Γ(2k − 3)

(4π)2k−3〈g, g〉L(1

2, πg × χD)

L(1, πg).

We can now argue as in [39] §5.3 (where M = 1) to see that this sum is log(Mk).

Note that the factor of [Sp4(Z) : Γ0(M)] cancels out the normalisation in vol(Γ0(M)\Hn),

but the Mk in the numerator and our ubiquitous assumption that k ≥ 6 give us (after

sacrificing a power of M to the 2m+3 in the denominator) the claimed bound.

Finally we use some representation theory to show that the Saito–Kurokawa lifts exhaust

all problematic cases. Suppose F ∈ Sk(N)# is such that πF has a local parameter with

absolute value p1/2 at some prime p | N . We will show that there exists an irreducible cus-

pidal automorphic representation π of PGL2, corresponding to a newform g ∈ S(1)2k−2(M),

such that Φ(F ) ∈ SK(π).

By our assumption (6.1) πF is CAP – in fact it follows from [53] Corollary 4.5 that πF

is associated to the Siegel parabolic P . Fix an additive character ψ of Q\A, and write

θ(·, ψ) for the theta lifting from SL2 to PGSp4. Then by [50] Theorem 2.2 πF = θ(π, ψ)

for some irreducible cuspidal automorphic representation π of SL2. The representation π

is not ψ-generic (c.f. [50] Theorem 2.4), which implies that it does not participate in the

138

theta correspondence with PGL2.

On the other hand, let S be the (finite) set of places at which πv is the non-generic element

in the fiber of the local Waldspurger correspondence between SL2 and PGL2. Replacing

πv with the generic element in the fiber we will obtain a globally ψ-generic representation

of SL2 which does have a non-vanishing theta lift to PGL2; write π for this lift. By the

definition in [62] (and multiplicity one for theta lifts from SL2) we have πF = Π(π× πS),

with S as above.

It remains to see that π in fact corresponds to a holomorphic newform g ∈ S(1)2k−2(M)

where M | N (the choice of S is then forced to be the one defining SK(π) by table (30) of

[63]). By examining Table 2 of [62] we easily deduce that π has the correct ∞-type (and

that∞ ∈ S) by knowing the∞-type πF . Similarly knowing that all the local components

of πF must have Iwahori-spherical vectors we deduce that π is nowhere supercuspidal.

Finally we see that the set of finite primes at which π is a discrete series is a subset of

the set of finite primes at which πF is not a principal series. Thus π corresponds to a

holomorphic newform g as above.

Remark 6.6. The preceding paragraph only shows that our problem cases are contained

in the Saito–Kurokawa cases. Certain Saito–Kurokawa representations may not be a prob-

lem: for example an elliptic modular form of squarefree level with all Atkin–Lehner eigen-

values equal to −1 will have small local factors at ramified primes. It will have large local

factors at unramified primes, but these are dealt with by Theorem 5.17.

Corollary 6.7. Let P = F ∈ Sk(N)#; (6.1) does not hold for πF. Then, for any δ > 0,

∑F∈P

ωF,N,k δ1

N3k2−δ

Proof. Let F ∈ P . By the preceding discussion we know that there exists an irreducible

cuspidal automorphic representation π of PGL2 corresponding to a newform g such that

ΦF ∈ SK(π). Thus F is a sum of the basis elements of BN from Lemma 6.3. Normalising

(recall ωF,N,k is invariant under rescaling) we may assume that the coefficient of SK(g)

(if non-zero) is one. Since all elements F ′ other than SK(g) of the basis clearly have

139

a(12;F ′) = 0, and hence ωF ′,N,k = 0, it follows that ωF,N,k is either zero (if the coefficient

of SK(g) is) or we have ωF,N,k = ωSK(g),N,k. The result then follows from Corollary 6.5

and the fact that ω·,N,k 1(N/M)3

ω·,M,k.

6.3 Low lying zeros

We now proceed with the proof of Theorem 6.1, beginning with the computations at

the archimedean place. If F ∈ Sk(N)∗ then the gamma factor of the L-function of the

representation πF is given by (6.2). As before let Φ be an even Schwartz function, and

now consider the expression

1

2π

∫R

(γ′

γ

(1

2+ it, πF

)+γ′

γ

(1

2− it, πF

))Φ

(t

2πlogCk,N

)dt

=1

logCk,N

∫R

(γ′

γ

(1

2+

2πix

logCk,N, πF

)+γ′

γ

(1

2− 2πix

logCk,N, πF

))Φ(x)dx.

Arguing from (6.2) as in [16] we see that

1

logCk,N

∫R

(γ′

γ

(1

2+

2πix

logCk,N, πF

)+γ′

γ

(1

2− 2πix

logCk,N, πF

))Φ(x)dx

= Φ(0)log k2

logCk,N+O

(1

logCk,N

).

Now setting h(x) = Φ(x logCk,N) (and hence h(t) = 1logCk,N

Φ(

tlogCk,N

)) in the explicit

formula (6.3), using the above archimedean computation and

Φ(0)log q(πf )

logCk,N+ Φ(0)

log k2

logCk,N+O

(1

logCk,N

)=

logCπflogCk,N

Φ(0) +O

(1

logCk,N

),

we get

∑ρ

Φ( γ

2πlogCk,N

)=

logCπFlogCk,N

Φ(0)− 2

logCk,N

∑p

m≥1

log(p)c(π, pm)p−m/2Φ

(m log p

logCk,N

)

+O

(1

logCk,N

).

140

Averaging over F ∈ Sk(N)# we therefore obtain

D(k,N ; Φ)

= Φ(0)− 1∑F ωF,k,N

∑F

2ωF,k,NlogCk,N

∑p

m≥1

log(p)c(πF , pm)p−m/2Φ

(m log p

logCk,N

)

+O

(1

logCk,N

) (6.8)

It remains to deal with the term involving the triple sum. It is not difficult to see that

for each m ≥ 3 the sum over primes converges (independently of k and N , and it does so

even without the cutoff provided by Φ), and therefore the whole term can be absorbed in

to the O(1/ logCk,N). Thus it suffices to estimate the sum over primes when m = 1 and

m = 2.

First consider m = 1. When p is an unramified prime we argue as in [39]: use the definition

(5.14) and Proposition 5.15 to see

1∑F ωF,k,N

∑F

ωF,k,Nc(πF , p)

=1∑

F ωF,k,N

∑F

ωF,k,N(U1,0p (ap(πF ), bp(πF )) + λpp

−1/2)

= λpp−1/2 +Oε(N

−1k−2/3p1+ε).

(6.9)

When p is a ramified prime, using Corollary 6.7 (and its notation)

∣∣∣∣∣∣ 1∑F∈Sk(N)# ωF,k,N

∑F∈Sk(N)#

ωF,k,Nc(πF , p)

∣∣∣∣∣∣≤ 4∑

F∈Sk(N)# ωF,k,N

∑F∈Sk(N)#

F /∈P

ωF,k,Npθ +

∑F∈Sk(N)#

F∈P

ωF,k,Np1/2

4pθ +

p1/2

N3k2−δ .

141

Thus, assuming that Φ is supported in [−α, α],

1∑F ωF,k,N

∑F

ωF,k,N2

logCk,N

∑p

log(p)c(πF , p)p−1/2Φ

(log p

logCk,N

)

=2

logCk,N

∑p-N

λp log p

pΦ

(log p

logCk,N

)+Oε

1

Nk2/3

∑p≤Cαk,N

p12

+ε

+O

∑p|N

log(p)p(θ−12)

.

(6.10)

We have left out the contribution at ramified primes from F ∈ P because this is clearly

negligible. For the remaining sum over ramified primes, the hypothesis θ < 1/2 and the

fact that #p | N = o(logN) show that the sum is o(logN). By the hypothesis (6.5)

this is in turn o(logCk,N), and so the sum over p | N is negligible due to the presence

of the 1logCk,N

factor in front. By choosing α small enough we will show that the second

term is negligible as well. For the first term note that λp takes the value 0 or 2 each on

sets of primes of asymptotic density 1/2, so by the prime number theorem

2

logCk,N

∑p

λp log p

pΦ

(log p

logCk,N

)= 2

∫ ∞1

Φ

(log x

logCk,N

)1

logCk,N

dx

x+ o(1)

= 2

∫ ∞0

Φ(x)dx+ o(1)

= Φ(0) + o(1)

where the last equality follows from the factor that Φ is even. Now the left hand side is

the same as the first sum in (6.10) except that we imposed the restriction p - N in the

latter: the difference between the two is easily seen to be O(

1logCk,N

)(remembering the

constant factor 1log(Ck,N )

in front), so we conclude

1∑F ωF,k,N

∑F

ωF,k,N2

logCk,N

∑p

log(p)c(πF , p)p−1/2Φ

(log p

logCk,N

)

= Φ(0) +Oε

1

logCk,NNk2/3

∑p≤Cαk,N

p12

+ε

+O

(1

logCk,N

).

Next consider m = 2. When p is unramified we again argue as in [39]: begin with the

formula

c(πF , p2) = U2,0

p (ap(πF ), bp(πF )) +λ√pU1,0p (ap(πF ), bp(πF ))

− τ(ap(F ), bp(F ))− 1− 1

p

(d

p

).

142

Averaging this over F with the help of Proposition 5.15 we have

1∑F ωF,k,N

∑F

ωF,k,Nc(πF , p2) = −1− 1∑

F ωF,k,N

∑F

ωF,k,Nτ(ap(F ), bp(F ))

+Oε

(p2+ε

Nk2/3

)+Oε

(p1+ε

Nk2/3

)+O

(1

p

).

Appealing to the definitions 5.14 and Proposition 5.15 with U0,1p we have that

1∑F ωF,k,N

∑F

ωF,k,Nτ(ap(F ), bp(F )) = Oε

(p

32

+ε

Nk2/3

).

For the ramified primes we argue as before and obtain the same result with pθ replaced by

p2θ in the first term on the RHS, and p1/2 replace by p in the second. Again the ramified

contribution from F ∈ P is clearly negligible and we obtain

1∑F ωF,k,N

∑F

ωF,k,N2

logCk,N

∑p

log(p)c(πF , p2)p−1Φ

(2 log p

logCk,N

)

=2

logCk,N

−∑p-N

λp log p

pΦ

(2 log p

logCk,N

)+Oε

1

Nk2/3

∑p≤Cα/2k,N

p1+ε

+O

∑p|N

log(p)p(θ−1)

.

The sum over p | N is even more negligible than before. We postpone choosing α suffi-

ciently small for a little longer and consider the main term, which similarly to before is

a negligible distance from

− 2

logCk,N

∑p

λp log p

pΦ

(2 log p

logCk,N

)= −1

2Φ(0) + o(1)

(using the prime number theorem as before). Thus

1∑F ωF,k,N

∑F

ωF,k,N2

logCk,N

∑p

log(p)c(πF , p2)p−1Φ

(2 log p

logCk,N

)

= −1

2Φ(0) +Oε

1

logCk,NNk2/3

∑p≤Cα/2k,N

p1+ε

+O

(1

logCk,N

).

Finally it remains to choose α small enough such that the two sums

1

logCk,NNk2/3

∑p≤Cα/2k,N

p1+ε = O

(Cα+εk,N

logCk,NNk2/3

)

143

and

1

logCk,NNk2/3

∑p≤Cαk,N

p12

+ε = O

C3α2

+ε

k,N

logCk,NNk2/3

are, say, O(1/ logCk,N). Since Ck,N N2k2 we can do this with α < 2/9.

Putting this altogether we have

D(k,N ; Φ) Φ(0)︸︷︷︸archimedean contribution

− Φ(0)︸︷︷︸p contribution

+1

2Φ(0)︸ ︷︷ ︸

p2 contribution

, (6.11)

which proves Theorem 6.1 since∫R

Φ(x)W (Sp)(x)dx = Φ(0)− 1

2Φ(0).

Note that it is probably possible to improve the range of α (which is typically desirable

in low-lying zeros questions) with better estimation in the above. If one were to study

families with orthogonal symmetry then one would require α > 1 to distinguish the type

of orthogonal symmetry (c.f. [29] §1 Remark D), but our α = 2/9 is large enough to bear

witness to the symplectic-type distribution of the low-lying zeros of our weighted family

of L-functions.

6.4 Discussion of results and future research

The upshot of Theorem 6.1 is that the low-lying zeros of the L-functions attached to

Siegel cusp forms of increasing weight and level weight as above exhibit a symplectic-

type distribution (at least for the one-level density). This is in stark contrast to what

one observes in the same constant weight in place of ωF,N,k. In fact it follows from the

main result of [66], since the Frobenius–Schur indicator of the tautological representation

of GSp4(C) is −1, that we see even orthogonal symmetry in this case. This statement

holds in either the weight or level aspect version of our problem, and should hold in both

simultaneously. Thus the difference in symmetry type must be due to the weighting ωF,k,N .

This can be explained in terms of Bocherer’s conjecture. This is a conjecture on the

relationship between Fourier coefficients and central values of L-functions, in the spirit

144

of the celebrated formula of Waldspurger. The ur statement of Boecherer’s conjecture is

that for any F ∈ Sk there exists a constant cF such that for all d > 0

|a(d,1;F )|2

dk−1w(−d)2〈F, F 〉= cF × L

(1

2, πF × χ−d

), (6.12)

where χ−d is the Hecke character associated to Q(√−d)/Q by class field theory, and

a(d,1;F ) is defined by (5.7). The constant cF should of course be independent of d, but

is not further specified. In fact even in the refined versions we have today the constant is

still not completely specified.

In order to simultaneously refine and generalise this, let d > 0, set K = Q(√−d),

and let Λ be any character of the ideal class group of K. Then Λ corresponds to a

automorphic representation of GL1(AK). By the process of automorphic induction we

obtain an automorphic representation Θ(Λ) of GL2(AQ); in classical language this is

corresponds simply to forming the Hecke theta series for the Hecke character Λ. Then a

refined version of Bocherer’s conjecture is that a formula

|a(d,Λ;F )|2

dk−1w(d)2〈F, F 〉= c× L

(1

2, πF ×Θ(Λ)

)(6.13)

should hold, where c is some constant. Of course c should certainly independent of d, it

may depend on F but the expected formula (1.4) of [24] gives us an inkling what to expect.

In order to determine the constant c we would need to pass the generic representation

πgen in the L-packet of πF (such a πgen always exists, at least conjecturally) and compute

a Whittaker period of the newform ϕ in πgen, in much the same way that the left hand

side of (6.13) is essentially a Bessel period for ΦF . The constant c is therefore expected to

be positive, which implies that the L-value on the right hand side is non-negative. That

this is a desirable feature is seen by analogy with the Waldspurger formula, where the

corresponding non-negativity has been immensely useful in analytic number theory. The

refinement (6.13) recovers (6.12) when we take Λ = 1, since in that case

L

(1

2, πF ×Θ(1)

)= L

(1

2, πF

)L

(1

2, πF × χ−d

),

so the two constants, whatever they are, are related by

c = L

(1

2, πF

)cF .

145

The formula (6.13) is the goal of a project of Furusawa–Martin–Shalika ([25], [21], [23],

[22]), who have developed various relative trace formulas which should eventually lead to

a proof.

Bocherer’s conjecture is related to the multiplicity one problem for degree two Siegel

modular forms. In fact if the level N is one, a simple argument due to Saha ([56]) shows

that a weak version of (6.13) implies that multiplicity one does hold. More precisely, if

we assume the hypothesis

“for all d > 0 and Hecke characters Λ of AQ(√−d), we have L(1/2, πF ×

Θ(Λ)) 6= 0 if and only if |a(d,Λ;F )| 6= 0”

then we can prove the statement

“if F1, F2 ∈ Sk(Sp4(Z)) are non-zero Hecke eigenforms such that λ(p;F1) =

λ(p;F2) and λ1(p2;F1) = λ1(p2;F2) for all primes p, then F1 is a scalar multiple

of F2”.

Although this is stated for full level it should be the case that it can be extended to

handle certain levels.

We finish by giving a rough heuristic for how Bocherer’s conjecture can be used to explain

the discrepancy between Theorem 6.1 and the results of Shin–Templier. The calculation is

essentially identical to the caseN = 1 which is included in [39]. Since Theorem 6.1 involves

an average over F it is necessary to use the refined version (6.13). Firstly, substituting in

(6.13) and using Dirichlet’s class number formula we obtain

ωF,N,k = γ(F )L(

12, πF)L(

12, πF × χ

)L(1, χ−d)

, (6.14)

for some fudge factor γ(F ) = γ(F,N, k) about which we will soon make some assumptions

based on Bocherer’s conjecture. Comparing the three contributions of (6.11) and the

corresponding contributions in [66] (§12.12, the contribution of p and p2 are (12.26) and

(12.30) respectively) we see that the difference between the result arises from the sum

over p. It is therefore natural to insert Bocherer’s conjecture in to the sum over p. We are

146

led to consider ∑F

γ(F )L(

12, πF)L(

12, πF × χ

)L(1, χ)

c(πF , p),

and compare this to the result of (6.9), which was computed using the local equidistri-

bution results in the guise of Proposition 5.15. Note that the L-functions L(s, πF ) and

L(s, πF × χ−4) both have analytic conductor about Nk2, so applying the approximate

functional equations1 to both we have∑F

γ(F )L(

12, πF)L(

12, πF × χ

)L(1, χ)

c(πF , p)

≈ 1

L(1, χ)

∑m,n≤

√Nk

∑F

γ(F )λ(m;F )λ(n;F )χ(n)λ(p;F ).

Now under Bocherer’s conjecture we can assume that γ(F ) is positive and has little

bearing on the sum over F .2 In particular, even with the presence of γ(F ), the main

term should be the contribution from the cases when m = np or n = mp (these are the

“diagonal terms”, after using multiplicativity of the coefficients of the Dirichlet series).

Thus the main term is∑F

γ(F )L(

12, πF)L(

12, πF × χ

)L(1, χ)

c(πF , p)

≈ 1

L(1, χ)

(∑F

γ(F )

) ∑m≤√Nk

χ(mp)

m√p

+∑

n≤√Nk

χ(n)

n√p

=

1

L(1, χ)

(∑F

γ(F )

)1 + χ(p)√p

∑m≤√Nk

χ(p)

p

≈

(∑F

γ(F )

)1 + χ(p)√p

.

Thus, including the normalisation factor, the above heuristic calculation shows that the

hypothesis (6.14) leads to the conclusion

1∑F ωF,N,k

∑F

ωF,N,kc(πF , p) ≈1 + χ(p)√p

=λp√p,

which is exactly the same result as (6.9). Since we have given a rigorous proof of Theorem

6.1, the above heuristic calculation can be interpreted as evidence for the validity of a

formula such as (6.14).

1To make this precise one should include the smooth cutoffs.2It should be some powers of 2 and π multiplied by a Whittaker period.

147

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