L-functions in Number Theory

by

Yichao Zhang

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto

Copyright c 2010 by Yichao Zhang Abstract

L-functions in Number Theory

Yichao Zhang

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2010

As a generalization of the Riemann zeta function, L-function has become one of the central objects in Number Theory. The theory of L-functions, which produces a large family of consequences and conjectures in a unified way, concerns their zeros and poles, functional equations, special values and the connections between objects in different

fields. Although most generalizations are largely conjectural, there are many existing results that provide us the evidence.

In this thesis, we shall consider some L-functions and look into some problems men- tioned above. More explicitly, for the L-functions associated to newforms of fixed square- free level, we will consider an average version of the fourth moments problem. The final bound is proven by considering definite rational quaternion algebras and divisor functions in them, generalizing Maass Converse Theorem and one of Duke’s result and eventually applying the solution to Basis Problem.

1 We then consider the problem of expressing the special value at 2 of the Rankin- Selberg L-function associated to two newforms in terms of the Pertersson inner product, where one of the newforms is twisted by the derivative of some Eisenstein series.

Finally, we consider the Artin L-functions attached to irreducible 4-dimensional S5- Galois representations and deal with the modularity problem. One sufficient condition on the modularity is given, which may help to find an affirmative example for Strong

Artin Conjecture in this case.

ii Acknowledgements

The author is greatly indebted to his advisor, Professor Henry H. Kim, who suggested all the problems treated in this thesis. His constant guidance and support have been invaluable.

The author would also like to thank Professor Stephen Kudla for many valuable and inspiring conversations and comments and Professor James Arthur and Professor Valentin

Blomer for the help during the work of this thesis.

The author is also grateful to Professor William Duke for agreeing to be the external examiner for this thesis.

The author thanks all of the faculty and the staff of the Department of Mathematics at University of Toronto for their wonderful help. Special thanks go to Ms. Ida Bulat, our graduate administrator, without whom the life at University of Toronto would never be easy and joyful for the author.

Lastly, the author would like to thank his wife, Hailing Liu, for all the sacrifices she has made these years. Without her company and support, this thesis could not have been completed.

iii List of Symbols

C the field of complex numbers

R the field of real numbers

+ R the field of complex numbers

N the set of non-negative rational integers

Z the ring of rational integers

+ Z the set of positive rational integers

AF the ring of adeles of F

× AF the group of ideles of F

OF the ring of integers of F

Fv the completion of F at v

OF,v the completion of OF at v

R× the group of units in the ring R

× GL2(R) the group of 2 × 2 matrices over the ring R with determinant in R

SL2(R) the group of 2 × 2 matrices over the ring R of determinant one

+ GL2(R) elements of GL2(R) with positive determinant

iv Γ(N) {γ ∈ SL2(Z): γ ≡ I2×2 mod N}

Γ1(N) {γ ∈ SL2(Z): aγ ≡ dγ ≡ 1 mod N, cγ ≡ 0 mod N}

Γ0(N) {γ ∈ SL2(Z): cγ ≡ 0 mod N}

H the upper half plane of C

cχ the conductor of the character χ

f d f d(z) = f(dz)

+ f|kγ weight k action of γ ∈ GL2(R)

q e(z) = exp(2πiz)

Mk(Γ) the space of modular forms of weight k for the group Γ

Mk(N, χ) the subspace of Mk(Γ1(N)) that Γ0(N) acts by Dirichlet character χ

Sk(Γ) the space of cusp forms of weight k for the group Γ

Sk(N, χ) Sk(Γ1(N)) ∩ Mk(N, χ)

new Sk (N, χ) the subspace of newforms in Sk(N, χ)

old Sk (N, χ) the subspace of oldforms in Sk(N, χ)

Tk(n) n-th Hecke operator of weight k

T, Tk the Hecke algebra of fixed weightk

ζ(s) the Riemann zeta function

ζO(s) the zeta function associated to an order O

ζF (s) the zeta function associated to the number field F

L(s, χ) the L-function attached to χ

L(s, f) the L-function attached to the modular form f

L(s, ρ) the L-function attached to the Galois representation ρ

v Lv(s, ·) the local factor at v of the L-function L(s, ·)

AF a quaternion algebra over the field F

(a, b)F the quaternion algebra generated as a vector space by {1, i, j, k},

whose ring structure is given by i2 = a, j2 = b, ij = −ji = k

Av the localization of A at v, that is A ⊗F Fv

AA the ring of adeles of A

A× the group of ideles of A A A× the group of finite ideles of A A,f

Ol(M) the left order of M

Or(M) the right order of M

av the localization of a at v, that is a ⊗OF OF,v

b |l a b divides a from the left

a ∼l b a and b are left-equivalent

H,H(A) the class number of A

α the conjugate of α

T r, T rA/F the reduced trace of AF

N,NA/F the reduced norm of AF

d(a) |{(b, c): a = bc}|, the divisor function

a(n) the number of integral ideals (of fixed left order) of norm n

2 Cn+1 the Clifford algebra generated by {i1, ··· , in} subject to ih = −1,

and ihik = −ikih(k 6= h)

n+1 V {x = x0 + x1i1 + ··· + xnin : xi ∈ R} ⊂ Cn+1

vi n+1 n+1 H {x ∈ V : xn > 0} n n+1 X ∂ 1−n ∂ ∆n+1 xn (xn ) ∂xh ∂xh h=0 Pm the space of spherical harmonics of degree m (in n variables) X∗ the summation over any orthonormal basis of Pm

Pm

The following symbols are used in limiting process, say, as x → a:

f  g when x is close enough to a, f(x) ≤ A|g(x)| for some positive

constant A

f = O(g) f  g f(x) f = o(g) lim = 0 x→a g(x)

f  g when x is close enough to a, A1|g(x)| ≤ f(x) ≤ A2|g(x)| for some

positive constants A1 and A2

f ∼ g f − g = o(g)

vii Contents

Abstract ii

Acknowledgements iii

List of Symbols iv

1 Introduction 1

1.1 Generalized Lindel¨ofHypothesis and Moments Problem ...... 1

1.2 Special Values and Non-vanishing Properties ...... 4

1.3 Strong Artin Conjecture ...... 5

2 L-functions 7

2.1 L-functions ...... 7

2.2 Modular Forms and L-functions ...... 9

2.2.1 Modular Forms ...... 9

2.2.2 The Theory of Hecke Operators ...... 11

2.2.3 The Theory of Newforms ...... 13

2.2.4 L-functions Attached to Modular Forms ...... 14

2.2.5 Automorphic Representations and L-functions ...... 15

2.3 Galois Representations and Artin L-functions ...... 17

3 Divisor Function 21

viii 3.1 Introduction ...... 21

3.1.1 General Notions ...... 21

3.1.2 Local and Global Theory ...... 24

3.2 Divisor Function over Quaternion Algebras ...... 27

4 Fourth Moments 39

4.1 Introduction ...... 39

4.2 Brandt Matrices and Newforms of Level N ...... 40

4.3 Maass Correspondence Theorem ...... 45

4.4 Application to Fourth Moments of L-functions ...... 50

5 Critical Values 57

5.1 Introduction ...... 57

5.2 General Equality ...... 58

5.3 Applications ...... 65

6 Modularity 67

6.1 Introduction ...... 67

6.2 Modularity of ρ and σ ...... 68

6.3 An Open Problem ...... 70

Appendix: Character Tables 72

Bibliography 77

ix Chapter 1

Introduction

1.1 Generalized Lindelof¨ Hypothesis and Moments

Problem

Conjecture 1.1.1 (Generalized Lindelof¨ Hypothesis) Assume L(s) is an L-function

(over Q) which admits meromorphic continuation to the whole s-plane and a functional equation under s ↔ 1 − s. Then for any  > 0,

1 L( + it) = O(t). 2

1  Lindel¨ofconjectured this for the Riemann zeta function, namely, ζ( 2 + it) = O(t ). Note that the Riemann Hypothesis, which asserts that all of the non-trivial zeros of ζ(s)

1 lie on the vertical line Re(s) = 2 , implies the Lindel¨ofHypothesis. Actually the former is much stronger. Define

µ(σ) = inf{a > 0 : ζ(σ + it) = O(ta)},

1 and Lindel¨ofHypothesis states µ( 2 ) = 0. In 1908, Lindel¨of proved that µ is convex and 1 1 it follows that µ( 2 ) ≤ 4 , which is called the convexity bound. On the other hand, one can show that the Lindel¨ofHypothesis for Riemann zeta

1 Chapter 1. Introduction 2 function is equivalent to the statement that the 2k-th integral moment

Z T 1 |ζ( + it)|2kdt = O(T 1+), as T → ∞, 0 2 for all positive integers k and all positive real numbers . The latter is known for k = 1, 2,

but for k ≥ 3, it is still open.

For a general L-function L(s), moments problems are the problems of providing non-

trivial bounds for the moments of L(s), namely, a result would take the form

Z T 1 |L( + it)|2kdt = O(T a). 0 2 In Chapter 3 and 4, we will consider an average version of fourth moments problem

for L-functions attached to newforms. This is first done by Duke [7] in the case of level

N = 2, then generalized by Kim and the author [13] to general prime level N = p. In

this thesis, we further generalize this result to general square-free level N.

Let us review Duke’s idea briefly. Take a maximal order O in the Hamiltonian quater-

nion algebra A = (−1, −1)Q, and construct its zeta function ζO(s). From this he obtained

2 a Dirichlet series φ, which is essentially ζO(s). This φ, as Duke proved, is a Dirichlet series of specific signature in the sense of Maass [17], that is, there is an automorphic

function f associated to φ. With an asymptotic assumption on the summatory function

of φ, he was able to give a (sharp) bound, as T → ∞, for

X X∗ Z T |φ(1 + it)|2dt, −T 0

On the other hand, since A is definite, there is a two-by-two complex matrix repre-

sentation of A, denoted by X1. Let Xm be the symmetric m-th power of X1 and the construction of Brandt matrices applies, so does that of theta (matrix) series (see [8] or √ [24]). It is known that the entries of m + 1Xm, considered as a function of the coordi- nate variables of (−1, −1)R with respect to the canonical basis, constitute an orthonormal Chapter 1. Introduction 3

basis of Pm. Moreover, by analyzing the divisor function d of A, he obtained an identity, which is a quaternion prototype of the classical famous identity

∞ X ζ4(s) d(n)2n−s = . ζ(2s) n=1 Such an identity provides the asymptotic assumption on the summatory function in

Theorem 2 of [7]. The solution to Basis Problem given by Eichler [8] for square-free level

and Theorem 2 mentioned above are applied to prove that, as T → ∞,

X X Z T k |L( + it, f)|4dt  T 3 log4 T. new −T 2 2

class number of a maximal order O in the definite quaternion algebra A that ramifies

precisely at ∞ and all prime divisors of N is bigger than 1 in general.

We need first generalize the Maass Correspondence Theorem to a family of Dirichlet

series and a family of automorphic functions, and then Duke’s Theorem 2. This is done

in [13] and we include this part in Chapter 4 for completeness.

Secondly, we need to verify a similar asymptotic assumption as in Theorem 2 above.

This is done by redefining the divisor function using ideals and working out the exact

expression and hence an analogous identity as before, namely,

X ζ4 (s) d(a)2N(a)−s = O . ζ (2s) a O Note that such an identity is true in general and we include this result for any maximal

order in general quaternion algebras over general number fields in Chapter 3.

Finally, we need a result on the exact modular forms that appear in the diagonal after

diagonalization. Although Eichler and others did not use general definite quaternion

algebras to solve the Basis Problem, they did calculate the traces of the Hecke operators

and Brandt matrices in general. By comparing the traces of Hecke operators and Brandt

matrices in the case of square-free level, we are able to see that the non-zero diagonal

entries (after diagonalization) form precisely the set of newforms of level N. Chapter 1. Introduction 4

Therefore, these efforts generalize Duke’s result to square-free level N. Please see

Chapter 3 for the consideration of quaternion algebras, divisor functions and the deriva- tion of the identity mentioned above. Chapter 4 contains the generalization of Maass

Correspondence Theorem, that of Duke’s Theorem 2 and eventually the application to fourth moment problem associated to newforms of square-free level N.

1.2 Special Values and Non-vanishing Properties

It turns out that much arithmetic information is encoded in the values or behavior of

L-functions at points where their series representations do not converge and also in the distribution of zeros and poles, provided that they can be meromorphically continued.

For example, Deligne conjectured that the special value of an automorphic L-function at a critical point is algebraic up to a specified transcendental number. In particular, for the Riemann zeta function ζ(s), it is a well known fact that ζ(2k)/π2k is a rational number for any k ∈ Z+. On the other hand, that ζ(s) is non-vanishing on the line Re(s) = 1 implies the Prime

Number Theorem. Recall that Prime Number Theorem says

x π(x) ∼ , as x → ∞, log(x)

where π(x) is the number of rational primes less than or equal to x.

Another conjectural example is the Birch and Swinnerton-Dyer conjecture, one of the

Millennium Prize Problems. It predicts that the rank of the abelian group of rational

points of an elliptic curve E is equal to the order of the zero of the associated L-function

L(s, E) as s = 1. (L(s, E) has a functional equation under s ↔ 2 − s.) In particular, if

L(1,E) 6= 0, there would be only finitely many rational points.

1 In Chapter 5, following Moreno ([20]), we will show that the value L( 2 , ρ1 ⊗ ρ2)

for two-dimensional irreducible modular Galois representation ρ1 and ρ2 with the same determinant character is related to the Petersson inner product. Specifically, there exists Chapter 1. Introduction 5 some constant c 6= 0, 1 hf ∗f, hi = cL( , ρ ⊗ ρ ), 2 1 2

where f, h are the newforms corresponding to ρ1 and ρ2 respectively, via Strong Artin Conjecture and f ∗ is the derivative of an Eisenstein series. By further decomposing

ρ1 ⊗ ρ2 in concrete cases, we may get information of the resulting L-functions.

There are two cases; ρ1 and ρ2 both are odd or both are even. In either case the proof is carried out by calculating one integral in two ways. The integral is essentially equal

to the Rankin-Selberg L-function of f and h.

Finally, at the end of Chapter 5, we give an easy application of this type of equalities

in the case of image being Se4.

1.3 Strong Artin Conjecture

Conjecture 1.3.1 (Strong Artin Conjecture) Let ρ : Gal(F /F ) → GLn(C) be a (continuous) Galois representation. Then there exists an automorphic representation

π(ρ) of GLn over F such that for almost all p, Lp(s, ρ) = Lp(s, π(ρ)).

We call ρ modular if the above conjecture is true. This conjecture is a special case

of a much more general conjecture of Langlands, known as the Langlands functoriality

conjecture. If n = 2 and F = Q, this conjecture can be reformulated using classical language. In particular, for odd Galois representations, it takes the following form.

Conjecture 1.3.2 (Strong Artin Conjecture) If ρ : Gal(Q/Q) → GL2(C) is an odd Galois representation with determinant character  and conductor N, then there exists a

newform fρ of weight 1,level N and central character  such that L(s, fρ) = L(s, ρ).

Note that the converse of this conjecture is a theorem of Deligne and Serre (cf. [6]).

Finite groups in P GL2(C) are classified by Klein into 5 classes, cyclic, dihedral (D2n),

tetrahedral (A4), octahedral (S4) and icosahedral (A5) (see [15]). We call ρ the same name Chapter 1. Introduction 6 if its image in P GL2(C) falls into the corresponding class. All the cases are proved except the icosahedral one; the cyclic and dihedral case was proved by Artin as part of the class

field theory, Langlands [16] proved the tetrahedral case and Tunnell [34] finished the octahedral case.

In this thesis, we shall consider a class of 4-dimensional Galois representations with image isomorphic to S5 and provide a sufficient condition on their modularity. Given such a Galois representation ρ, let K be the number field fixed by ker(ρ) and

Gal(K/Q) ' S5. Let Se5 be one of the two nontrivial central extensions by C2, say, the one where a transposition in S5 has preimages of order 2. Assume that K/Q has a double cover K/e Q, that is, K/Ke is quadratic and Gal(K/e Q) ' Se5.

Let E/Q be the quadratic subextension of K/Q determined by A5; hence Gal(K/Ee ) '

Ae5, providing an 2-dimensional icosahedral Galois representation σ of GE. Our main theorem in Chapter 6 tells us that the modularity of σ implies that of ρ.

The proof given there is elementary. By using character tables, we can show that ρ is

a (degree-one) character twist of As(σ), the Asai lift of σ. Then a result of Ramakrishnan

[28] applies to show that As(σ) is modular, so is ρ.

In the last section of Chapter 6, we shall state an open problem which asks for an

affirmative example of modular 4-dimensional S5-Galois representation. The above result may help to attack this problem. Chapter 2

L-functions

In this chapter, we will provide some necessary background for the subsequent chapters.

As well as to list some basic facts in corresponding theories, this chapter also serves to

fix the notation to be used in the sequel. More specifically, we will give the notion of L- functions, sketch the theory of modular forms and their L-functions and then summarize some basic facts on Galois representations and Artin L-functions. Readers who are acquainted with these subjects are urged to skip this chapter.

2.1 L-functions

The theory of L-functions started with the consideration of the Riemann zeta function and then of the Dirichlet series attached to Dirichlet characters.

Let N ∈ Z+. Recall that a Dirichlet character mod N is a mapping χ : Z → C induced from a homomorphism χ0 :(Z/NZ)× → C× by lifting first to integers relative prime to N and then to Z by filling with 0’s elsewhere. We call N the modulus of χ. Note that the correspondence χ0 → χ is bijective.

Suppose M ∈ Z+, M | N and χ is a Dirichlet character mod M induced from χ0.

By composing with the projection (Z/NZ)× → (Z/MZ)×, we can obtain a character ψ0, hence a Dirichlet character ψ mod N. Define the conductor cχ of χ to be the smallest

7 Chapter 2. L-functions 8 positive integer m such that some Dirichlet character mod m induces χ. If a Dirichlet

character χ mod N has conductor N, we say that χ is primitive.

Definition 2.1.1 The Riemann zeta function is defined to be

∞ X 1 ζ(s) = , ns n=1

and for a Dirichlet character χ, the associated Dirichlet L-function is defined similarly

as ∞ X χ(n) L(s, χ) = . ns n=1

Both series defining them converge absolutely for Re(s) > 1 and they both have Euler

products Y Y ζ(s) = (1 − p−s)−1,L(s, χ) = (1 − χ(p)p−s)−1. p p Moreover, they also have meromorphic continuations to the s-complex plane and func-

tional equations.

We follow Iwaniec and Sarnak [10] on the definition of general L-functions. Fix a

number field F , denote its prime ideals by p and let N(a) be the absolute norm of a, for

a general ideal a.

Definition 2.1.2 An L-function takes the form of a product of degree m ≥ 1 over all

prime ideals p of F Y L(s) = Lp(s), p where for almost all p the local factors are

m Y −s −1 Lp(s) = (1 − αi(p)(Np) ) , i=1

for some complex numbers αi(p), such that as a function of s, the product converges absolutely for Re(s) > 1. Chapter 2. L-functions 9

Note that if we multiply out the product, we can get the series of the form

X L(s) = c(a)N(a)−s, a6=0

where the sum is over integral ideals a.

The Riemann zeta function and Dirichlet L-functions are degree one L-functions over

Q. Other examples can be constructed from many mathematical objects. For example, given a number field F , its Dedekind zeta function ζF (s) is an L-function. Recall that

0 X −s Y −s −1 ζF (s) = N(a) = (1 − N(p) ) , a p

where P0 is over all non-zero integral ideals of F . In the following sections, we will see

L-functions constructed from modular forms and Galois representations and We shall

also consider zeta functions for quaternion algebras in Chapter 3.

2.2 Modular Forms and L-functions

Now let us recall the concept of modular forms and their attached L-functions. For our

purpose, we shall focus on the case of congruence subgroups Γ0(N). For modular forms

with respect to SL2(Z), see Serre’s famous short book [30] and refer to Miyake’s book [18] or Shimura’s book [32] in the case of congruence subgroups or general groups.

2.2.1 Modular Forms

+ Let H = {z ∈ C : Imz > 0} be the upper half plane and GL2(R) be the group of

+ 2 × 2 real matrices with positive determinant. Define the action of GL2(R) on H by fractional linear transformations, namely   a b aγz + bγ γ γ + γ(z) = , if z ∈ H, γ =   ∈ GL2(R) . cγz + dγ   cγ dγ Chapter 2. L-functions 10

+ For the weight k action of GL2(R) on functions on H, we use the traditional notation

k/2 −k (f|kγ)(z) = det(γ) (cγz + dγ) f(γ(z)).

For an integer N, the following congruence subgroups are defined:

Γ0(N) = {γ ∈ SL2(Z): cγ ≡ 0 mod N},

Γ1(N) = {γ ∈ Γ0(N): aγ ≡ dγ ≡ 1 mod N},

Γ(N) = {γ ∈ SL2(Z): γ ≡ I2×2 mod N}.

And generally, a congruence subgroup is a subgroup of SL2(Z) which contains Γ(N) for some N.

Definition 2.2.1 Let Γ ⊃ Γ(N) be a congruence subgroup. A holomorphic function f : H → C is called a modular form of weight k (∈ Z) with respect to Γ if it satisfies

1. Automorphy condition: (f|kγ)(z) = f(z) for any γ ∈ Γ;

2. Regularity at cusps: f is regular at all cusps z ∈ Q ∪ i∞.

Roughly speaking, the regularity condition above is the same as to say that for all γ ∈

1/N SL2(Z), f|kγ admits a Fourier expansion over only non-negative powers of q with q = e(z) = exp(2πiz).

A modular form is called a cusp form or cuspidal if it vanishes at all cusps, namely the above Fourier expansions contain only positive powers of q1/N .

The complex vector space of all modular (resp. cusp) forms of weight k with respect to Γ is denoted by Mk(Γ) (resp. Sk(Γ)). The space Sk(Γ) can be made into an inner product space by introducing the Petersson inner product, which is defined as

Z 1 k dxdy hf, gi = f(z)g(z)y 2 , [Γ(1) : Γ(N)] Γ(N)\H y Chapter 2. L-functions 11 where R means integration over a fundamental domain for Γ(N)\ . Recall that Γ(N)\H H a fundamental domain for Γ\H is a open connected subset of H such that distinct points represent distinct orbits in Γ\H and its closure represents all the orbits. Let χ be a Dirichlet character mod N and put

Mk(N, χ) = {f ∈ Mk(Γ1(N)) : f|kγ = χ(dγ)f for all γ ∈ Γ0(N)}, and

Sk(N, χ) = Mk(N, χ) ∩ Sk(Γ1(N)).

We then say forms in Mk(N, χ) are of level N and central character χ. One has the following decomposition

M M Mk(Γ1(N)) = Mk(N, χ), Sk(Γ1(N)) = Sk(N, χ), χmodN χmodN where both χ run through all Dirichlet characters mod N.

2.2.2 The Theory of Hecke Operators

For Γ = Γ(1) = SL2(Z), recall that the discriminant modular form ∆(z) is the cusp form of smallest weight k = 12. It is defined as

∞ ∞ Y X ∆(z) = q (1 − qn)24 = τ(n)qn. n=1 n=1 The Fourier coefficients τ(n), known as the Ramanujan tau function, have the follow- ing multiplicative properties:

τ(mn) = τ(m)τ(n), for (m, n) = 1,

τ(pr+1) = τ(p)τ(pr) − p11τ(pr−1), p a prime number, r ≥ 1.

This is true in general by the theory of Hecke operators. We sketch now the theory for the space Sk(N, χ), following Miyake [18] or Shimura [32]. Define first

∆0(N) := {γ ∈ M2(Z): cγ ≡ 0 modN, (aγ,N) = 1, det(γ) > 0} . Chapter 2. L-functions 12

` For any α ∈ ∆0(N), assume Γ0(N)αΓ0(N) = ν Γ0(N)αν and this double coset acts on

Sk(N, χ) via

k/2−1 X f |k Γ0(N)αΓ0(N) = det(α) χ(αν)f |k αν. ν

+ For any n ∈ Z , define the operator T (n) on Sk(N, χ) by

X T (n) = Γ0(N)αΓ0(N), det(α)=n where the summation is taken over all double cosets Γ0(N)αΓ0(N) with α ∈ ∆0(N) and det(α) = n.

We collect the main results in the following theorem:

Theorem 2.2.2 (1) If (m,n)=1, then T (m)T (n) = T (mn);

(2)   T (pr+1) + pk−1T (pr−1) if p - N, T (p)T (pr) = ;  T (pr+1) if p | N

(3) As operators on Sk(N, χ), T (n) with (n, N) = 1 are normal and generate a com- mutative algebra;

(4) The space Sk(N, χ) has a basis consisting of common eigenfunctions of all T (n) with (n, N) = 1.

P∞ n (5) Assume f(z) = n=0 a(n)q ∈ Sk(N, χ) is a common eigenfunction of all T (n)

with (n, N) = 1 and f|kT (n) = λ(n)f with λ(n) ∈ C. Then a(n) = λ(n)a(1).

So the Fourier coefficients of a common eigenfunction satisfy analogous multiplicative properties as ∆(z) does. In (3), (4) or (5) of the above theorem, we cannot (in general) eliminate the condition (n, N) = 1. However, on a subspace, we are able to do it. This is the theory of newforms which we shall explain in next subsection. Chapter 2. L-functions 13

2.2.3 The Theory of Newforms

The theory of newforms was established by Atkin and Lehner ([2]).

Given a function f, denote by f d the function f d(z) = f(dz).

Let M,N ∈ Z+ be such that M | N and let χ be a Dirichlet character mod M. It is

easy to see that χ is also defined mod N and Sk(M, χ) ⊂ Sk(N, χ). In general, for any

N d positive d | M and f ∈ Sk(M, χ), f ∈ Sk(N, χ). As usual, let cχ be the conductor of χ.

old Definition 2.2.3 The subspace of oldforms in Sk(N, χ), denoted by Sk (N, χ), is defined to be the subspace spanned by the set

[ [ d {f : f ∈ Sk(M, χ)} M d

where M runs over all positive integers such that cχ|M, M|N, M 6= N and d runs over all

N positive divisors of M . Then the subspace of newforms is defined to be its orthogonal new complement with respect to the Petersson inner product. We denote it by Sk (N, χ).

new By convention, we shall reserve the notion newforms for special elements in Sk (N, χ).

old new Lemma 2.2.4 (1) Sk (N, χ) and Sk (N, χ) both are stable under Hecke operators T (n) ((n, N) = 1);

P∞ n new (2) Let f(z) = n=1 anq ∈ Sk (N, χ) be a common eigenfunction of T (n) for all n

relatively prime to some integer L, then a1 6= 0.

P∞ n new Definition 2.2.5 We call an element f(z) = n=1 anq ∈ Sk (N, χ) a newform if

f(z) is a common eigenfunction of all T (n) with (n, N) = 1 and a1 = 1.

P∞ n new Theorem 2.2.6 (1) A newform f(z) = n=1 a(n)q ∈ Sk (N, χ) is a common eigenfunctions for all T (n) with eigenvalues a(n);

new (2) Sk (N, χ) has a basis consisting of newforms. Chapter 2. L-functions 14

2.2.4 L-functions Attached to Modular Forms

P∞ n Let f(z) = n=0 a(n)q ∈ Mk(N, χ) and define formally the Dirichlet series ∞ X L(s, f) = a(n)n−s. n=1

Denote by wN the following matrix   0 −1     . N 0

Since wN normalizes Γ0(N), it is not hard to see that f → f|kwN induces the isomor- phisms

Mk(N, χ) 'Mk(N, χ), Sk(N, χ) 'Sk(N, χ),

new new old old Sk (N, χ) 'Sk (N, χ), Sk (N, χ) 'Sk (N, χ). √ −s P∞ n Now define ΛN (s, f) = (2π/ N) Γ(s)L(s, f) and assume f|kwN (z) = n=0 b(n)q .

Theorem 2.2.7 (Hecke) The function ΛN (s, f) can be meromorphically continued to the whole s-plane, satisfy the functional equation

k ΛN (s, f) = i ΛN (k − s, f|kwN ),

and the function a(0) ikb(0) Λ (s, f) + + N s k − s is holomorphic on the whole s-plane and bounded on any vertical strip.

P∞ n new Theorem 2.2.8 Assume f(z) = n=1 a(n)q is a newform in Sk (N, χ). (1) L(s, f) has a Euler product and Y L(s, f) = (1 − a(p)p−s + χ(p)pk−1−2s)−1; p

new (2) g(z) = f(−z) is a newform in Sk (N, χ) and

f|kwN (z) = cg(z), for some constant c ∈ C.

Given a modular form f, a newform or not, we shall call L(s, f) the L-function

associated to f. Chapter 2. L-functions 15

2.2.5 Automorphic Representations and L-functions

The theory of automorphic forms or representations gives a natural generalization of the

theory of modular forms. In this subsection, we shall briefly mention this theory with

little details. See Cogdell, Kim and Murty’s lecture notes [5] for a reference.

Let F be a number field and OF its ring of integers. As usual, Fv denotes the completion of F with respect to a place v. For a non-Archimedean place v, let Ov be the ring of integers in Fv, $v a uniformizer and qv the order of the corresponding residue

field. Denote by A = AF the adele ring of F and for simplicity, we will assume G = GLn.

We know that GLn(A) is the direct limit of the topological groups Y Y GLn(A)S = GLn(Fv) × GLn(Ov), v∈S v∈ /S where S ranges over all finite sets of valuations of F containing all of the Archimedean valuations. Denote by K the standard maximal compact subgroup of GLn(A), i.e., Y Y Y K = U(n, C) × O(n, R) × GLn(Ov), v complex v real v discrete and let Kv be the corresponding local factor of K. (see [36] for details.)

For a fixed unitary central character ω : F ×\A× → C×, define

2 2 L (ω) = L (GLn(F )\GLn(A), ω)

to be the Hilbert space of square integrable functions ϕ modulo the center, i.e., ϕ(zg) =

ω(z)ϕ(g) for z ∈ Z(A) and Z |ϕ(g)|2dg < ∞. Z(A)GLn(F )\GLn(A)

Here Z is the center of GLn and dg is the Haar measure.

2 Denote by R, the right regular representation of GLn(A) on L (ω), i.e., (R(h)ϕ)(g) =

2 ϕ(gh) for g, h ∈ GLn(A) and ϕ ∈ L (ω). Define

2 2 L0(ω) = L0(GLn(F )\GLn(A), ω)      Z I x  2 r  = ϕ ∈ L (ω) ϕ  g dx = 0 , for all 1 ≤ r < n . r(n−r)  (A/F ) 0 In−r  Chapter 2. L-functions 16

2 This is a closed R-invariant subspace and denote the action on L0(ω) also by R.

Definition 2.2.9 An automorphic representation of GLn(A) is an irreducible con-

2 stituent of R on L (ω) for some ω and a cuspidal representation of GLn(A) is an

2 irreducible constituent of R on L0(ω) for some ω.

Theorem 2.2.10 Let π be a cuspidal representation of GLn(A). Then there is a re- stricted tensor product decomposition

0 π = ⊗v πv

into irreducible unitary representations of the local groups GLn(Fv). Hence for almost all the finite places v, πv has a GLn(Ov)-fixed vector.

A representation πv of GLn(Fv) is called spherical if it has a GLn(Ov)-fixed vector.

Assume that πv is an irreducible admissible spherical representation of GLn(Fv). Then it is known that

π = IndGLn(Fv)(µ ⊗ · · · ⊗ µ ), v B(Fv) 1,v n,v i.e., the induced represenation from the Borel subgroup B(Fv) of n uniquely determined

× unramified characters µi,v of Fv . Thus the set of n complex numbers

{µ1,v($v), ··· , µn,v($v)}

completely determines πv, since µi,v’s are unramified.

Definition 2.2.11 For a finite v where πv is spherical, we define the local L-factor to be

n Y −s −1 Lv(s, π) = L(s, πv) = (1 − µi,v($v)qv ) . i=1 Let S be a finite set of places containing all the Archimedean places and all the non- spherical places. We define the partial L-function as

Y LS(s, π) = Lv(s, π). v∈ /S Chapter 2. L-functions 17

Note that with more efforts we can define local L-factors L(s, πv) for all the places, hence get a completed L-function L(s, π) by formally taking the infinite product of all the local L-factors. Since the partial L-function suffices our purpose in this thesis, we will skip this.

2.3 Galois Representations and Artin L-functions

In general, there are three types of Galois representations, namely, mod p, p-adic and complex Galois representations, corresponding to representation spaces over finite fields, p-adic fields and C, respectively. But we note here that in the sequel, we only consider complex Galois representations.

Let F be a number field and F = Q be one algebraic closure. With the Krull topology, the absolute Galois group Gal(F /F ) is a compact and totally disconnected topological

group.

Now let (ρ, V ) be a (complex) n-dimensional Galois representation, namely, a continuous homomorphism ρ : Gal(F /F ) → GL(V ). Note that the continuity is the same as to say that Im(ρ) is finite, for there is no nontrivial group contained in a sufficiently small neighborhood of the identity in GLn(C). Denote the determinant character

det(ρ) by . In the case of F = Q and n = 2, we call ρ odd if (c) = −1, where c is the complex conjugation in Gal(Q/Q) and even if (c) = 1. Let L/F be a finite Galois extension such that Gal(F /F ) ⊂ ker(ρ), so we can factor

ρ through Gal(L/F ). Take any prime ideal p of F and Gal(L/F ) permutes all the prime

ideals lying above p. Let P be a prime ideal of L lying above p and the decomposition

group DP for P | p is defined to be the subgroup of Gal(L/F ) that stabilizes P. Moreover

the inertia group IP for P | p is the subgroup of DP whose elements induce the identity automorphism on the residue field, i.e.,

IP = {σ ∈ DP : σx ≡ x mod P for all x ∈ OL}. Chapter 2. L-functions 18

Similarly we define the ramification groups Gi = GP,i (i ≥ 0) by

i+1 Gi = {σ ∈ G0 : σx ≡ x mod P for all x ∈ OL}.

In particular, G0 = IP.

We know that DP/IP ' Gal(kL,P/kF,p), where the right-hand side is the cyclic Galois group of the corresponding residue fields extension generated by the Frobenius automor- phism x 7→ xN(P).

Now for any p, take any P lying above p and any element σP ∈ DP that projects to

IP the Frobenius element. Let V be the subspace of V stabilized by IP. Then one can

IP show that the characteristic polynomial of σP on V is independent of the above choices of L, P and then σP, so the following definition is well-defined.

Definition 2.3.1 Define the local factor of the Artin L-function attached to ρ by

−s −1 Lp(s, ρ) = detV IP (I − ρ(σP)N(p) ) ;

and the Artin L-funtion attached to ρ is

Y L(s, ρ) = Lp(s, ρ). p

Note that L(s, ρ) is defined on Re(s) > 1. If χρ is the character corresponding to

the Galois representation ρ of the finite group Gal(L/F ), we shall also write L(s, χρ) = L(s, ρ). We summarize some fundamental properties of Artin L-functions.

Theorem 2.3.2 (1) If ρ1 and ρ2 are two Galois representations over F , then

L(s, ρ1 ⊕ ρ2) = L(s, ρ1)L(s, ρ2);

(2) Let ρ be a Galois representation over L with L/F an arbitrary finite extension

F and let IndL (ρ) be the induced Galois representation over F . Then

F L(s, ρ) = L(s, IndL (ρ)). Chapter 2. L-functions 19

Let us recall the notion of Artin conductor. Fix a Galois representation (ρ, V ), a prime ideal p of F and a prime ideal P of L above p. Let us define

∞ X gi n(ρ, p) = codim(V Gi ), g i=0 0

where gi is the order of Gi, i ≥ 0. It is easy to see that the summation is finite and n(ρ, p) is independent of the choice of P. Artin proved that this rational number n(ρ, p)

is actually an integer. Note also that if p is unramified in L/F , n(ρ, p) = 0. These facts

allow us to define the Artin conductor f(ρ) of ρ as follows:

Y f(ρ) = pn(ρ,p). p

In the case of F = Q, we also call the positive generator of f(ρ) the Artin conductor. We end this section with the functional equation of L(s, ρ). We need to complete the

L-function by adding some γ-factors.

Put γ(s) = π−s/2Γ(s/2). For any complex place v of F , define

v dim V γρ (s) = (γ(s)γ(s + 1)) .

Consider a real place v. Take any place w of L lying above v and the decomposition

+ − group Dw is of order 1 or 2. Decompose V = Vv ⊕ Vv corresponding the the eigenvalues

1 and −1 of ρ(σw) and we put

+ − v dim Vv dim Vv γρ (s) = γ(s) γ(s + 1) .

v Q v We note here that γρ (s) does not depend on the choice of w. Finally let γρ(s) = v γρ (s), where the product is over all the infinite places of F .

dim V Let A(ρ) = |dF | NF/Q(f(ρ)) where dF is the absolute discriminant of F . Define

s/2 Λ(s, ρ) = A(ρ) γρ(s)L(s, ρ). Then we have the following theorem:

Theorem 2.3.3 The function Λ(s, ρ) has a meromorphic continuation to the whole com- plex plane and satisfies the functional equation

Λ(1 − s, ρ) = W (ρ)Λ(s, ρ), Chapter 2. L-functions 20 where ρ is the dual (or contragredient) representation of ρ and W (ρ) is a constant of absolute value 1.

The W (ρ) in the above theorem is known as the Artin root number. Chapter 3

Divisor Function over Quaternion

Algebras

As a preparation for the next chapter, we will define and investigate the divisor functions for quaternion algebras. In this chapter and the next, except some slight generalizations, we will mainly follow the joint paper [13] with H.H.Kim.

3.1 Introduction

We review the basic knowledge and some important facts on quaternion algebras in this section. See Reiner’s book [29] or Weil’s book [36] for a reference on the general theory.

For the theory over Q, see also the introduction of Pizer’s paper [24].

3.1.1 General Notions

Let K be a finite extension of Q or Qv for some place v of Q. A quaternion algebra A over K is a central simple algebra of dimension 4 over K, i.e., a 4-dimensional K-algebra with center K and no nontrivial double-sided ideals.

Example. Let a, b ∈ K× be arbitrary and define A as the vector space spanned by

21 Chapter 3. Divisor Function 22

{1, i, j, k} whose ring structure is subject to i2 = a, j2 = b and ij = −ji = k. Then A is a quaternion algebra over K, denoted by (a, b)K . Actually, it is known that any quaternion algebra over K is isomorphic to one of (a, b)K . Note that (−1, −1)R is the Hamiltonian quaternion algebra.

Assume K is not R or C and denote by S its ring of integers. A full S-lattice M in A is a finitely generated S-submodule which contains a K-basis of A. We will just say lattice for short, since we will only deal with full S-lattices. The notion ideal will share the same definition as lattice, which should cause no confusion since we are not interested in ideals of A itself. Indeed, unless A ' M2(K), there are no non-trivial ideals.

An order O in A is a subring of A with the same identity element such that it is also a lattice. An order is called maximal if it is not contained in any other order.

For a lattice M, define the left order Ol(M) and the right order Or(M) of M as follows:

Ol(M) = {α ∈ A : αM ⊂ M}, Or(M) = {α ∈ A : Mα ⊂ M}.

It is easy to check that they are orders. Given an order O and a lattice M, we call

1 M a left O-ideal if O = Ol(M). The obvious analogous definitions hold for right

O-ideal and two-sided O-ideal. An integral ideal a is a lattice such that a ⊂ Ol(a) or equivalently a ⊂ Or(a). An ideal a is normal, if Ol(a) is maximal or equivalently

Or(a) is maximal. An integral ideal a is a maximal ideal if a is a maximal left ideal in

Ol(a), or equivalently, a is a maximal right ideal in Or(a). An ideal is called principal

× if a = Ol(a)α or equivalently a = αOr(a) for some α ∈ A . If M,N are two lattices of A, define the product MN as

( n ) X + MN = αiβi : αi ∈ M, βi ∈ N, n ∈ Z , i=1

1 In the literature, a different definition of left O-ideal is given by O ⊂ Ol(M). However, if O is maximal, they coincide. Chapter 3. Divisor Function 23

which is certainly a lattice. The product MN is called proper if Or(M) = Ol(N). If

a, b are two normal ideals, then ab is normal with Ol(ab) = Ol(a) and Or(ab) = Or(b). A proper product of integral normal ideals is again integral.

For a lattice M in A, define the inverse of M as

M −1 = {α ∈ A : MαM ⊂ M}.

−1 −1 Then M is a lattice. If a is a normal ideal, a is also normal, with left order Or(a)

and right order Ol(a); moreover if b is also normal and ab is proper,

−1 −1 −1 −1 −1 −1 −1 aa = Ol(a), a a = Or(a), (a ) = a, (ab) = b a .

As a consequence, the cancellation law holds for proper products of normal ideals.

Let a, b be two ideals. We call b divides a from the left, denoted b |l a, if there exists an integral ideal c such that bc is proper and equal to a. If a, b are normal of the

same left order, then a ⊂ b if and only if b |l a. The divisibility from the right is similarly defined.

Let α ∈ A be any element and α is algebraic over K whose minimal polynomial

2 mα(X) over K is of degree 2 if α∈ / K. Assume mα(X) = X − tX + n with t, n ∈ K.

We call the other root of mα(X) as the conjugate of α, denoted by α, and define the reduce trace T r(α) = α + α = t and the reduced norm N(α) = αα = n. In the case

2 of α ∈ K, N(α) = α and T r(α) = 2α. Explicitly, in (a, b)K , let α = x + yi + zj + wk be any element. We have α = x − yi − zj − wk, and

T r(α) = 2x, N(α) = x2 − ay2 − bz2 + abw2.

The quadratic form

q(X,Y,Z,W ) = X2 − aY 2 − bZ2 + abW 2

is called the norm form of (a, b)K . If M is a lattice, define its conjugate M = {α :

α ∈ M}. Thus, Ol(M) = Or(M) and Or(M) = Ol(M). Chapter 3. Divisor Function 24

Define the reduced norm of a lattice M, N(M), be to the (fractional) ideal in K generated by the set {N(α): α ∈ M}. For any two normal ideals a, b with ab proper, it is true that

N(ab) = N(a)N(b),N(a−1) = N(a)−1,N(a) = N(a).

If a is an integral normal ideal of A, N(a) is an integral ideal of K.

Example. If O is a maximal order and α ∈ A×, then α−1Oα is a maximal order and

Oα is a normal ideal. Specifically, Oα is a left O-ideal and a right α−1Oα-ideal with

N(Oα) = SN(α). Moreover, Oα is integral if and only if α ∈ O.

3.1.2 Local and Global Theory

From now on till the end of this chapter, we fix a number field F and a quaternion algebra

A over F . To avoid confusion with orders in A, we use R = RF to denote the ring of

integers of F . If v is a place of F , then Fv is the completion of F at v and if v is finite,

Rv is the ring of integers in Fv. Set Av = A ⊗F Fv and Av is a quaternion algebra over

Fv. For a finite place v = vp, when there is no danger of confusion, we will also denote

Rv ∩ F by Rv and the maximal ideal in Rp, namely pRp, by p. Let N(p) = NF/Q(p) denote the cardinality of R/p.

Let us first see the classification of quaternion algebras. Locally, if Fv is C, then there

is a unique quaternion algebra over Fv up to isomorphism, namely the two-by-two matrix

algebra M2(Fv). If Fv is R or p-adic, there are precisely two quaternion algebras over Fv

up to isomorphism, namely, the two-by-two matrix algebra M2(Fv) and a unique division quaternion algebra. This division quaternion algebra can be constructed explicitly as

follows. If Fv = R, the division algebra is given by     α β  (−1, −1) '   : α, β ∈ . R   C  −β α 

If Fv is a p-adic field, let L/Fv be the unique unramified field extension of degree 2.

Assume Gal(L/Fv) = hσi and let $ be a uniformizer of Fv. Then the division quaternion Chapter 3. Divisor Function 25

algebra over Fv is isomorphic to the subalgebra of M2(L) given by     α β      : α, β ∈ L ,  $βσ ασ  where ασ = σ(α).

Globally, for a quaternion algebra A over F , we say that A is unramified or splits at a place v if Av is isomorphic to M2(Fv), and ramified otherwise. From the theory of Hilbert symbols, we know that the number of ramified places of A over F must be finite and even. Conversely, given any finite set S of real or finite places of F of even cardinality, there is a unique quaternion algebra, up to isomorphism, that ramifies precisely at places in S. So we obtain a one-to-one correspondence between the set of isomorphism classes of A over F and the set of even finite sets of real or finite places of F .

For a lattice M in A and a finite place p of F , denote Mp = M ⊗R Rp, called the localization of M at p. Obviously Mp is a lattice in Ap. Moreover, given any two lattices

M,N, we have Mp = Np for almost all p. There is a local-global correspondence in the context of lattices; namely, if M is a lattice of A and N(p) is a lattice in Ap for every p such that for almost all p, Mp = N(p), then there exists a unique lattice N of A such that T Np = N(p). Actually N = p(A ∩ N(p)). (cf. [36, Theorem 2] on page 84.) Replacing lattice with order, we obtain the local-global correspondence for orders.

If M is a lattice, then

Ol(Mp) = Ol(M)p, Or(Mp) = Or(M)p.

Q If a is a normal ideal, N(ap) = N(a)p, hence N(a) = p N(ap). If b is also normal, we have (ab)p = apbp and ab is proper if and only if apbp is proper for any p.

Any order of A or Ap is contained in a maximal order. An order O of A is maximal if and only if Op is maximal for all p. Locally, if Ap is a division algebra, there is a unique maximal order; if, on the other hand, Ap = M2(Fp), all maximal orders are conjugate to

× M2(Rp) by elements in Ap . Globally, there are non-conjugate maximal orders; actually the number of conjugacy classes of maximal orders is called the type number of A. Chapter 3. Divisor Function 26

Locally, let Op be a maximal order in Ap. All the one-sided ideals of Op are principal, all maximal ideals have norm p; there is a unique maximal two-sided ideal mp and mp generates the set of two-sided Op-ideals as an infinite cyclic group. Specifically, if Ap =

M2(Fp) and Op = M2(Rp), then mp = Op$; if, on the other hand, Ap is the division algebra constructed above, and Op is the unique maximal order, that is,     α β    Op =   : α, β ∈ RL ,  $βσ ασ  then   0 1 m   p = Op   . $p 0

It follows that N(mp) = p and all one-sided ideals of Op are also double-sided, hence integer powers of mp. Let us consider the global situation now. For convenience, let us introduce the adeles Q0 and ideles over A. The ring of adeles AA = v Av, the restricted product of Av over all the places of F with respect to {Op} for any given order O. Similarly, the group of ideles A× = Q0 A× is the restricted product of A× with respect to {O×}. They are A v v v p independent on the order O. Denote the finite part of the adeles and ideles by AA,f and A× , respectively. Now if a is left O-ideal, then there exists α ∈ A× for each p, such that A,f p p a = O α and for almost all p, α ∈ O×. So α = (α ) ∈ A× . By abuse of language, we p p p p p p A,f write a = Oα. If a = Oα and b = O0β with α, β ∈ A× , then ab is proper if and only if A,f O0 = α−1Oα, in which case ab = Oαβ.

Two left O-ideals a, b are called left-equivalent, if there exists α ∈ A× such that a = bα, denoted a ∼l b. The equivalence relation ∼r for right O-ideals is similarly defined. For an order O, the map a → a−1 induces a bijection between the set of left O- ideal classes and the set of right O-ideal classes. We define the class number of O as the cardinality of the set of left or right O-ideal classes, denoted by H(O). Maximal orders have the same class number and call this the class number of A, denoted by H = H(A). Chapter 3. Divisor Function 27

3.2 Divisor Function over Quaternion Algebras

We fix a number field F and a quaternion algebra A over F . Let all the notations be the

same as those in the introduction of this chapter.

Hereafter, all ideals of A will be normal integral ideals and all products of ideals will

be proper unless otherwise stated.

Fix any maximal order O in A.

Definition 3.2.1 Define the divisor function d : {integral ideals of A} → Z+ by

d(a) = |{(b, c): a = bc}|.

For any non-zero integral ideal n in R, put

A(n) = {a : O = Ol(a),N(a) = n},

and define a(n) = |A(n)|.

Lemma 3.2.2 Assume N(a) = nm with n, m relatively prime. Then there exists a unique

pair (b, c) of integral ideals such that a = bc, N(b) = n and N(c) = m.

Proof. The existence follows trivially from the theorem in (22.28) of [29]. The theorem states that for any preassigned order in the factorization of N(a) into product of prime ideals, there exists a corresponding factorization of a into maximal ideals. Note that such a factorization is not unique in general.

For the uniqueness, assume a = b1c1 = b2c2. Because of the local-global correpon- dence for lattices, it suffices to show it locally, that is, b1,p = b2,p for all p. This is true, since for p | n both are equal to ap and for p - n both are trivial. Done. 2

Proposition 3.2.3 Both of the functions d and a are multiplicative, in the sense that if

n, m are relatively prime, then a(nm) = a(n)a(m), and if bc is proper and N(b), N(c)

are relatively prime, then d(bc) = d(b)d(c). Chapter 3. Divisor Function 28

Proof. For the function a, let us define a bijective function φ : A(nm) → A(n) × A(m) as

follows. Given a ∈ A(nm), by Lemma 3.2.2, we can find a unique pair (b, c) of integral

ideals such that bc is proper and equal to a and N(b) = n, N(c) = m. Assume b = Oβ and

c = (β−1Oβ)γ for some β, γ ∈ A× . Then define φ(a) = (Oβ, Oβγβ−1) ∈ A(n) × A(m). A,f It is easy to see that φ is surjective and the injectivity follows from the uniqueness in

Lemma 3.2.2.

For the divisor function, assume N(b) = n and N(c) = m. Let

A = {(a1, a2): a = a1a2}

and similarly B and C be the sets of decompositions for b and c, respectively.

Now we define a map φ : B × C → A as follows. For any pair of decompositions b =

b1b2 and c = c1c2, we have N(b2),N(c1) are relatively prime, and by Lemma 3.2.2, there

0 0 0 0 0 0 exist unique ideals b2 and c1 such that b2c1 = b2c1 and N(b2) = N(c1), N(c1) = N(b2). Now

0 0 a = bc = b1b2c1c2 = (b1b2)(c1c2)

gives us a decomposition of a.

Suppose b = b1b2 = d1d2 and c = c1c2 = e1e2 give the same decompositions of a.

0 0 0 0 0 0 Then we have b1b2 = d1d2 and c1c2 = e1e2. So N(b1)N(b2) = N(d1)N(d2) which, by

construction, is the same as N(b1)N(c1) = N(d1)N(e1). But since N(b) and N(c) are

0 0 relatively prime, we have N(b1) = N(d1). Now b1b2 = d1d2 implies, by Lemma 3.2.2,

that b1 = d1. Similarly we obtain c2 = e2. So we get the injectivity.

For the surjectivity of φ, let a = a1a2 be any decomposition. By Lemma 3.2.2, there

exist unique decompositions a1 = b1c1 and a2 = b2c2 where in N(b1) and N(b2) only

prime divisors of n appear and in N(c1) and N(c2) only prime divisors of m appear.

0 0 0 0 Let b2 and c1 be the unique ideals that b2c1 = c1b2. Then it is obvious that (a1, a2) =

0 0 φ((b1, b2), (c1, c2)). Done. 2 Proposition 3.2.3 is the key to work on the functions d and a and it says we only need Chapter 3. Divisor Function 29

to proceed locally.

Fix any finite place p where A ramifies, that is, Ap is the division algebra over Fp.

Let L/Fp be the unique unramified quadratic extension of Fp, $p be a uniformizer of p and σ be the generator of Gal(L/Fp). We have     a b  Ap '   : a, b ∈ L .  σ σ   $pb a 

Without loss of generality, we assume the equality in the above isomorphism, so we have     a b  Op =   : a, b ∈ RL ,  σ σ   $pb a 

where RL is the ring of integers in L.

Proposition 3.2.4 For any n ∈ N, a(pn) = 1, that is, there exists a unique integral ideal

n n n of norm p , which is given by mp . As a consequence, if a has norm p , then d(a) = n+1.

Proof. The statement on the divisor function d follows trivially from that on a. Note

that there exists a unique maximal left Op-ideal, mp, which is also a two-sided normal

ideal. So any integral proper left O-ideal is contained in, hence divisible by mp. This

n implies that all integral ideals take the form mp with n ∈ N and the proposition follows

since N(mp) = p. 2

Suppose now A splits at p, so Ap ' M(2,Fp), the two-by-two matrix algebra over Fp. Without loss of generality we assume that they are equal. One of the maximal orders is

× Op = M(2,Rp) and all others are conjugate to this one by elements in Ap . An integral

left Op-ideal a is primitive if Opp does not divide a from the left.

n Let us determine a(p ) using Op = M(2,Rp).

n Lemma 3.2.5 (1) All integral left Op-ideals of norm p are      $l b   p n−l  Op   : b ∈ Rp/p , 0 ≤ l ≤ n ;  n−l   0 $p  Chapter 3. Divisor Function 30 hence a(pn) = (N(p)n+1 − 1)/(N(p) − 1).

n (2) All primitive left Op-ideals of norm p are         1 b   $n 0   n [  p  Op   : b ∈ Rp/p Op    n     0 $p   0 1       $l b  [  p n−l ×  Op   : b ∈ (Rp/p ) , 1 ≤ l ≤ n − 1 ;  n−l   0 $p  the total number of them are N(p)n−1(N(p) + 1).

n In this lemma, b ∈ Rp/p means that b suns over a fixed complete representative set for

n n × Rp/p and similarly b ∈ (Rp/p ) means that b suns over a fixed complete representative

n × set for the group (Rp/p ) .

Proof. The second part follows from the first part trivially by noting that a = Opα is

× primitive if and only if at least one of the four entries of α is in Rp .

n Let a = Opα be any integral ideal of norm p and assume   a b   α =   , a, b, c, d ∈ Rp. c d

If vp(a) ≥ vp(c), then let   0 1 β =   ∈ O×   p 1 −c−1a and βα has zero left lower element. If on the other hand vp(a) < vp(c), then let   1 0 β =   ∈ O×   p −a−1c 1 and βα also has zero left lower element. So we may assume c = 0. Assume vp(a) = l and hence vp(d) = n − l. Let   a−1$l 0 p × β =   ∈ Op  −1 n−l  0 d $p Chapter 3. Divisor Function 31

l n−l l and βα has diagonal elements $p and $p respectively. We can assume a = $p and

n−l n−l n−l n−l d = $p . There is a unique x mod p such that b − x = −y$p ∈ p with y ∈ Rp. Let     1 y $l x × p β =   ∈ Op and βα =   .    n−l  0 1 0 $p Now it suffices to show that all ideals in (1) are distinct. Suppose       a b $l b $l0 b0 1 1 × p p β =   ∈ Op and β   =   .    n−l   n−l0  c1 d1 0 $p 0 $p

× Comparing the left lower elements gives us c1 = 0, hence a1, d1 ∈ Rp . Now equalities

0 of the diagonal elements imply that a1 = d1 = 1 and l = l . Finally the right upper elements equal and implies b ≡ b0 mod pn−l; so by the choices of b and b0, they must be the same. Done.2

Lemma 3.2.6 (1) The maximal left Op-ideals are precisely the norm p ideals;

(2) Any primitive left Op-ideal a has a unique decomposition into a proper product of maximal ideals; hence if N(a) = pm, d(a) = m + 1.

Proof. As before, assume the left order of a is M(2,Rp). We claim that a is left divisible by a unique norm p ideal; indeed, using the explicit generators in Lemma 3.2.5 and

passing to right divisibility of corresponding generators, this is easy to verify directly.

We will skip this calculation. As a consequence, the norm p ideals are precisely the

maximal ideals. The uniqueness of the lemma follows from the cancelation law. Done.

2

Lemma 3.2.7 Suppose N(a) is a p-power. Assume a = bc with b and c primitive but a non-primitive. If b = br ··· b1 and c = c1 ··· cs are decompositions into maximal ideals, then b1c1 = p, i.e., b1 = c1.

Here the p in b1c1 = p actually means the two-sided ideal in Ol(b1) generated by p. Chapter 3. Divisor Function 32

Proof. Let us first prove the lemma in the case of s = 1, that is, c is maximal, hence

primitive. First note that in the matrix algebra Op, α is nothing but the adjoint matrix of α. Then from Lemma 3.2.5, we obtain that cc = p. Now since a is not primitive, there

exists a normal integral ideal a1 such that a = pa1 = bc. Multiply the ideal c on the right

and we obtain pa1c = pb, hence a1c = b and c |r b. Since b is primitive, by Lemma 3.2.6,

b1 = c and b1c = p.

For the general case, let j be the smallest integer such that br ··· b1c1 ··· cj is not

primitive. So br ··· b1c1 ··· cj−1 is primitive. Apply the result on the case of s = 1, and

we obtain cj|rbr ··· b1c1 ··· cj−1. If j > 1, by the uniqueness in Lemma 3.2.6, cj = cj−1. It contradicts to the fact that c is primitive. So j = 1 and we are done. 2

2n+m n n We say an ideal is of signature (p; n, m) if N(a) = p and p kla. Here p kla means that pn is the highest power of p that divides a. For such an a, we put

c(a) = c(p; n, m) = |{(b, c): a = bc with b, c primitive}|.

We will see in the following lemma that this number only depends on the signature, which justifies the notation we used here. Similarly we will use d(p; n, m) = d(a). See also Lemma 3.2.11 below for the independence on the maximal order O.

Lemma 3.2.8 Let a be of signature (p; n, m).   N(p)n−1((m + 1)(N(p) − 1) + 2) n ≥ 1, c(p; n, m) =  m + 1 n = 0.

Proof. If n = 0, then a is primitive and by Lemma 3.2.6, we can write a = a1 ··· am as a unique decomposition into maximal ideals. So we m + 1 decompositions into product

of two factors and of course all of them are primitive. Hence c(p; 0, m) = m + 1.

If n = 1, then a = pb with b primitive. For any primitive decomposition a = a1a2 =

a1,ra1,r−1 ··· a1,1a2,1a2,2 ··· a2,s, since a1 and a2 are primitive but a is not, by Lemma 3.2.7,

we know that a1,1a2,1 = p. Canceling p from both sides, we get a primitive decomposition Chapter 3. Divisor Function 33

of b. Therefore, this defines a map from the set of all primitive decompositions (into

products of two factors) for a, to the set of all those decompositions for b.

Let us look at this map more closely. From the case n = 0, we know that b has m + 1

decompositions. Let b = b1b2 = b1,r ··· b1,1b2,1 ··· b2,s be any of them with both factors nontrivial and bi,j maximal. There are m − 1 of them. Now we can factor p = b1,0b2,0 and there are N(p) + 1 possible choices by Lemma 3.2.5. For each of them, we produce a1 = b1,r ··· b1,0, a2 = b2,0 ··· b2,s and a = a1a2. But this decomposition is primitive if and only if b1,0 6= b1,1 and b2,0 6= b2,1, hence if and only if b1,0 6= b1,1,b2,1, since b1,0 = b2,0.

That b is primitive implies b1,1 6= b2,1. So among those N(p) + 1 choices of b1,0, only

N(p) − 1 of them give primitive decompositions of a. So if both b1 and b2 are nontrivial,

(b1, b2) has N(p) − 1 preimages, (m − 1)(N(p) − 1) in total. If one of b1 and b2 is trivial, i.e., equals to the left or right order of b, we argue in the same way as above, except that in this case we only have one restriction on the factorization of p. Consequently, either of them has N(p) preimages and hence we have 2N(p) more choices. Finally, we have c(p; 1, m) = (N(p) − 1)(m − 1) + 2N(p) = (N(p) − 1)(m + 1) + 2.

Now we are left to show c(p; n, m) = N(p)c(p; n−1, m) if n > 1. Again let a = pb.Note

here b is not primitive anymore since n > 1. We follow exactly the same argument as

above, except that each decomposition of b will give us exactly N(p) preimages. (For

nontrivial decompositions of b, this is because b is not primitive, which implies b1,1 = b2,1, hence only excludes one choice among N(p) + 1 of them.) So in this case, the above map is N(p)-to-1 and surjective and our conclusion follows. Done. 2

Proposition 3.2.9 Let a be of signature (p; n, m).

N(p)n+1 − 1 2(n + 1) 2(N(p)n+1 − 1) d(a) = d(p; n, m) = (m + 1) − + . N(p) − 1 N(p) − 1 (N(p) − 1)2

n r t Proof. Any decomposition of a = p a1 can be written uniquely as a = (p b)(p c) with b and c primitive and r, t ≥ 0, r + t ≤ n. So it is produced by primitively decomposing Chapter 3. Divisor Function 34 n−r−t p a1, followed by assigning the r + t p-factors. So we have n X d(a) = (k + 1)c(m, n − k). k=0 By Lemma 3.2.8 and elementary calculations, we get the identity for d(a). Done. 2

Let us denote by a(p; n, m) the number of left Op-ideals with signature (p; n, m). Then

Proposition 3.2.10   N(p)m−1(N(p) + 1) m ≥ 1 a(p; n, m) =  1 0 Proof. It is obvious that a(p; 0, m) = a(p; n, m) because of the bijectivity of the map a → pna and the formula for a(p; 0, m) is given by Lemma 3.2.5. Done. 2

Lemma 3.2.11 (1) The divisor function d(a) depends only on the norm and the signa- ture of a at each p;

(2) The functions a(n), a(p; n, m) are independent of the fixed maximal order O.

Proof. Let O0 be another maximal order in A and let c = OO0. It is trivial that c is an ideal and the fact that O and O0 are maximal implies that c is normal with left order O and right order O0. Choose α ∈ A× such that c = Oα, so O0 = α−1Oα. A,f It is not hard to check that the map a → α−1aα induces a bijective map from the set of integral left O-ideals of signature (p; n, m) to the set of integral left O0-ideals of signature (p; n, m), hence a bijective map from the set of integral left O-ideals of norm n to the set of integral left O0-ideals of norm n. The independence of d(a), a(n) and a(p; n, m) on the maximal order O follows from this by easy verifications.

Part (1) follows by Proposition 3.2.9. Done. 2

There should be no danger of confusion between the norm of ideals of F and that of ideals of A. However, we will use NF/Q to denote the norm of ideals of F in the following. Define the zeta function for the maximal order O as follows:

X −s ζO(s) = NF/Q(N(a)) , a Chapter 3. Divisor Function 35 where the sum is over all integral left O-ideals. Since both norms are multiplicative, by

Lemma 3.2.2, we can write the zeta function as an Euler product

Y ζO(s) = ζO,p(s), p where

X −s ζO,p(s) = NF/Q(N(a)) , a and the sum is taken over all integral left O-ideals of norm p-powers. By Lemma 3.2.11,

ζO and its local factors do not depend on the choice of O.

Proposition 3.2.12

Y 1−s ζO(s) = ζF (s)ζF (s − 1) (1 − NF/Q(p) ), ramified p where ζF (s) is the Dedekind zeta function for F and the product is over all ramified finite places.

Proof. If p ramifies in A, by localization and by Proposition 3.2.4,

∞ X −ns −s −1 ζO,p(s) = NF/Q(p) = (1 − NF/Q(p) ) ; n=0 if p splits in A, by Lemma 3.2.5,

∞ n+1 X NF/ (p) − 1 ζ (s) = Q N (p)−ns = (1 − N (p)−s)−1(1 − N (p)1−s)−1. O,p N (p) − 1 F/Q F/Q F/Q n=0 F/Q Our proposition follows. Done. 2

The following theorem is the main theorem of this chapter, which is a quaternion analogue of the well-known formula

∞ X ζ(s)4 d(n)2n−s = , ζ(2s) n=1 where d(n) is the divisor function for the rational integers and ζ(s) is the Riemann zeta function. Chapter 3. Divisor Function 36

Theorem 3.2.13 4 X ζO(s) d(a)2N (N(a))−s = , F/Q ζ (2s) a O where the sum is over all integral left O-ideals.

Proof. By Proposition 3.2.12, we know that the right-hand side has an Euler product.

Explicitly, if p splits, the local factor at p is

−s 1−2s −s −3 1−s −4 (1 + NF/Q(p) )(1 − NF/Q(p) )(1 − NF/Q(p) ) (1 − NF/Q(p) ) , and if p ramifies, the local factor at p is

−2s −s −4 (1 − NF/Q(p) )(1 − NF/Q(p) ) .

We now show that the left-hand side also has an Euler product. The left-hand side equals 0 0 X X 2 −s X −s d(a) NF/Q(n) = anNF/Q(n) , n a∈A(n) n P 2 where n runs through the non-zero integral ideals in F and an = a∈A(n) d(a) . Now for any relatively prime pair (m, n), by Lemma 3.2.2, we have a bijection on the two sets:

[ {a : Ol(a) = O,N(a) = mn} −→ {(b, c): Ol(c) = Or(b),N(c) = n}. b∈A(m)

It follows that

X 2 amn = d(a) a∈A(mn) X X = d(bc)2

b∈A(m) Ol(c)=Or(b),N(c)=n X X = d(b)2d(c)2

b∈A(m) Ol(c)=Or(b),N(c)=n X X = d(b)2 d(c)2

b∈A(m) Ol(c)=Or(b),N(c)=n X X = d(b)2 d(c)2 b∈A(m) c∈A(n) = aman, Chapter 3. Divisor Function 37 where in the second last equality we used the Lemma 3.2.11. Then we have a Euler product for the left-hand side, that is,

∞ Y X X 2 −s ( d(a) NF/Q(N(a)) ). p m=0 a∈A(pm)

Hence to show the equality, it suffices to prove that the corresponding local factors coincide.

If p ramifies, we need to prove

∞ X X 2 −s −2s −s −4 d(a) NF/Q(N(a)) = (1 − NF/Q(p) )(1 − NF/Q(p) ) . m=0 a∈A(pm)

By Proposition 3.2.4,

∞ X X 2 −s d(a) NF/Q(N(a)) m=0 a∈A(pm) ∞ X 2 −ms = (m + 1) NF/Q(p) m=0 −2s −s −4 = (1 − NF/Q(p) )(1 − NF/Q(p) ) .

On the other hand, if p splits, we need to show

∞ X X 2 −s d(a) NF/Q(N(a)) m=0 a∈A(pm) −s 1−2s −s −3 1−s −4 = (1 + NF/Q(p) )(1 − NF/Q(p) )(1 − NF/Q(p) ) (1 − NF/Q(p) ) .

By Lemma 3.2.11, Proposition 3.2.9 and Proposition 3.2.10,

∞ X X 2 −s d(a) NF/Q(N(a)) m=0 a∈A(pm) ∞ ∞ X X X 2 −s = d(a) NF/Q(N(a)) m=0 n=0 a,(p;n,m) ∞ ∞ X X 2 −(2n+m)s = a(p; n, m)d(p; n, m) NF/Q(p) m=0 n=0 = ······

= (1 + N(p)−s)(1 − N(p)1−2s)(1 − N(p)−s)−3(1 − N(p)1−s)−4, Chapter 3. Divisor Function 38 where the last summation in the second term is over all integral left O-ideals of signature

(p; n, m) and the dots stand for a rather tedious but elementary process of calculations.

It is exactly the same calculations as in Duke ([7], p.829) and could be done by hand.

We checked it using the software Mathematica. Done. 2

Corollary 3.2.14 As x → ∞,

X d(a)2 ∼ Ax2 log3(x), N (N(a))≤x F/Q for some positive A.

Proof. Define the Dirichlet series f(s) by

∞ X −s X 2 −s f(s) = ann := d(a) NF/Q(N(a)) . n=1 a

Since ζF (s) only has a simple pole at s = 1 on the right half plane Re(s) > 1 − 1/N where N = [F : Q], ζO(s) only has a simple pole at s = 2 on the right half plane

Re(s) > 2 − 1/N. Moreover ζF (2s) is regular and non-vanishing there, so by Theorem 3.2.13, f(s) is regular on Re(s) > 2 − 1/N except a unique pole at s = 2 of order 4.

Apply Perron’s formula to f(s), and we obtain

X 2 X 2 3 d(a) = an ∼ Ax log (x), as x → ∞, N (N(a))≤x n≤x F/Q for some positive constant A. Done. 2

For a more precise asymptotic result for the summatory function in Corollary 3.2.14 in the case of F = Q, please see Corollary 17 of our paper [13]. In the following chapter, we shall see an application of this asymptotic result on a fourth moment problem of L-functions associated to newforms. Chapter 4

Fourth Moments of L-functions

Associated to Cusp Forms

In this chapter, we will follow Duke’s method ([7]) to obtain an bound for fourth moments of L-functions of newforms of level N on average, where N is square-free. As mentioned before, this chapter and Chapter 3 are mainly a joint work of Kim and the author ([13]).

4.1 Introduction

The identity for the divisor function of rational quaternion algebra, proved in the last chapter, provides us with one of the two main ingredients in Duke’s method. Due to the fact that the class number of the quaternion algebra is no longer one in general, we need to generalize Maass correspondence theorem to a system of automorphic functions and a system of Dirichlet series; this gives us the other main ingredient.

Throughout this chapter, let us fix a square-free positive integer N which has an odd number prime divisors, and fix A as the rational definite quaternion algebra that ramifies precisely at all prime divisors of N and ∞.

39 Chapter 4. Fourth Moments 40

4.2 Brandt Matrices and Newforms of Level N

Let O be a maximal order in A and denote by H be the class number of A. Let I1, ..., IH , not necessarily integral, be a complete set of representatives of all distinct left O-ideal

−1 −1 classes. Let Oj = Or(Ij), the right order of Ij. Then Ij I1, ..., Ij IH , is a complete set

−1 of representatives of all distinct left Oj-ideal classes. Here note that Ij Ij = Oj, and

{Oj} exhausts (with multiplicity in general) all types of maximal orders in A. Let ej be

the number of units in Oj.

Since A ramifies at ∞, A∞ ' (−1, −1)R, the Hamiltonian quaternion algebra. The

two dimensional complex representation of (−1, −1)R (see Section 3.1.2) induces a two

dimensional complex representation of A from the embedding A ,→ (−1, −1)R. Denote

the matrix representation by X = X1. By taking symmetric powers, for any m ∈ N, we

m obtain a (m+1)-dimensional matrix representation Xm = Sym (X1) with X0 = 1 for the trivial one dimensional representation. We know that the entries of Xm(α) are harmonic homogeneous polynomials P (x1, x2, x3, x4) of degree m where xi’s are the coordinates of

α with respect to the canonical basis of (−1, −1)R (see [8], Proposition 6, page 104). For any n ∈ Z+, m ∈ Z+ ∪ {0}, and 1 ≤ i, j ≤ H, define

m −1 X t bij (n) = ej Xm(α)

−1 −1 where the sum is over all α ∈ Ij Ii with N(α) = nuij and the superscript t denotes

0 −1 m “transpose”. Further, let bij(0) = ej and bij (0) = 0 for any m > 0.

Definition 4.2.1 Let the notations be as above. For m, n ∈ N, the Brandt matrix

Bm(n) is given by

m Bm(n) = Bm(n; N) = (bij (n))H(m+1)×H(m+1).

The theta series attached to the Brandt matrices is defined to be

∞ X m Θm(τ) = Θm(τ; N) = Bm(n) exp(nτ) = (θij (τ))H(m+1)×H(m+1), n=0 Chapter 4. Fourth Moments 41 where as usual exp(τ) = e2πiτ . Moreover, the shifted L-series attached to Brandt matrices are

∞ X m −(s+ 2 ) m Ψm(s) = Ψm(s; N) = Bm(n)n = (ψij (s))H(m+1)×H(m+1). n=1

It is known that the entries of Θm(τ) are modular forms of weight k = m + 2 and level N. Moreover if m > 0, they are cusp forms. Please refer to Theorem 1 on page 105 in [8] for details. Here are some properties of the Brandt matrices (Theorem 2, page 106,

[8]),

t −1 −1 −1 (1) Bm(n) = D Bm(n)D with D = diag{e1N(I1) Em+1, ··· , eH N(IH ) Em+1}, the

−1 partitioned diagonal matrix with blocks eiN(Ii) Em+1. Here Em+1 is the identity matrix of size m + 1.

(2) Bm(n1n2) = Bm(n1)Bm(n2) if (n1, n2) = 1.

(3) If p - N, then for any r, t ∈ N,

min{r,t} r t X (m+1)l r+t−2l Bm(p )Bm(p ) = p Bm(p ). l=0

(4) If p | N, then for any r, t ∈ N,

r t r+t Bm(p )Bm(p ) = Bm(p ).

So the Brandt matrices for fixed m generate a commutative semisimple algebra. Hence they can be simultaneously diagonalized. A striking result is the connection between the

Hecke operator and Brandt matrices. Observe first that the above identities for Brandt matrices are exactly the identities satisfied by the Hecke operators. Moreover, The action of Tm+2(n), (n, N) = 1, on entries of Θm(τ) is given formally by left multiplication by

m Bm(n), namely, Tm+2(n)Θm(τ) = (Tm+2(n)θij (τ)) = Bm(n)Θm(τ) as matrices (Propo- sition, page 138, [8]). We can diagonalize Θm(τ) and get diag{θ1(τ), ··· , θ(m+1)H (τ)}.

Denote by Sm the direct sum of the one dimensional complex vector spaces spanned by Chapter 4. Fourth Moments 42

non-zero θi(τ). So Tm+2(n) acts on this space for any n relatively prime to N through

the diagonalization of Bm(n), hence θi, if not zero, is an eigenfunction for all Tm+2(n) with (n, N) = 1.

Denote by T = Tm+2 the Hecke algebra generated by Tm+2(n) with (n, N) = 1.

Remark. (1) In Eichler’s fundamental work [8], on page 138, the T-module θl(D,H) should be replaced by the direct sum of the one-dimensional complex vector spaces

spanned by the non-zero theta series θi, where θi’s are the cuspidal diagonal entries

of the matrix obtained from θl(z; D,H) by diagonalization. This is due to the fact that

θi’s could be linearly dependent in general. See Pizer’s observation on this, page 112 of [23].

(2) If k = m + 2 = 2, then all the diagonal entries of the diaganolized matrix are

non-zero. This can be shown by comparing the dimensions. While if k > 2, some of the

θi’s might be zero. For example, when N = 2 and k = 4, we have H = 1. Since we

know that there is no nonzero cusp form of level 2 weight k, the two cusp forms θ1 and

θ2 are zeros. This can also be shown by comparing the dimensions in general. So some of the “one-dimensional” vector spaces in Theorem 2.28 and Corollary 2.29 in [24] could

be zero spaces.

We present a theorem as follows. See [26] for the weight two case, namely m = 0.

∼ new Theorem 4.2.2 Fix any even m > 0 and we have Sm = Sm+2(N) as T-modules. In

particular, the θi’s are newforms of level N and of weight m + 2.

Proof. Since the Hecke algebra T is semisimple, to show that the two finitely generated

T-modules are isomorphic, it suffices to show that the traces of T coincide for any element

0 new T ∈ T. Let Tm+2(n) denote the action of Tm+2(n) on the space of newforms Sm+2(N)

00 0 and let Tm+2 be the action of Tm+2(n) on Sm. So we only need to show tr(Tm+2(n)) =

00 tr(Tm+2(n)) for any (n, N) = 1. Since the action of Tm+2 on Sm is given formally

00 0 by Bm(n; N) and tr(Tm+2(n)) = tr(Bm(n; N)), we only need to show tr(Tm+2(n)) =

tr(Bm(n; N)) for (n, N) = 1. Chapter 4. Fourth Moments 43

The traces of Hecke operators and Brandt matrices are well-known, see [8] for example.

Here we use Pizer’s notations and quote the formulas in Theorem 1 and 2 in [22] where you can see the meaning of the notations. Recall that k = m+2 > 2 and N is square-free.

X X Y √ k − 1 Y 1 tr T (n) = − a (s) b(s, f) c0(s, f, p) + δ( n) N (1 + ), N k k 12 p s f p|N p|N

and

X X Y √ k − 1 Y 1 trB (n; N) = a (s) b(s, f) c(s, f, p) + δ( n) N (1 − ). k−2 k 12 p s f p|N p|N

0 Let us first calculate trN Tk(n). From the theory of newforms in [2], we know that

M M new d Sk(Γ0(N)) = Sk (Γ0(M)) , M|N N d| M

d d 0 where Sk(Γ0(M)) = {f : f ∈ Sk(Γ0(M))}. For any two positive divisors d, d of N/M, as H-modules, new d ∼ new d0 Sk (Γ0(M)) = Sk (Γ0(M)) ,

hence Tk(n) has the same traces on them. Define λ(n) to be the number of distinct prime divisors of n. We obtain

X X 0 X λ(N)−λ(M) 0 trN (Tk(n)) = trM (Tk(n)) = 2 trM (Tk(n)). M|N N M|N d| M Apply M¨obiusinversion formula, we have

0 X λ(N)−λ(M) X λ(N)−λ(M) trN (Tk(n)) = 2 µ(N/M)trM (Tk(n)) = (−2) trM (Tk(n)). M|N M|N

0 By above expression of trN (Tk(n)), we have trN (Tk(n)) = S1 + S2 where

X λ(N)−λ(M) X X Y 0 S1 = (−2) (− ak(s) b(s, f) c (s, f, p)) M|N s f p|M

and X √ k − 1 Y 1 S = (−2)λ(N)−λ(M)δ( n) M (1 + ). 2 12 p M|N p|M Chapter 4. Fourth Moments 44

By explicit verification using the tables in [22], we have c(s, f, p) + c0(s, f, p) = 2 for

any p and any pair of (s, f). Now

X X X λ(N)−λ(M) Y 0 S1 = − ak(s) b(s, f) (−2) c (s, f, p) s f M|N p|M X X Y 0 = − ak(s) b(s, f) (c (s, f, p) − 2) s f p|N X X Y = ak(s) b(s, f) c(s, f, p), s f p|N

where the last equality is true because λ(N) is odd. Similarly,

√ k − 1 X Y S = δ( n) (−2)λ(N)−λ(M) (p + 1) 2 12 M|N p|M √ k − 1 Y = δ( n) (p + 1 − 2) 12 p|N √ k − 1 Y 1 = δ( n) N (1 − ). 12 p p|N

0 Now it is clear that tr(Tm+2(n)) = tr(Bm(n; N)) for (n, N) = 1. Done. 2 Let us end this section with a description of the action of the canonical involution

(given by wN ) on the theta series above. See Pizer’s work [25] (Section 9) on this. First, let us define an two-sided integral ideal as follows. Assume O is the maximal

order such that Op = M2(Zp) if p - N and     α β    Op =   : α, β ∈ RL , if p | N.  pβσ ασ 

For the meaning of these notations, please see Section 3.1.2. Define ω ∈ A× by ω = 1 A,f p if p - N and   0 1   ωp =   , if p | N. p 0 Let J = Oω be the integral left O-ideal, determined by the local-global correspondence.

It is easy to see that J is two-sided and N(J) = N. For other maximal orders, J is also

defined by conjugation of some element in A× . A,f Chapter 4. Fourth Moments 45

H Now fix as before a complete representative set of left O-ideal classes, {Ii}i=1. Since

H J is two-sided, {JIi}i=1 gives another representative set. So JIi = Iσ(i)αi for a unique

× σ(i) and some αi ∈ A . Obviously, σ is a permutation of {1, ··· ,H} of order 2. We

−m/2 define an H(m + 1) × H(m + 1) matrix Wfm(N) by setting Wfm(N) = N (ρij), where

ρij is the (m + 1) × (m + 1) matrix   t  Xm(αi) if j = σ(i) ρij =  0 otherwise. We will need the following results in Section 4.4.

Proposition 4.2.3 (1) Wfm(N) commutes with Bm(n) for any n ∈ N;

2 (2) Wfm(N) = I, the identity matrix of size H(m + 1);

(3) Θm |wN (τ) = −Wfm(N)Θm(τ), where wN acts on Θm(τ) by acting on each entry.

Proof. The verifications of (1) and (2) are straightforward. For (3), although Pizer’s treatment in [25] carries the restriction that A ramifies at only one finite prime, the same procedure can be applied in our situation. We shall omit the identical proof. Done. 2

4.3 Maass Correspondence Theorem

In this section, we generalize Maass correspondence theorem [17] and Duke’s result [7] to a system of Dirichlet series and a system of automorphic functions.

Let us denote Pm the space of all degree m spherical harmonic polynomials, and let P∗ denote the summation over any orthonormal basis for P with respect to the metric Pm m on Sn−1.

n Definition 4.3.1 For i = 1, ..., M, let Λi ⊂ R be a full lattice, and

0 X −2s φi(s) = ai(β)|β| ,

β∈Λi be a Dirichlet series. Let C = (cij) be an M × M constant matrix and r ∈ R a constant.

Then we say φ(s) = (φ1(s), ..., φM (s)) has signature hΛ1, ..., ΛM , n, r, Ci if Chapter 4. Fourth Moments 46

n+ir n−ir (1) (s − 2 )(s − 2 )φi(s) is entire and bounded in vertical strips, and for any Pm ∈

Pm, the twist of φi(s) by Pm,

X 0 β φ (s, P ) = a (β)P ( )|β|−2s, i m i m |β| β∈Λi is entire and bounded in vertical strips for m ≥ 1.

(2) For any Pm with m ≥ 0 and m + ir m − ir R (s, P ) = π−2sΓ(s + )Γ(s + )φ (s, P ), i m 2 2 i m we have M X n R (s, P ) = (−1)m c R ( − s, P 0 ) i m ik k 2 m k=1 0 where the conjugate Pm is defined to be

0 Pm(w1, w2, ··· , wn) = Pm(w1, −w2, ··· , −wn).

When M = 1, the above Dirichlet series were first considered by Maass [17] who proved a non-holomorphic, (n+1)-dimensional version of Hecke correspondence. Namely, there is one to one correspondence between the functions of signature hΛ, n, ri and certain automorphic functions on hyperbolic (n + 1)-space. We note that Spin(1, n) acts on the hyperbolic (n + 1)-space. It is easy to generalize to a system of Dirichlet series. We use the notation in [7].

Let Cn+1 be the Clifford algebra, generated by i1, ..., in subject to the relations ihik =

2 −ikih for k 6= h, ih = −1, and no others. The (n + 1)-dimensional subspace

n+1 V = {x = x0 + x1i1 + ··· + xnin : xi ∈ R},

n+1 n+1 contains H = {x ∈ V : xn > 0}, a model for hyperbolic (n + 1)-space when

2 −2 2 n 0 endowed with the metric ds = xn |dx| . Let Λ be a lattice in V , and let Λ be its dual lattice with respect to the inner product Re(¯xy). Let Γ be the group of isometries of

Hn+1 generated by x → x + α for all α ∈ Λ0 and x → −x−1. We say that f ∈ C2(Hn+1) is an automorphic function for Γ if Chapter 4. Fourth Moments 47

2 n2 n+1 Pn ∂ 1−n ∂ (1) ∆n+1f + (r + )f = 0 for some r ∈ , where ∆n+1 = x (x ), 4 R n h=0 ∂xh n ∂xh

A −A (2) f(x)  xn as xn → ∞, and f(x)  xn as xn → 0+ uniformly in x0, ..., xn−1,

(3) f(γx) = f(x) for all γ ∈ Γ.

Then we have the following Fourier expansion of f(x):

0 n X 2 2πiRe(βx¯ ) f(x) = u(xn) + a(β)xn Kir(2π|β|xn)e , β∈Λ where  n n 2 +ir 2 −ir  a1xn + a2xn if r 6= 0, u(xn) = n  x 2 (a1 + a2 log xn) if r = 0.

For the constants a1 and a2, please see [17] or [7].

Now let us consider the case of a family of automorphic functions. Let Λ1, ..., ΛM be

n 0 0 lattices in V , and let Λ1, ..., ΛM be its dual lattices. Let Γi be the group of isometries

n+1 0 −1 of H generated by x → x + α for all α ∈ Λi and x → −x for each i. We say that

(f1, ..., fM ) is a system of automorphic functions for hΓ1, ..., ΓM , n, r, Ci where r ∈ R and

C = (cij) is a constant M × M matrix, if

2 n2 (1) for each i, ∆n+1fi + (r + 4 )fi = 0,

A −A (2) for each i, fi(x)  xn as xn → ∞, and fi(x)  xn as xn → 0+ uniformly in

x0, ..., xn−1,

0 (3) fi(x + αi) = fi(x) for all αi ∈ Λi, and

PM −1 (4) fi(x) = k=1 cikfk(−x ).

Now we can prove

Theorem 4.3.2 (Maass Correspondence Theorem for a system of Dirichlet series) Fix any constant matrix C = (cij)M×M and suppose, for i = 1, ..., M,

0 n X 2 2πiRe(βx¯ ) fi(x) = ui(xn) + ai(β)xn Kir(2π|β|xn)e ,

β∈Λi Chapter 4. Fourth Moments 48

and 0 X −2s φi(s) = ai(β)|β| .

β∈Λi The following are equivalent:

(1) (φ1(s), ..., φM (s)) has signature hΛ1, ..., ΛM , n, r, Ci,

(2) (f1, ..., fM ) is a system of automorphic functions for hΓ1, ..., ΓM , n, r, Ci.

Proof. This is a straightforward generalization of [17]. For i = 1, ..., M, y ∈ R+, Let

X 0 n Fmi(y, Pm) = umi(y) + ai(β)Pm(β)y 2 Kir(2π|β|y),

β∈Λi

where   ui(y) if m = 0, umi(y) =  0 if m > 0.

PM −1 We can show as in [17] that fi(x) = k=1 cikfk(−x ) if and only if Fmi(y, Pm) = m PM 1 0 (−1) k=1 cikFmk( y ,Pm) for each m. By the integral formula

Z ∞ 2s dx ir ir 4 Kir(2x)x = Γ(s + )Γ(s − ), 0 x 2 2

we have Z ∞ 2s+m− n dy Ri(s, Pm) = 4 (Fmi(y, Pm) − umi(y))y 2 . 0 y By Mellin inversion, Z m− n 1 s −s 4(Fmi(y, Pm) − umi(y))y 2 = Ri( ,Pm)y ds. 2πi Re(s)0 2

n Note that Ri(s, Pm) is entire for m > 0. In that case, by moving the contour to Re(s) = 4 , we have Z 1 s n −m−s 4Fmi(y, Pm) = Ri( ,Pm)y 2 ds. 2πi n 2 Re(s)= 4 m PM 1 0 Hence the functional equation Fmi(y, Pm) = (−1) k=1 cikFmk( y ,Pm) and Ri(s, Pm) = m PM n 0 (−1) k=1 cikRk( 2 − s, Pm) are equivalent. Chapter 4. Fourth Moments 49

When m = 0, the integrand has a pole. By calculating the residues, we have Z −1 1 s n −s 4(Fi(y, 1) − ui(y) − ui(y )) = Ri( , 1)y 2 ds. 2πi n 2 Re(s)= 4 We again obtain the same result. 2

It is straightforward to generalize Duke’s result ([7], Theorem 2) to a system of Dirich- let series and we skip the proof.

Theorem 4.3.3 Suppose (φ1(s), ..., φM (s)) has signature hΛ1, ..., ΛM , n, r, Ci, n is even, and that M 0 X X 2 n c n c−1 |ai(β)| = Ax log x + O(x log x).

i=1 |β|≤x,β∈Λi π − Then as θ → 2 ,

M ∗ Z ∞ X X X 2 n 2 0 c+1 c |M(2it, θ)| |φi( + it, Pm)| dt = A log (sec θ) + O(log (sec θ)), −∞ 2 m>0 Pm i=1 for some constant A0, where

m n i n Γ( )Γ(a)Γ(b) n −s− 2 m −s 2 2 M(s, θ) = π (tan θ) (cos θ) n F (a, b; m + , − tan θ), 4 Γ(m + 2 ) 2

n n s+m+ 2 +ir s+m+ 2 −ir a = 2 , b = 2 .

If PM P 0|a (β)|2  xn logc x, then we have i=1 |β|≤x,β∈Λi i

M ∗ Z ∞ X X X 2 n 2 c+1 |M(2it, θ)| |φi( + it, Pm)| dt  log (sec θ). −∞ 2 m>0 Pm i=1 Letting T = tan θ, as in [7], page 823, we have

Corollary 4.3.4 Suppose (φ1(s), ..., φM (s)) has signature hΛ1, ..., ΛM , n, r, Ci, n is even, and that PM P 0|a (β)|2  xn logc x. Then as T → ∞, i=1 |β|≤x,β∈Λi i

M ∗ Z T X X X n 2 n c+1 |φi( + it, Pm)| dt  T log T. −T 4 0

In the application to the fourth moments of L-functions of newforms, n = 4; the main work is to relate the Dirichlet series φ(s, Pm) to the Dirichlet series formed from quaternion algebras and establish the estimate

M 0 X X 2 4 3 4 2 |ai(β)| = Ax log x + O(x log x).

i=1 |β|≤x,β∈Λi

4.4 Application to Fourth Moments of L-functions

Let N ∈ Z+ be square-free, which has an odd number of prime divisors, and let A be a rational definite quaternion algebra that ramifies precisely at the prime divisors of N.

We maintain all the notations as before.

−1 For any 1 ≤ i, j ≤ H, we may also define the divisor function for any γ ∈ Ij Ii by

H X −1 −1 −1 d(γ) = el #{(α, β) ∈ Il Ii × Ij Il : βα = γ}. l=1

−1 −1 It is easy to see that γ ∈ Ij Ii if and only if Ii Ijγ is integral, and if and only if

−1 γIi Ij is integral. The following lemma shows that two divisor functions coincide.

−1 −1 0 −1 0 Lemma 4.4.1 Let γ ∈ Ij Ii and let a = Ii Ijγ, a = γIi Ij. Then d(γ) = d(a) = d(a ).

Proof. The above remark says that a is integral and d(a) makes sense. Here we only show d(γ) = d(a) and the other equality follows in the same way.

It is enough to show that for any l,

−1 −1 −1 −1 el #{(α, β) ∈ Il Ii × Ij Il : βα = γ} = #{b ∼l Ii Il : b|la}.

We define a map φ:

−1 −1 −1 {(α, β) ∈ Il Ii × Ij Il : βα = γ} −→ {b ∼l Ii Il : b|la}

−1 (α, β) 7−→ Ii Ilα Chapter 4. Fourth Moments 51

−1 −1 −1 First, φ is well-defined. Indeed, α ∈ Il Ii implies Ii Ilα is integral; β ∈ Ij Il and

−1 −1 −1 −1 −1 −1 βα = γ imply that Il Ijβ is integral and so is α (Il Ijβ)α. So (Ii Ilα)(α (Il Ijβ)α) = a.

−1 Furthermore, φ is surjective. For any b|la with b ∼l Ii Il, assume a = bc and

−1 −1 −1 b = Ii Ilα. Then αcα has left order Ol. So for a unique k and for some β ∈ Il Ik,

−1 −1 −1 × αcα = Il Ikβ. Then a = Ii Ikβα, which implies k = j and there exists  ∈ Oj , such that γ = βα. So φ((α, β)) = a.

Finally, it is enough to show that φ is el-to-1. Actually, the preimage set of any element is nonempty by surjectivity and if (α, β) is one, then so is (α, β−1) for any

× −1 −1  ∈ Ol . Conversely, if (α1, β1) and (α2, β2) have the same image, then Ii Ilα1 = Ii Ilα2 × and there exists  ∈ Ol such that α1 = α2. Done. 2 Let the Brandt matrices, theta series and shifted L-functions be defined as in Defini-

tion 4.2.1, and we shall drop the superscript 0 if m = 0 as we always do in the following,

0 that is, θij(τ) = θij(τ) and

0 1 −1 −1 bij(n) = bij(n) = |{α ∈ Ij Ii : N(α) = nuij }| ej −1 = |{a : N(a) = n, a ∼l Ii Ij}|.

We know θij(τ) is a modular form of weight 2 on Γ0(N), and ∞ X −s −1 −s X −s X −s ψij(s) = bij(n)n = ej uij N(α) = N(a) , n=1 −1 −1 α∈Ij Ii a∼lIi Ij −1 which is exactly the zeta function ζij(s) for the ideal class Ii Ij.

m N s 2 Let Φm(s) = (φij (s)) = ( 4 ) (Ψm(s)) , and in particular for m = 0 H N X φ (s) = ( )s ψ (s)ψ (s). ij 4 il lj l=1 Lemma 4.4.2 N X N X φ (s) = ( )s d(a)N(a)−s = e−1( )s d(I−1I α)N(I−1I α)−s ij 4 j 4 i j i j −1 −1 a∼lIi Ij α∈Ij Ii X −2s = aij(β)|β|

β∈Λij Chapter 4. Fourth Moments 52

p −1 −1 −1 −1 p −1 where Λij is the lattice 2 uijN Ij Ii and aij(β) = ej d(Ii Ijβ/(2 uijN )). More- over, as x → ∞, for each i = 1, ..., H, H X X 2 4 3 aij(β)  x log x.

i,j=1 β∈Λij ,|β|≤x

−1 −1 −1 Proof. Given any pair b ∼l Ii Il and c ∼l Il Ij, we know that b = Ii Ilα and −1 −1 −1 c = Il Ijβ for some α, β ∈ A; actually, to make them integral, α ∈ Il Ii and β ∈ Ij Il.

−1 −1 So b(α cα) ∼l Ii Ij. Since α is unique up to units in Ol, we get a well-defined map H [ −1 −1 −1 {b : b ∼l Ii Il} × {c : c ∼l Il Ij} −→ {a : a ∼l Ii Ij}. l=1 −1 −1 Let a ∼l Ii Ij and a = bc be any decomposition into integral ideals. Then b ∼l Ii Il for

−1 −1 a unique l, hence b = Ii Ilα for some α. Then (b, αcα ) has image a and the preimage set of a are in one to one correspondence with the set of decompositions of a. So this

−1 map is surjective and for any a ∼l Ii Ij, there are precisely d(a) preimages. This proves the first part.

Now H H X X 2 X X −2 −1 2 aij(β) = ej d(Ii Ijα) j=1 β∈Λ ,|β|≤x j=1 −1 −1 2 ij α∈Ij Ii,N(α)≤puij x /4 H X −1 X 2 = ej d(a) . j=1 −1 2 a∼lIi Ij ,N(a)≤Nx /4

Since ej is bounded, by Corollary 3.2.14, we conclude that

H H X X 2 X X 2 4 3 aij(β)  d(a) ∼ Ax log x. 2 i,j=1 β∈Λij ,|β|≤x i=1 Ol(a)=Oi,N(a)≤Nx /4 2

H2 2 2 Let {vij : 1 ≤ i, j ≤ H} be a basis for R and let C be the H × H matrix that −1 represents the linear transformation which sends vij to ej eivji.

Proposition 4.4.3 The system of the Dirichlet series (φ11(s), ··· , φij(s), ··· , φHH (s)) is of signature

hΛ11, ··· , Λij, ··· , ΛHH , 4, 0,Ci. Chapter 4. Fourth Moments 53

m Proof. Direct calculation gives us φij (s) = φij(s, Xm). The fact that ψij(s) has only one

2 pole, that is a simple pole at s = 2, implies that (s − 2) φij(s) is entire and bounded in

m vertical strips. For m ≥ 1, it follows from [33] (on page 54) that all entries of ψij (s) are

entire and bounded on vertical strips, hence so are those of φij(s, Xm).

Now it suffices to show the functional equations. Denote L(s, Θm(τ)) to be the matrix

whose entries are L-functions for corresponding entries of Θm(τ), and Θm(τ)|wN the m resulting matrix where wN acts on each entry. Hence Ψm(s) = L(s + 2 , Θm). Since

entries of Θm(τ) are modular forms of level N and weight m + 2,

m −s m+2 1−s+ 2 s−m−2 (2π) Γ(s)L(s, Θm) = i N (2π) Γ(m + 2 − s)L(m + 2 − s, Θm|wN ).

m By changing the variable s → s + 2 , we get

m m m m (2π)−sΓ(s + )L(s + , Θ ) = im+2N 1−s(2π)s−2Γ(2 − s + )L(2 − s + , Θ | ). 2 2 m 2 2 m wN

By taking squares,

m m (2π)−2sΓ(s + )2L(s + , Θ )2 2 2 m m m = (−1)mN 2−2s(2π)2s−4Γ(2 − s + )2L(2 − s + , Θ | )2. 2 2 m wN

2 By Proposition 4.2.3, Θm|wN = −Wfm(N)Θm and Wfm(N) = I, which gives us

L(s, Θm|wN ) = −Wfm(N)L(s, Θm). Since Wfm(N) commutes with all Bm(n), we have 2 2 L(s, Θm|wN ) = L(s, Θm) . Then the above equation implies

m m m m (2π)−2sΓ(s + )2L(s + , Θ )2 = (−1)mN 2−2s(2π)2s−4Γ(2 − s + )2L(2 − s + , Θ )2, 2 2 m 2 2 m

and m m π−2sΓ(s + )2Φ (s) = (−1)mπ2s−4Γ(2 − s + )2Φ (2 − s), 2 m 2 m

m m since Ψm(s) = L(s + 2 , Θm). This implies Rij(s, Pm) = (−1) Rij(2 − s, Pm) for any i, j −2s m 2 and any Pm, where Rij(s, Pm) = π Γ(s + 2 ) φij(s, Pm).

Since entries of Xm form a basis for Pm (see [35], page 160), it is enough to show

−1 0 0 that φij(s, Xm) = ej eiφji(s, Xm), where Xm is the one obtained by taking conjugation Chapter 4. Fourth Moments 54

on each entry. Indeed,

X β φ (s, X0 ) = a (β)X0 ( )|β|−2s ji m ji m |β| β∈Λji N X α = e−1( )su−s d(I−1I α)X0 ( )N(α)−s i 4 ji j i m |α| −1 α∈Ii Ij N X X α = e−1( )su−s d(a) X0 ( )N(α)−s i 4 ji m |α| −1 −1 a∼lIj Ii α,a=Ij Iiα N X X α = e−1( )su−s d(a) X ( )N(α)−s. i 4 ji m |α| −1 −1 a∼lIj Ii α,a=Ij Iiα

By changing α → α in the second summation and then a → a in the first summation,

we have

N X X α φ (s, X0 ) = e−1( )su−s d(a) X ( )N(α)−s ji m i 4 ji m |α| −1 −1 a∼lIj Ii α,a=Ij Iiα N X X α = e−1( )su−s d(a) X ( )N(α)−s i 4 ji m |α| −1 −1 a∼lIj Ii α,a=αIiIj N X X α = e−1( )su−s d(a) X ( )N(α)−s. i 4 ji m |α| −1 −1 a∼rIiIj α,a=αIiIj

−1 It is obvious that d(a) = d(a) by Lemma 3.2.11; moreover, since Ij = Ij N(Ij) and

−1 −1 −1 −1 −1 −1 −1 Ii = IiN(Ii) , we have IjIi = uijIj Ii and α ∈ IjIi if and only if β = uij α ∈ Ij Ii. So

N X −1 α φ (s, X0 ) = e−1( )su−s d(I I α)X ( )N(α)−s ji m i 4 ji i j m |α| −1 α∈Ij Ii N X α = e−1( )su−s d(I−1I u−1α)X ( )N(α)−s i 4 ji i j ij m |α| −1 −1 uij α∈Ij Ii −1 −1 = ei ejφij(s, Xm), by setting β = uij α.

Done. 2

Now we prove our theorem on the fourth moment of L-functions associated to new-

forms as a corollary of Corollary 4.3.4. Chapter 4. Fourth Moments 55

Theorem 4.4.4 As T → ∞,

X X Z T k |L( + it, f)|4dt  T 3 log4 T. new −T 2 2

π − So as θ → 2 ,

∗ Z ∞ X X X 2 2 4 |M(2it, θ)| |φij(1 + it, Pm)| dt  log (sec θ). −∞ m>0, m even Pm i,j √ As we know (see [35], page 160), entries of m + 1Xm constitute an orthonormal

m √ 1 basis of Pm, so entries of φij (s) are exactly those m+1 φij(s, Pm), where Pm’s are entries √ of m + 1Xm. Hence Z ∞ X X 2 m m ∗ LHS = (m + 1) |M(2it, θ)| tr(φij (1 + it)(φij (1 + it)) ) dt, m>0, m even i,j −∞ which can be further simplified to

Z ∞ X 2 ∗ (m + 1) |M(2it, θ)| tr(Φm(1 + it)(Φm(1 + it)) ) dt, m>0, m even −∞ then to

Z ∞ X 2 2 2 ∗ (m + 1) |M(2it, θ)| tr(Ψm(1 + it)(Ψm(1 + it)) ) dt. m>0, m even −∞

For fixed m > 0, the Brandt matrices Bm(n)(n = 0, 1, 2...) are simultaneously diago- nalizable and the resulting non-zero diagonal entries are the n-th coefficients of all new

forms of level N, weight m + 2, by Theorem 3.2.2. So there is a unitary matrix U, such

−1 new that UΘm(τ)U = diag{f1, ··· , fH(m+1)} and the subset of non-zero fi’s is Sm+2(N). As a result,

m m m UΨ (s)U −1 = UL(s + , Θ )U −1 = diag{L(s + , f ), ··· ,L(s + , f )}, m 2 m 2 1 2 H(m+1)

so we have (setting k = m + 2)

X X Z ∞ k LHS = (k − 1) |M(2it, θ)|2|L( + it, f)|4 dt. new −∞ 2 k>2, k even f∈Sk (N) Chapter 4. Fourth Moments 56

By setting T = tan θ, as in [7], page 823, we obtain

Z T X X k 4 4 4 (k − 1) |Lf ( + it)| dt  T log T. new −T 2 2

Critical Values of Two-dimensional

Artin L-functions

In this chapter, we will generalize a result of Moreno ([20]), which relates the critical value of some Artin L-functions to the Petersson inner product by means of Strong Artin

Conjecture.

5.1 Introduction

Let K/Q be a Galois extension with Gal(K/Q) = S3 and ρ : S3 −→ GL2(C) be the irreducible two dimensional representation. Then in [20, Theorem 7], Moreno gave the identity (the notations are slightly different from his original paper) for some nonzero constant c

∗ cL(1/2, χ2)L(1/2, ρ) =< f fρ, fρi,

where χ2 is the nontrivial linear character of S3, fρ is the cusp form associated to ρ by Strong Artin Conjecture, f ∗ is the derivative of a Eisenstein series and the inner product is the Petersson inner product. See Section 5.2 for details.

So such an equality connects the vanishing of the value of an L-function at 1/2 with

57 Chapter 5. Critical Values 58

∗ the existence of fρ in the spectral decomposition of the function f fρ. Although he assumed that ρ is odd, Moreno showed, in another paper [19], that the technique also works in the even case.

In this paper, we follow the same idea and formulate the general equality in Section

5.2. Then in Section 5.3 we apply it to the case of Se4 and we remark here that all other cases can be similarly treated.

5.2 General Equality

Before we state the general equality, let us first set up the notations.

Let H be the upper half plane, S = {z ∈ H : |x| ≤ 1/2} and let N be a positive integer.

Choose a fundamental domain of Γ0(N)\H in S and denote it by F(N). Then we ` can choose a complete set of representatives {σ} for Γ0(N)∞\Γ0(N) s.t. S = σ σF(N)

in the sense that two sides differ by a measure zero set. Here Γ0(N)∞ is the stabilizer of

the cusp ∞ in Γ0(N), namely     a b    Γ0(N)∞ =   : ad = 1, a, b, d ∈ Z .  0 d 

Let E(z, s, Γ0(N)) be the Eisenstein series defined by

X s+1 E(z, s, Γ0(N)) = Im(σz) 2 .

σ∈Γ0(N)∞\Γ0(N)

∗ 2 ∗ ∗ Recall that E(z, s, Γ0(N)) = f (z)s + O(s ) where f (z) = f (z; N) satisfies

1 f ∗(σz) = f ∗(z), for any σ ∈ Γ (N), and ∆f ∗ = f ∗. 0 4

For a modular form f, let us introduce the K operator, i.e., K(f)(z) = f(−z). If

f, h ∈ S1(N, ) , then the Petersson inner product takes the form

1 Z dxdy hf, hi = yf(z)h(z) 2 ; [Γ(1) : Γ0(N)] F(N) y Chapter 5. Critical Values 59

If f and h are two Maass cusp forms of weight 0 and level N, then the Petersson inner product is 1 Z dxdy hf, hi = f(z)h(z) 2 . [Γ(1) : Γ0(N)] F(N) y

Let K/Q be a finite Galois extension with Galois group G. Assume ρ : G → GL2(C) is an irreducible Galois representation. Let N be the Artin conductor and  = det(ρ).

Then we know that its Artin L-function is of the form:

∞ X Y Y L(s, ρ) = a(n)n−s = (1 − a(p)p−s)−1 (1 − a(p)p−s + (p)p−2s)−1. n=1 p|N p-N

Q Denote Lp(s, ρ) the p-factor above and let LM (s, ρ) = p|M Lp(s, ρ), for any positive Q −s −1 integer M. Similarly ζM (s) = p|M (1 − p ) . Let us first work on the calculations on the automorphic side. There are two lemmas.

P∞ n P∞ n Lemma 5.2.1 Let f(z) = n=1 a(n)q ∈ S1(N1, 1) and h(z) = n=1 b(n)q ∈ S1(N2, 2).

∗ ∗ Let N = N1N2 and f (z) = f (z; N). Then

(a) if 1 = 2,

∞ − 1+s 1 + s X − 1+s ∗ 2 (4π) 2 Γ( ) a(n)b(n)n 2 = [Γ(1) : Γ (N)]hf f, his + O(s ); (5.1) 2 0 n=1

−1 (b) if 1 = 2 ,

∞ − 1+s 1 + s X − 1+s ∗ 2 (4π) 2 Γ( ) a(n)b(n)n 2 = [Γ(1) : Γ (N)]hf f, K(h)is + O(s ). (5.2) 2 0 n=1

Proof. Consider the following integral

Z 1+s dxdy 1+ 2 I = y f(z)h(z) 2 . S y Chapter 5. Critical Values 60

On one hand,

Z ∞ Z 1/2 ∞ ∞ 1+ 1+s X n X n dxdy I = y 2 ( a(n)q )( b(n)q ) y2 0 −1/2 n=1 n=1 ∞ ∞ Z ∞ Z 1/2 X X 1+ 1+s m n dy = y 2 a(m)b(n) q q dx y2 m=1 n=1 0 −1/2 ∞ Z ∞ X 1+s −4πny dy = a(n)b(n) y 2 e y n=1 0 ∞ − 1+s 1 + s X − 1+s = (4π) 2 Γ( ) a(n)b(n)n 2 2 n=1 which is the same as the left-hand side of (4.1).

On the other hand, Z X 1+ 1+s dxdy I = y 2 f(z)h(z) y2 σ σF(N) Z X 1+ 1+s dxdy = Im(σz) 2 f(σz)h(σz) y2 σ F(N) where σ runs through the chosen set of representatives for Γ0(N)∞\Γ0(N). By automorphy of f and h,

y Im(σz)f(σz)h(σz) = (d)(cz + d)f(z)(d)(cz + d)h(z) |cz + d|2 = yf(z)h(z).

−2 Moreover, we know the measure y dxdy is invariant under Γ0(N), hence Z X 1+s dxdy I = Im(σz) 2 yf(z)h(z) y2 σ F(N) Z dxdy = E(z, s, Γ0(N))yf(z)h(z) 2 F(N) y Z ∗ dxdy 2 = ( y(f f)(z)h(z) 2 )s + O(s ) F(N) y which is right-hand side of (4.1).

We can prove (4.2) along the same lines above except that we use the integral

Z 1+s dxdy 0 1+ 2 I = y f(z)h(−z) 2 , S y Chapter 5. Critical Values 61

instead of I above. We are done. 2

0 s+1 Next lemma deals with the real analytic case. Let s = 2 .

Lemma 5.2.2 Let

X 1/2 2πinx f(z) = a(n)y K0(2π|n|y)e n6=0 X 1/2 2πinx h(z) = b(n)y K0(2π|n|y)e n6=0 1 be two Maass cusp forms (weight 0) of eigenvalues 4 with central characters 1, 2, and ∗ ∗ levels N1, N2, respectively. Let N = N1N2) and f (z) = f (z; N). Assume further that f, h have the same parity, i.e., a(−n)b(−n) = a(n)b(n). Then

(a) if 1 = 2,

s0 4 ∞ (Γ( )) X 0 2 a(n)b(n)n−s = [Γ(1) : Γ (N)]hf ∗f, his + O(s2); (5.3) 4πs0 Γ(s0) 0 n=1 −1 (b) if 1 = 2 , s0 4 ∞ (Γ( )) X 0 2 a(n)b(n)n−s = [Γ(1) : Γ (N)]hf ∗f, K(h)is + O(s2). (5.4) 4πs0 Γ(s0) 0 n=1 Proof. Consider the integral Z s0 dxdy J = f(z)h(z)y 2 . S y As we did in the holomorphic case, we shall compute J in two different ways.

On one hand, Z ∞ Z 1/2 J = ( f(z)h(z)dx)ys0−2dy 0 −1/2 Z ∞ Z 1/2 X s0−1 2πi(m−n)x = a(m)b(n)y K0(2π|m|y)K0(2π|n|y)( e dx)dy 0 m6=0,n6=0 −1/2 Z ∞ X s0−1 = a(n)b(n) K0(2π|n|y)K0(2π|n|y)y dy n6=0 0 Now the identity, with Re(α) > 0 and Re(1 − ρ ± r ± r0) > 0, Z ∞ −ρ Kr(αt)Kr0 (αt)t dy 0 αρ−1 1 − ρ + r + r0 1 − ρ + r − r0 1 − ρ − r + r0 1 − ρ − r − r0 = Γ( )Γ( )Γ( )Γ( ) 2ρ+2Γ(1 − ρ) 2 2 2 2 Chapter 5. Critical Values 62 implies s0 4 ∞ (Γ( )) X 0 J = 2 a(n)b(n)n−s 4πs0 Γ(s0) n=1 which is the left-hand side of (4.3). This is where we need the parity condition.

On the other hand, Z X 0 dxdy J = f(z)h(z)ys y2 σ σF(N) Z X 0 dxdy = f(σz)h(σz)Im(σz)s y2 σ F(N) Z X cz + d cz + d 0 dxdy = (d)( )−1f(z)(d)( )−1h(z)Im(σz)s |cz + d| |cz + d| y2 σ F(N) Z X 0 dxdy = f(z)h(z)Im(σz)s y2 σ F(N) Z dxdy = E(z, s, Γ0(N))f(z)h(z) 2 F(N) y Z ∗ dxdy 2 = ( (f f)(z)h(z) 2 )s + O(s ) F(N) y which is exactly the right-hand side of (4.3). So we are done with (a).

For (4.4), we carry out the same process except we replace J with Z 0 s0 dxdy J = f(z)h(−z)y 2 . S y We are done. 2

On the Galois side, we have an analogous lemma. Let us first recall an elementary result:

Lemma 5.2.3 Assume we have

∞ X n −1 −1 A(n)T = (1 − α1T ) (1 − α2T ) , n=1 ∞ X n −1 −1 B(n)T = (1 − β1T ) (1 − β2T ) . n=1 Then we have

∞ X n 2 Y −1 A(n)B(n)T = (1 − α1α2β1β2T ) (1 − αiβjT ) . n=1 i,j=1,2 Chapter 5. Critical Values 63

Proof. See Lemma 1.6.1 in Bump’s book [4]. 2

Lemma 5.2.4 Let ρ1, ρ2 : Gal(Q/Q) → GL2(C) be two irreducible Galois representa- tions with determinant characters 1 and 2 respectively. Assume ∞ ∞ X −s X −s L(s, ρ1) = a(n)n and L(s, ρ2) = b(n)n . n=1 n=1 Then

(a) if 1 = 2, there is a nonzero constant c, s.t. ∞ X − 1+s 2 a(n)b(n)n 2 = cL(1/2, ρ1 ⊗ ρ2)s + O(s ); (5.5) n=1

−1 (b) if 1 = 2 , there is a nonzero constant c, s.t. ∞ X − 1+s 2 a(n)b(n)n 2 = cL(1/2, ρ1 ⊗ ρ2)s + O(s ). (5.6) n=1

Proof. Assume 1 = 2. Let N be the least common multiple of the conductors of ρ1,ρ2 and ρ1 ⊗ ρ2. For p - N, by Lemma 5.2.3, we have the formal identity ∞ 1 X a(pn)b(pn)T n = det(I − ρ ⊗ ρ (σ )T )−1. 1 − T 2 1 2 p n=1 By setting T = p−s above, we have

∞ −1 X −s ζ(2s)ζN (2s) a(n)b(n)n n=1 ∞ ∞ Y 1 Y X Y X = ( a(pn)b(pn)p−ns) ( a(pn)b(pn)p−ns) 1 − p2s p-N p|N n=1 p-N n=1 ∞ Y 1 X = L (s, ρ , ρ ) ( a(pn)b(pn)p−ns) N 1 2 1 − p2s p-N n=1 Y −1 = LN (s, ρ1, ρ2) det(I − ρ1 ⊗ ρ2(σp)T ) p-N

LN (s, ρ1, ρ2) = L(s, ρ1 ⊗ ρ2) LN (s, ρ1 ⊗ ρ2)

Q P∞ n n −ns where LN (s, ρ1, ρ2) = p|N ( n=1 a(p )b(p )p ). Again by the Lemma 5.2.3, we know it is a product of factors in the form of (1−λp−s)−1 where by basic representation theory

|λ| = 1. Chapter 5. Critical Values 64

Let L (s, ρ , ρ )ζ (2s) C(s) = N 1 2 N , LN (s, ρ1 ⊗ ρ2) so again C(s) is a product of factors in the form of (1 − λp−s)−1 or (1 − λp−s). Since

1 |λ| = 1, so they have no zeros at 2 . And

∞ X −s −1 a(n)b(n)n = C(s)ζ(2s) L(s, ρ1 ⊗ ρ2). (5.7) n=1

−1 2 1+s We know that ζ(1 + s) = s + O(s ), and by changing variable s → 2 and setting s = 0 in other factors in the right-hand side of (4.7), we obtain

∞ X − 1+s 2 a(n)b(n)n 2 = C(1/2)L(1/2, ρ1 ⊗ ρ2)s + O(s ). n=1 Set c = C(1/2) which is not zero and we are done with (a). Proof of (b) is exactly the

same as that of (a). Done. 2

Putting together what we have done above, we obtain the following theorem.

Theorem 5.2.5 Let ρ1, ρ2 : Gal(Q/Q) −→ GL2(C) be two irreducible representations with determinant characters 1, 2, respectively. Assume they are both odd or even and both modular, corresponding to newforms f and h respectively. Let N be any positive

∗ ∗ integer divisible by the conductors of ρ1 and ρ2, and f (z) = f (z; N).

(a) If 1 = 2, then ∃ c 6= 0, s.t.,

∗ hf f, hi = cL(1/2, ρ1 ⊗ ρ2);

−1 (b) If 1 = 2 , then ∃ c 6= 0, s.t.,

∗ hf f, K(h)i = cL(1/2, ρ1 ⊗ ρ2);

Proof. If ρ1, ρ2 are both odd, i.e. f and h are holomorphic, and 1 = 2, then Lemma

5.2.1(a) and Lemma 5.2.4(a) together imply that ∃ c1 6= 0 s.t.

∗ −1/2 [Γ(1) : Γ0(N)]hf f, hi = c1(4π) Γ(1/2)L(1/2, ρ1 ⊗ ρ2). Chapter 5. Critical Values 65

Here all factors except the inner product and the L-function factor are nonzero and we

obtain the existence of c, which proves one of the two cases in (a).

All other cases can be shown similarly by simply combining Lemma 5.2.1, Lemma

5.2.2 and Lemma 5.2.4. Done. 2

5.3 Applications

Now we can apply Theorem 5.2.5 to any modular irreducible 2-dimensional Galois rep-

resentation ρ. In particular, it applies to all cases except the icosahedral case, the last

unproven case in Strong Artin Conjecture. Since all these cases are similar, we shall only

deal with the example when the image of ρ is isomorphic to Se4.

Let us look at the character table of Se4 in Appendix A.3.

2 2 0 0 00 0 0 0 It is trivial to check that χ3 = Sym (χ2), Sym (χ2) = χ2 + χ1, χ2 = χ2χ1, χ3 = χ3χ1

0 and χ4 = χ2χ2. Explicitly, χ2 corresponds to ρ where √ ρ : GL2(F3) −→ GL2(Z[− 2]) ⊂ GL2(C)     −1 1 −1 1       7−→   −1 0 −1 0     1 −1 1 −1       7−→  √ √  . 1 1 − −2 −1 + −2

00 0 So det χ2 = det χ2 is not trivial, hence must be χ1. Consequently,

2 0 00 0 χ2 = χ3 + χ1 and χ2χ2 = χ3 + χ1.

0 Moreover, the representation corresponds to χ2 is obtained by the irreducible 2-dimensional

representation of S3 via the surjective homomorphisms: Se4 → S4 → S3. The determinant

0 0 of that of S3 is not trivial and we know det(χ2) = χ1 too. We know that the strong Artin conjecture is true for all those two dimensional rep-

resentations. Apply Theorem 5.2.5, and we obtain Chapter 5. Critical Values 66

Theorem 5.3.1 Let fi (i = 1, 2, 3) be the three modular forms (either all holomorphic

0 00 or all real analytic) for χ2, χ2 and χ2, resp.. Then there exist non-zero constants cj, j = 1, 2, 3, such that

∗ 0 hf f1, f1i = c1L(1/2, χ1)L(1/2, χ3),

∗ 0 hf f1, f3i = c2L(1/2, χ3),

∗ hf f1, f2i = c3L(1/2, χ4).

Proof. By the above observations, we can directly apply Theorem 5.2.5 for each pair,

(f1, f1), (f1, f3) and (f1, f2). Using the above decompositions and grouping all non-zero constants together, the theorem follows. 2

1 In applying these results to non-vanishing result at 2 , calculating the Petersson in- ner products on the left hand sides becomes crucial. It is totally mysterious how the multiplication of f ∗ affects the spectrum decomposition. Chapter 6

Modularity of S5-Galois Representations

In this chapter, we shall give a sufficient condition for the modularity of irreducible

4-dimensional Galois representations ρ with image S5, the permutation group on five letters. Roughly speaking, it reduces the modularity of ρ to the modularity of an icosa-

hedral representation over a quadratic field. Please see Appendix for the character tables

involved.

6.1 Introduction

Given a number field F , denote by GF the absolute Galois group of F , that is GF = Gal(F¯ /F ).

Let ρ : GF → GL4(C) be a four dimensional Galois representation with image isomor-

phic to S5. This makes sense since we know there are two four dimensional representations

for S5 (see Table A.2 in Appendix).

Let K/F be the S5-Galois extension fixed by ker(ρ), so ker(ρ) = GK . We say K/F

has a double cover Ke, if K/Ke is Galois and Gal(K/Ke ) = Se5, where Se5 is the one of the

two C2-central extensions in which a transposition in S5 has preimages of order 2 in Se5.

67 Chapter 6. Modularity 68

Here C2 is the group of order 2. Assume this is the case, and denote E the subfield of K which is quadratic over F .

Then Gal(K/E) = A5 and K/Ee is Galois with Gal(K/Ee ) ' Ae5, the unique double cover

of A5. Hence we have a two dimensional (icosahedral) Galois representation σ of GE with ker(σ) = G . Here is the picture: Ke

Ke 2 tt tt tt K Ae5 KK A5 KK KK S5 E sss sss2 F s In this chapter, we consider the modularity of ρ and our main theorem says that the modularity of σ implies that of ρ. The idea of the proof is to connect ρ with the Asai lift of σ by simple computations of characters.

In the last section, we will discuss an open problem, namely, how to construct an affirmative example for the Langlands Functoriality Conjecture in this case.

6.2 Modularity of ρ and σ

Before we get to our main theorem, let us first recall the concept of Asai lift (see [28] or

[11] for details). Let E/F be a quadratic extension of number fields with ring of adeles

AE, AF , respectively.

On the Galois side, let σ : Gal(F /E) → GL2(C) be an irreducible representation. The Asai lift of σ, denoted by As(σ), is given by

2 F F ∧ (IndE(σ)) = (As(σ) ⊗ ωE/F ) ⊕ IndE(det(σ)),

where ωE/F is the quadratic gr¨osencharacter of F attached to E/F by class field theory.

On the automorphic side, let G = ResK/F GLn be quasi-split group obtained by Chapter 6. Modularity 69 restriction of scalars and we know the L-group is

L G = (GL2(C) × GL2(C)) o Gal(E/F ), where the generator θ of Gal(E/F ) acts by changing the two factors. One defines a representation

L 2 2 ∼ L r : G → GL4(C) = GL(C ⊗ C ) = GL4, by setting, for x, y ∈ C2,

r(g, g0; 1)(x ⊗ y) = g(x) ⊗ g0(y); r(1, 1; θ)(x ⊗ y) = y ⊗ x.

Let π = ⊗vπv be a cuspidal automorphic representation of G(AF ) = GL2(AE) where

L each Πv has the local parametrization φv : WFv × SL2(C) → G by local Langlands correspondence. Let As(πv) be the irreducible admissible representation of GL4(Fv) attached to r◦φv, and then As(π) = ⊗vAs(πv) is an irreducible admissible representation of GL4(AF). The following theorem is first proved by Ramakrishnan [28] and then by Krishnamurthy [14] and Kim [11] using different methods.

Theorem 6.2.1 (Ramakrishnan [28]) The Asai lift of π is automorphic.

Let the notations be the same as those in the introduction of this chapter. Now we can state and prove our main theorem.

Theorem 6.2.2 Assume K has a double cover. Then if σ is modular, so is ρ.

Proof. Now σ is one of the two dimensional representations of Ae5 corresponding to χ2

0 Se5 or χ2 in Table A.4, Appendix. Direct calculation shows that Ind σ has character χ4 in Ae5 e Table A.5, Appendix. Moreover, Λ2(IndSe5 σ) has character row Ae5

6 6 2 3 3 0 0 2 -1 -1 1 1 . Chapter 6. Modularity 70

2 Of course det(σ) = Λ (σ) is trivial since the only one dimensional character of Ae5 is the trivial one (see Table A.4). So we can get the character row of IndSe5 (det(σ)) as Ae5 follows:

220220020022 .

0 As we know(here χ1 is the one in Table A.5),

0 2 Se5 Se5 Λ (Ind σ) = As(σ) ⊗ χ1 ⊕ Ind (det(σ)). Ae5 Ae5

0 And finally we get the character row of As(σ) ⊗ χ1

4 4 2 1 1 0 0 0 -1 -1 -1 -1

which is exactly χ4 of Se5. So if we know that σ is modular, then so is As(σ) and ρ, since the character of ρ is

0 0 either χ4 or χ4 = χ4χ1. We are done. 2

6.3 An Open Problem

The proof of the modularity of general ρ may be too much to ask for the moment.

However, just as Buhler did for an icosahedral representation in [3], it is natural and also

interesting to ask the following question:

Open Problem: Can we find an affirmative example, that is, a modular four-dimensional

S5-Galois representation over Q?

Theorem 6.2.2 suggests that we find an S5-field K that has double cover and then it reduces to show that the corresponding two dimensional icosahedral representation is

modular. Let us see some results that may be helpful for solving this problem.

Let us assume from now on F = Q and let f ∈ Q[X] be a irreducible quintic polyno- ∼ mial whose splitting field is K and Gal(K/Q) = S5. Assume L is one of the root fields for f, so K is the Galois closure of L. Chapter 6. Modularity 71

Let QL be the trace form of L/Q. Let dL denote the discriminant of the field L/Q. We quote the following result from [31], which is Proposition 1 therein.

Theorem 6.3.1 (Serre) Ke exists if and only if QL is equivalent to I3 ⊕ h2, 2dLi.

Suppose Ke exists and we want to attach a Hilbert modular form to the resulting

σ. We need dL > 0, since the quadratic subfield needs to be real. Then based on the corollary to Proposition 7 in [31] and some simple calculations, we know that for the

splitting field of the quintic polynomials X5 + aX + b with a, b ∈ Q to have a double

cover, dL must be negative. So we need more terms in the polynomials. Among some examples that we worked on, let us note down the following:

f(X) = X5 − 6X3 − X2 + 4X − 1.

It is irreducible and has splitting field an S5-field with double cover. The discriminant is √ 36497; 36497 is a prime and the narrow class number of Q( 36497) is one. Now the main difficulty is the modularity of σ and we do not have many results on

the general theory or explicit constructions. The idea of Achimescu and Saha in [1] seems

not feasible in practice. Appendix: Character Tables

In this appendix, we will present some character tables for some finite groups that appear in this thesis. In those tables, the first row gives some typical representative of each conjugacy class and the second row consists of the sizes of all conjugacy classes.

A.1 : Character Table of A5

As usual, A5 is considered as the group of even permutations on five letters. Here √ √ 1+ 5 1− 5 u = 2 , v = 2 . This table is available in many references, for example [9].

Class 1 (123) (12)(34) (12345) (13452)

Size 1 20 15 12 12

χ1 1 1 1 1 1

χ4 4 1 0 -1 -1

χ5 5 -1 1 0 0

χ3 3 0 -1 u v

0 χ3 3 0 -1 v u

72 Chapter 6. Modularity 73

A.2 : Character Table of S5

As usual, S5 is the group of permutations on five letters. This table is also available in [9].

Class 1 (12) (123) (1234) (12345) (12)(34) (12)(345)

Size 1 10 20 30 24 15 20

χ1 1 1 1 1 1 1 1

0 χ1 1 -1 1 -1 1 1 -1

χ4 4 2 1 0 -1 0 -1

0 χ4 4 -2 1 0 -1 0 1

χ6 6 0 0 0 1 -2 0

χ5 5 1 -1 -1 0 1 1

0 χ5 5 -1 -1 1 0 1 -1 Chapter 6. Modularity 74

A.3 : Character Table of Se4

Here S4 is the permutation group on 4 letters and Se4 is the degree 2 non-trivial central extension of S4, up to isomorphism, where a transposition has only preimages of order 2. Degree 2 non-trivial central extension means that

1 → C2 → Se4 → S4 → 1

is a non-split exact sequence and C2 is in the center of Se4. Here C2 is the cyclic group of order 2. It turns out that Se4 ' GL2(F3). See [12] for this table.

            1 1 2 1 1 0 0 −1 1 2 2 2 class I −I             0 1 0 2 0 −1 1 0 1 1 1 2 size 1 1 8 8 12 6 6 6

χ1 1 1 1 1 1 1 1 1

0 χ1 1 1 1 1 −1 1 −1 −1 √ √ χ2 2 −2 −1 1 0 0 − −2 −2

0 χ2 2 2 −1 −1 0 2 0 0 √ √ 00 χ2 2 −2 −1 1 0 0 −2 − −2

χ3 3 3 0 0 1 −1 −1 −1

0 χ3 3 3 0 0 −1 −1 1 1

χ4 4 −4 1 −1 0 0 0 0 Chapter 6. Modularity 75

A.4 : Character Table of Ae5

Ae5 is the unique degree 2 non-trivial central extension of A5, up to isomorphism. It

turns out that Ae5 ' SL2(F5). See [9] for the computation of this table and also Table App7.1 in [3]. Note that there are some misprints in the latter table.

5 5 0 0 0 2 2 2 0 Class (1 ) (1 ) (5)1 (5)2 (5)1 (5)2 (12 ) (13 ) (13 ) Order 1 1 12 12 12 12 30 20 20

χ1 1 1 1 1 1 1 1 1 1

χ5 5 5 0 0 0 0 1 -1 -1

χ6 6 -6 1 1 -1 -1 0 0 0

χ3 3 3 u v u v -1 0 0

0 χ3 3 3 v u v u -1 0 0

χ4 4 4 -1 -1 -1 -1 0 1 1

0 χ4 4 -4 -1 -1 1 1 0 -1 1

χ2 2 -2 -u -v u v 0 1 -1

0 χ2 2 -2 -v -u v u 0 1 -1

√ √ 1+ 5 1− 5 As before u = 2 , v = 2 . We use the subscripts 1,2 to distinguish the possible two 0 classes in A5 belonging to the given partition and use the superscript to distinguish the possible two classes in Ae5 with the same projection in A5. Chapter 6. Modularity 76

A.5 : Character Table of Se5

Se5 is one of the two degree 2 non-trivial central extension of S5, up to isomorphism,

+ where a transposition has only preimages of order 2. Some authors denote it by Se5 and

− denote the other one by Se5 . The character tables of them are closely related and can be computed from each other easily. See [21] for the computations.

Class (15) (15)0 (132) (123) (123)0 (14) (14)0 (122) (23) (23)0 (5) (5)0

Order 1 1 20 20 20 30 30 30 20 20 24 24

χ1 1 1 1 1 1 1 1 1 1 1 1 1

0 χ1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1

χ4 4 4 2 1 1 0 0 0 -1 -1 -1 -1

0 χ4 4 4 -2 1 1 0 0 0 1 1 -1 -1

χ5 5 5 1 -1 -1 -1 -1 1 1 1 0 0

0 χ5 5 5 -1 -1 -1 1 1 1 -1 -1 0 0

χ6 6 6 0 0 0 0 0 -2 0 0 1 1

χe4 4 -4 0 2 -2 0 0 0 0 0 1 -1 0 √ √ χe4,1 4 -4 0 -1 1 0 0 0 3 − 3 1 -1 0 √ √ χe4,2 4 -4 0 -1 1 0 0 0 − 3 3 1 -1 0 √ √ χe6,1 6 -6 0 0 0 2 − 2 0 0 0 -1 1 0 √ √ χe6,2 6 -6 0 0 0 − 2 2 0 0 0 -1 1 Bibliography

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