Mock-Modular Forms of Weight One

UCLAUCLA Electronic Theses and Dissertations

TitleMock-Modular Forms of Weight One

Permalinkhttps://escholarship.org/uc/item/0d84g565

AuthorLi, Yingkun

Publication Date2013-01-01 Peer reviewed|Thesis/dissertation

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Mock-Modular Forms of Weight One

A dissertation submitted in partial satisfaction

of the requirements for the degree

Doctor of Philosophy in Mathematics

by

Yingkun Li

2013

c© Copyright by

Yingkun Li

2013

Abstract of the Dissertation

Mock-Modular Forms of Weight One

by

Yingkun Li

Doctor of Philosophy in Mathematics

University of California, Los Angeles, 2013

Professor William Duke, Chair

In this thesis, we will study mock-modular forms of weight one and their Fourier coef-

ficients. In particular, we will concentrate on the mock-modular forms whose shadows are

dihedral newforms arising from ray class group characters of imaginary quadratic fields. We

will show that certain linear combinations of their Fourier coefficients are logarithms of CM

values of the modular j-function We will also make a conjecture about the algebraicity of

the individual Fourier coefficients.

ii

The dissertation of Yingkun Li is approved.

Don Blasius

Haruzo Hida

Yingnian Wu

William Duke, Committee Chair

University of California, Los Angeles

2013

iii

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction and Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 Integral Weight Modular Forms . . . . . . . . . . . . . . . . . . . . . 6

1.2.2 Half-Integral Weight Modular Forms . . . . . . . . . . . . . . . . . . 7

1.2.3 Mock-modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1 Existence of Mock-Modular Forms of Weight One . . . . . . . . . . . . . . . 10

2.2 Principal Part Coefficients of Harmonic Maass Forms . . . . . . . . . . . . . 17

2.3 Transformation Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Commutation Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.2 Trace Down Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Weight One Space Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.1 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.2 Applications of the Projection Operators . . . . . . . . . . . . . . . . 31

3 Weight One Newforms of Imaginary Dihedral Type . . . . . . . . . . . . . 37

3.1 Newforms of Weight One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Structure of Pic(O) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Modular Forms of Imaginary Dihedral Type . . . . . . . . . . . . . . . . . . 41

3.3.1 Ray Class Group and Its Characters . . . . . . . . . . . . . . . . . . 41

3.3.2 Genus Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.3 Relationship to Characters of Pic(Op) . . . . . . . . . . . . . . . . . . 46

iv

3.4 Mock-Modular Forms with Imaginary Dihedral Shadow . . . . . . . . . . . . 49

3.4.1 Petersson Inner Product of Newforms . . . . . . . . . . . . . . . . . . 49

3.4.2 Principal Part Coefficients of Mock-Modular Forms . . . . . . . . . . 54

4 Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1 Counting Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1.2 Counting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Fourier Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3 Fourier Coefficients and Values of Modular Functions . . . . . . . . . . . . . 85

4.3.1 Borcherds Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3.2 Automorphic Green’s Function . . . . . . . . . . . . . . . . . . . . . 86

4.3.3 Holomorphic Projection . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3.4 Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 91

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

v

Acknowledgments

First, I would like to express my deepest gratitude to my advisor, Prof. Bill Duke, for

providing such an interesting thesis topic and for his generosity with time and ideas, without

which this thesis would not exist. I would also like to heartily thank Prof. Don Blasius, Prof.

Haruzo Hida and Prof. Chandrashekhar Khare for teaching many classes and seminars, from

which I learned a great deal of modern number theory. I am grateful to Prof. Yingnian Wu

for serving on my thesis committee.

Throughout my graduate studies, I have benefited from discussing mathematics with the

students and postdocs in the math department, and am very thankful for their fellowships.

The staffs at the department of mathematics at UCLA has been a great source of support,

both technical and non-technical and I am very appreciative for all their helps.

Finally, I would like to thank my family for their supports, sacrifices and constant en-

couragements.

vi

Vita

2005–2009 B.S. (Mathematics) Caltech, Pasadena, California.

2009–2011 M.A. (Mathematics), UCLA, Los Angeles, California.

Publications

K. Bringmann, Y. Li, R. Rhoades, Asymptotics for the number of row-Fishburn matrices,

submitted (2013)

W. Duke, Y. Li, Harmonic Maass Forms of Weight One, submitted (2012)

J. Brown, Y. Li, Level lowering for half-integral weight modular forms, Proc. Amer. Math.

Soc. 138 (2010), 1171-1173

J. Brown, Y. Li, Distribution of the Powers of the Partition Function modulo `j, J. Number

Theory 129 (2009), 2557-2568

vii

CHAPTER 1

Introduction

1.1 Introduction and Setup

The main object we are interested in studying, mock-modular forms, started with the last

letter of Ramanujan to Hardy, dated January 1920. In this letter, Ramanujan described a

class of q-series, which he called “Mock ϑ function”, by writing down several examples and

stating interesting combinatorial and asymptotic properties. In contrast to other q-series,

such as the generating function of the partition function, these examples cannot be made

holomorphic and modular, hence the name “mock ϑ function”. Nevertheless, their properties

were quite similar to those of the modular theta functions. Unfortunately, Ramanujan passed

away before he could give the definition of his mock ϑ function, and left the world 17 such

examples and a big mystery. Over the next eighty years, many people, including G. E.

Andrews, L. Dragonette, A. Selberg and Watson, have studied these special examples in the

absence of a uniform theory of mock-modular forms [1, 20, 47, 55, 56].

In 2003, Dutch mathematician Sander Zwegers gave a defining property of mock-modular

forms in his thesis [63], by realizing them as the holomorphic part of a non-holomorphic

modular form f(z). Furthermore, these non-holomorphic modular forms have poles and are

annihilated by the weight k Laplacian operator

∆k = ξ2−k ◦ ξk, ξk := 2iyk∂z. (1.1.1)

Fittingly, these non-holomorphic modular objects are called harmonic weak Maass forms.

The differential operator ξk is anti-holomorphic and commutes with the slash operator by

changing the weight from k to 2 − k and conjugating the nebentypus character. So the

1

function ξkf is a holomorphic modular form of weight 2− k and is called the shadow of the

mock-modular form.

The case of weight k = 1 has always been mysterious and important in the theory of

modular forms. One of the mysteries is the number of weight one modular forms. Unlike

in the cases when k ≥ 2, the dimension of the space of modular forms over the complex

numbers is unknown as the Riemann-Roch theorem yields trivial information. Using analytic

techniques, various people have obtained asymptotic bounds on the dimensions of these

spaces (see for example [4, 21, 41]). Over finite field, there are more weight one modular

forms, some of which does not even come from reduction of modular forms over Q (see [45]).

Another important features of weight one modular forms is their connection to Galois rep-

resentations. By the Deligne-Serre’s theorem, one could attach to each weight one newform

f an odd, irreducible Artin representation ρf of Gal(Q/Q)

ρf : Gal(Q/Q) −→ GL2(C).

Since ρf is continuous, it has finite image and the field fixed by ker ρf is an algebraic number

field over Q. Let ρf be the composition of ρf and the surjection GL2(C) −→ PGL2(C).

Then the image of ρf , which is finite, is isomorphic to one of the following groups

• Dihedral, or D2n,

• Tetrahedral, or A4,

• Octahedral, or S4,

• Icosahedral, or A5.

We call modular forms with the last three types of projective images exotic and use this

classification to denote the types of weight one newforms and mock-modular forms. Fol-

lowing Langlands’ philosophy and works by Deligne-Serre, Langlands, Tunnell and Khare-

Wintenberger (see [34, 35, 39, 54]), this correspondence is functorial and bijective, and pro-

vides a bridge between weight one modular forms, which are a priori complex analytic, with

2

algebraic number fields. This correspondence also enables one to check Stark’s conjecture

on L-series attached to weight one modular forms [51].

In this thesis, we will study the Fourier coefficients of weight one mock-modular forms,

whose shadows are newforms. The case where the newform arises from a class group character

of an imaginary quadratic field has been treated in [24, 26, 58]. Here, we will generalize the

techniques in [24] to treat the case when the newform arises from a ray class group character

of an imaginary quadratic field. The main goal is to relate the linear combinations of these

Fourier coefficients to logarithms of CM values of modular functions

To be precise, let D ≡ 1 (mod 4) be an odd, negative fundamental discriminant and

χD =(D·

)the associated Dirichlet character. Let p be an odd prime with χD(p) = 1. It

splits into pp in the imaginary quadratic field K = Q(√D). Let φ : Gal(K/K) −→ C× be a

ray class group character with modulus p such that the induced representation

ρφ := IndQK(φ) : Gal(Q/Q) −→ C×

is odd and irreducible. Then det(ρφ) = χDφ1 with φ1 a character of conductor p defined by

φ1 : (Z/pZ)× −→ Ip/Pp,1φ−→ C×.

Denote the weight one newform associated to ρφ by fφ(z). Then it is a newform of level |D|p

and character χ−7φ1. For example, suppose D = −7, p = 11 and φ1(2) = ζ5. Then there

exists a newform in fφ(z) ∈ S1(77, χDφ1) with the Fourier expansion

fφ(z) = q +(−ζ3

5 − ζ5 − 1)q2 +

(−ζ2

5 − 1)q4 + ζ5q

7 +(ζ3

5 + ζ25 + ζ5 + 1

)q8 + ζ3

5q9 + ζ2

5q11

+(ζ3

5 + 1)q14 +

(ζ2

5 + 1)q18 +

(−ζ3

5 − ζ25 − 1

)q22 +

(−ζ3

5 − ζ25 − 1

)q23 +O(q25).

In general, the dimension of S1(|D|p, χDφ1) is expected to be the class number of D plus the

number of exotic forms in this space.

For a discriminant D′ < 0, let C(D′) be the set of positive definite binary quadratic forms

Q = [A,B,C] with discriminant D′. To each Q, one could associate a point τQ ∈ H. The

group SL2(Z) has an action on C(D′), which translates into linear fractional transformation

on τQ. We use wQ to denote the size of the stabilizer of this action on Q ∈ C(D′). A binary

3

quadratic form is called primitive if gcd(A,B,C) = 1. This property is preserved by the

action of SL2(Z). Let C(D′) be the group of equivalence classes of primitive binary quadratic

forms of discriminant D′. One could evaluate modular functions, such as j(z), at these τQ

and obtain algebraic values by the theory of complex multiplication. Furthermore, this value

only depends on the equivalence class of Q. We could now state a simple case of the main

result.

Theorem 1.1.1. Suppose D < −5 is a prime, fundamental discriminant such that the space

S1(|D|p, χDφ1) is one dimensional. Then there exists a unique mock-modular form fφ(z)

with shadow fφ(z) ∈ S1(|D|p, χDφ1) and Fourier expansion

fφ(z) = cφ(−1)q−1 +∑n>1

χD(n)=−1

cφ(n)qn (1.1.2)

at the cusp infinity. Furthermore, for any fundamental discriminant D′ < 0 satisfying

χD(D′) = −1, we have∑k∈Z

cφ

(p2DD′ − k2

4

)φ1

(k

2

)δD(k) = −4

∑Q∈C(Dp2)Q′∈C(D′)

ψ2(Q) log |j(τQ)− j(τQ′)|2/wQ′ , (1.1.3)

where ψ : Pic(Op) −→ C× is the ring class group character associated to φ as in Prop.

(3.3.4).

Remark 1.1.2. The case that dimS1(|D|p, χDφ1) = 1 happens when the class number of D

is 1, φ1 has order greater than two and there is no exotic form in S1(|D|p, χD, φ1). For the

general version of the main result, see Theorem 4.3.4.

This is result is the analogue of Theorem 1.2 in [24]. Its generalization will be helpful for

the future when we study mock-modular forms with shadows arising from ray class group

characters of real quadratic field.

Suppose Q has discriminant Df 2 with D an odd, negative fundamental discriminant. By

the theory of complex multiplication, the quantity∏Q′∈C(D′)

(j(τQ)− j(τQ′))

4

is an algebraic number lying in the ring class field of K of conductor f , denoted by Hf . In

[28], Gross and Zagier gave a factorization of the rational norm of this quantity when f = 1

and gcd(D,D′) = 1. They gave two proofs of this factorization, one algebraic, one analytic.

The algebraic proof in fact gives the valuation of this quantity at various primes in the Hilbert

class field of K. Later, people have given various generalizations of this factorization, both

to non-fundamental discriminants (see [19, 40]) and to Hilbert modular functions (see [15]).

These factorizations prompt us to make the following conjecture regarding the individual

coefficient cφ(n).

Conjecture 1.1.3. Let D be an odd, negative fundamental discriminant, K = Q(√D) and

φ : Gal(K/K) −→ C× a character modulo p such that φ1 : (Z/pZ)× −→ C× has order greater

than two. Let fφ,1(z) ∈ S1(|D|p, χDφ1) be the cusp form associated to fφ(z) as in Eq. (3.3.13).

Then there exists κ ∈ Z, u(n,A′) ∈ OHp and a mock-modular form fφ,1(z) =∑

n≥−n0cφ(n)qn

with shadow fφ(z) such that

cφ(n)φ1(√n) =

1

κ

∑A∈Pic(Op)

ψ2(A′) log |u(n,A′)|,

and σC′(u(n,A′)) = u(n,A′C ′−1), where σC′ ∈ Gal(Hp/K) is associated to C ′ ∈ Pic(Op) via

class field theory.

The structure of the thesis is as follows. In Chapter 1, we will give a brief introduction to

mock-modular forms. In Chapter 2, some preliminary results on existence of mock-modular

forms and modular form transformations are given. In Chapter 3, we give the facts on weight

one newforms, such as their associated Galois representation and Petersson inner products.

In Chapter 4, we will give the proof of Theorem 4.3.4, from which Theorem 1.1.1 can be

deduced.

1.2 Modular Forms

In this section, we will give some basic background on modular forms, following the reference

[36].

5

1.2.1 Integral Weight Modular Forms

For any commutative ring R, denote the 2 × 2 matrices with entries in R by M2(R). The

groups GL2(R),GL+2 (R) and SL2(Z) are defined by

GL2(R) :={

( a bc d ) ∈M2(R) : ad− bc ∈ R×},

GL+2 (R) := {( a bc d ) ∈M2(R) : ad− bc > 0} ,

SL2(Z) := {( a bc d ) ∈M2(Z) : ad− bc = 1} .

When R = Z and M ∈ N, we can define the congruence subgroups Γ0(M) by

Γ0(M) := {( a bc d ) ∈ SL2(Z) : c ≡ 0 (mod M)} .

Let H := {z ∈ C : Im(z) > 0} be the upper half plane. The group GL+2 (R) acts on H via

linear fractional transformation, i.e. for γ ∈ GL2(R) and z ∈ H, we have

γz :=az + b

cz + d.

Similarly, Γ0(M) also act on H by linear fractional transformation. Modulo this action, the

upper half plane becomes an open Riemann surface possibly with pinched points. It can be

compactified by adding P1(Q) modulo the action of Γ0(M), which is a finite set. We call

points in this set cusps.

Given a function f(z) on H and a integer k, the group GL2(R) acts on f(z) by the weight

k slash operator defined by

(f |k γ) := (det γ)k/2(cz + d)−kf(γz),

where γ = ( a bc d ) ∈ GL+2 (R). For a Dirichlet character χ : (Z/MZ)∗ −→ C×, one could view

it as a character of Γ0(M) via χ(γ) := χ(d). Now, we can define modular forms of integral

weight.

Definition 1.2.1. Let k ≥ 0 be an integer, M a positive integer and χ : (Z/MZ)∗ −→ C×

a character such that χ(−1) = (−1)k. Then a function f(z) on the upper half plane is

called a modular form of level M and nebetypus character χ if it satisfies the following three

conditions

6

(1) f(z) is holomorphic.

(2) The following equation holds for all γ = ( a bMc d ) ∈ Γ0(M)

(f |k γ)(z) = χ(d)f(z).

(3) f(z) does not have poles on H ∪ P1(Q).

We denote the C-vector space of function by Mk(M,χ). Suppose f(z) satisfies the fol-

lowing stronger version of (3)

(3)’ f(z) does not have poles on H and vanishes at all cusps,

then f(z) is called a cuspform and the C-vector space of all such functions is denoted by

Sk(M,χ).

1.2.2 Half-Integral Weight Modular Forms

When the weight k above is a half-integer, then the weight k slash operator is not well-defined

for GL+2 (R). Different branch choices are the reasons for this problem. To overcome it, we

define the group G, which is a four-sheeted cover of GL+2 (R), by

G :={

(α, φ(z)) | α = ( a bc d ) ∈ GL+2 (R), φ(z) : H −→ C holomorphic , φ(z)2 = ± cz+d√

detα

}.

The group law is defined by

(α, φ(z)) · (β, ψ(z)) = (αβ, φ(βz)ψ(z)).

For any element γ = ( a bc d ) ∈ Γ0(4), define the automorphy factor j(γ, z) by

j(γ, z) :=(cd

)ε−1d

√cz + d,

where εd = 1, resp. i, if d is congruent to 1, resp. 3, modulo 4. Then there is a natural copy

of Γ0(4) in G via γ 7→ γ = (γ, j(γ, z)). We will denote the image of any congruence subgroup

Γ0(4M) in G by Γ0(4M).

7

Let k be an integer. For a function f(z) defined on H, (α, φ(z)) ∈ G acts on it with

weight k2

by

f |k/2 (α, φ(z)) := φ(z)−kf(αz).

With the half-integral weight slash operator defined, we can give the definition of modular

forms of half-integral weight.

Definition 1.2.2. Let k ≥ 1 be an integer, M an integer and χ : (Z/4Z)× −→ C× a char-

acter. A function f(z) on H is called a modular form of weight k2, level 4M and nebentypus

character χ if the following conditions are satisfied.

(1) f(z) is holomorphic.

(2) The following equation holds for all (α, φ(z)) ∈ Γ0(4M)

(f |k/2 (α, φ(z)))(z) = χ(α)f(z).

(3) f(z) does not have poles on H ∪ P1(Q).

We denote the C vector space of such functions by Mk/2(4M,χ). Similarly, we call f(z) a

cuspform of weight k2, level 4M and character χ if condition (3) is replaced by condition (3)’

above, and denote the space of cuspforms by Sk/2(4M,χ).

1.2.3 Mock-modular forms

In this section, we will give some background information on mock-modular forms. Let

k ∈ Z, M ∈ Z+ and χ : (Z/MZ)× −→ C× be as before. Denote by Fk(M,χ) the space of

smooth functions f : H → C such that

(f |k γ)(z) = χ(γ)f(z)

for all γ ∈ Γ0(M). Recall from Eq. (1.1.1) the differential operator ξk and the weight k

hyperbolic Laplacian ∆k. If z = x+ iy, then ∆k can be written as

∆k = y2(∂2

∂x2+ ∂2

∂y2

)− iky

(∂∂x

+ i ∂∂y

).

8

We say that f(z) ∈ Fk(M,χ) is a harmonic weak Maass form of weight k, level M and

character χ (or more briefly, a weakly harmonic form) if it satisfies the following properties.

(i) f(z) is real-analytic.

(ii) ∆k(f) = 0.

(iii) The function f(z) has at most linear exponential growth at all cusps of Γ0(M).

Let Hk(M,χ) be the space of weakly harmonic forms of weight k, level M and character

χ, whose image under ξk is a holomorphic modular form. Denote by M !k(M,χ) the usual

subspaces of weakly holomorphic modular forms. It also contains Mk(M,χ) and Sk(M,χ)

as subspaces as well. A mock-modular form is a formal Laurent series in q,

g(z) =∑

n�−∞

c+(n)qn,

such that there exists g(z) =∑

n≥0 c(n)qn ∈M2−k(M,χ) satisfying∑n�−∞

c+(n)qn −∑n≥0

c(n)βk(n, y)q−n ∈ Hk(M,χ).

The form g(z) is called the shadow of g(z). The expression∑

n<0 c+(n)qn is called the

principal part of g(z). Let Mk(M,χ) be the subspace of mock-modular forms whose shadows

are in M2−k(M,χ). Since every weakly harmonic form can be written uniquely as the sum

of a holomorphic part and a non-holomorphic part, the spaces Hk(M,χ) and Mk(M,χ) are

canonically isomorphic to each other.

With some computations, one could verify that ξk commutes with the slash operator as

follows

ξk (f |k γ) = (ξkf) |2−k γ (1.2.1)

for all γ ∈ GL2(R). Property (ii) and Eq. (1.1.1) then gives the following map

ξk : Hk(M,χ) −→M2−k(M,χ),

whose kernel is exactly M !k(M,χ). When k 6= 1, the map above is surjective as shown in [10]

and [12]. When k = 1, one can still prove surjectivity by analytically continuing the weight

one Poincare series, the same family as in [10] for k = 1, via spectral expansion. We will

carry this out in §2.1.

9

CHAPTER 2

Preliminary Results

2.1 Existence of Mock-Modular Forms of Weight One

Given a cusp form of weight one, we will show the existence of a weight one mock-modular

forms with it as shadow in this section. There are several different approaches to proving this

result, such as a geometric approach in [12], or another approach found by Zwegers using the

holomorphic projection trick. For completeness and since it might have some independent

interest, we prove the existence by analytically continuing weight one Poincare series via the

spectral expansion.

From now on we fix k = 1 and write | for the slash operator |1. The notations H1(M,χ),

M !1(M,χ), M1(M,χ) and S1(M,χ) are the same as in §1.2. Let M1(M,χ) be the space of

mock-modular forms, which is canonically isomorphic to H1(M,χ).

To proceed, we will construct two families of Poincare series Pm(z, s), Qm(z, s), where

the first family is similar to the one used in [10]. They are a priori defined for Re(s) > 1

and will be analytically continued to Re(s) > 0 through their spectral expansions. Unlike

the cases k ≥ 2, this will only be a statement about existence, and not a formula that could

be used to calculate the Fourier coefficients of the preimage explicitly. To prove the analytic

continuation, we will refer to results in [42] and [44].

Given any positive integer m, define

φ∗m(z, s) := e2πimx(4π|m|y)−1/2M sgn(m)2

,s−12(4π|m|y), (2.1.1)

ϕ∗m(z, s) := e2πimx(4π|m|y)s−1/2e−2π|m|y. (2.1.2)

10

Here Mµ,ν(y) is the M -Whittaker function defined by

Mµ,ν(z) := zν+1/2ez/21F1

(12

+ µ+ ν; 1 + 2ν;−z), (2.1.3)

with 1F1(α; β; z) being the generalized hypergeometric function. Averaging them over the

coset representatives of Γ∞\Γ0(M), we can define the following Poincare series

Pm(z, s, χ) :=∑

γ∈Γ∞\Γ0(M)

χ(γ)(φ∗m | γ)(z, s),

Qm(z, s, χ) :=∑

γ∈Γ∞\Γ0(M)

χ(γ)(ϕ∗m | γ)(z, s).

For convenience, we shall omit χ and write Pm(z, s) and Qm(z, s) instead. Both families are

absolutely convergent for Re(s) > 1 and define a holomorphic function in s. Also, Pm(z, s)

is an eigenfunction of −∆1 with eigenvalue (s−1/2)((1−s)−1/2) and belongs to F1(M,χ).

Let ∆′k be the differential operator defined by

−∆′k := −y2(∂2

∂x2+ ∂2

∂y2

)+ iky ∂

∂x.

It is related to ∆k by the following equation

∆′k + k2

(1− k

2

)= yk/2∆ky

−k/2.

Define the space D1(M,χ) by

D1(M,χ) := {y1/2f(z) : f(z) ∈ F1(M,χ) is smooth with compact support on H} (2.1.4)

Let D1(M,χ) be the completion of D1(M,χ) with respect to the Petersson norm

||g||2 = 〈g, g〉 :=

∫Γ0(M)\H

|g(z)|2 dxdyy2.

Satz 3.2 in [44] implies that −∆′k has a self-adjoint extension −∆′k from D1(M,χ) to

D1(M,χ). Also, −∆′k has a countable system of Maass cusp forms {en(z)}n∈N ⊂ D1(M,χ)

forming the discrete spectrum. Each Maass cusp form en(z) has eigenvalue λn and each

eigenvalue has finite multiplicity. For any f ∈ D1(M,χ), the discrete spectrum contributes

the following sum to its spectral expansion

∞∑n=1

〈f, en〉en(z),

11

which converges absolutely and uniformly for all z ∈ H [44, Satz 8.1].

Let ι be a cusp of Γ0(M), σι ∈ GL2(R) the scaling matrix sending∞ to ι, and Γι ⊂ Γ0(M)

the subgroup fixing ι. One can define the weight one real analytic Eisenstein series Eι(z, s)

by

Eι(z, s) := y1/2

2

∑γ∈Γι\Γ0(M)

χ(γ)cz+d

(Im(σ−1ι γz))s−1/2.

Selberg showed that the Eisenstein series has meromorphic continuation to the whole complex

plane in s. For ι ranging over the cusps of Γ0(M), the Eisenstein series {Eι(z, s)}ι make

up the continuous spectrum of −∆′1. So for any f ∈ D1(M,χ), the contribution of the

Eisenstein series to the spectral expansion of f is

1

4π

∑ι

∫ ∞−∞〈f(·), Eι(·, 1

2+ ir)〉Eι(z, 1

2+ ir)dr.

If f is smooth, then the integral in r converges absolutely and uniformly for z in any fixed

compact subset of H [44, Satz 12.3]. Applying the completeness theorem [44, Satz 7.2 ], we

have the spectral expansion for any smooth f ∈ D1(M,χ) in the form

f(z) =∞∑n=1

〈f, en〉en(z) +∑ι

1

4π

∫ ∞−∞〈f(·), Eι(·, 1

2+ ir)〉Eι(z, 1

2+ ir)dr, (2.1.5)

where the sum over n ∈ N and the integral in r both converge uniformly and absolutely for

z in any compact set of H.

By setting f(z) = y1/2Qm(z, s) and comparing Pm(z, s) to Qm(z, s) as in [42], we can

deduce the following proposition.

Proposition 2.1.1. For any positive integer m, the Poincare series Qm(z, s) and Pm(z, s)

have analytic continuation to Re(s) > 0. At s = 1/2, Pm(z, s) has at most a simple pole.

Furthermore, the residues of Pm(z, s) at s = 1/2 generate S1(M,χ).

Proof. First, we will prove the analytic continuation Qm(z, s). For any m > 0, the function

Qm(z, s) := y1/2Qm(z, s) is square integrable for Re(s) > 1, hence contained in D1(M,χ).

12

So we can write out its spectral expansion

Qm(z, s) = D(z, s) + C(z, s), (2.1.6)

D(z, s) :=∞∑n=1

〈Qm(·, s), en(·)〉en(z),

C(z, s) := 14π

∑ι

∫ ∞−∞〈Qm(·, s), Eι(·, 1

2+ ir)〉Eι(z, 1

2+ ir)dr,

where D(z, s) and C(z, s) are the contributions from the dicrete and continuous spectrum

of −∆′1 respectively.

By Satz 5.2 and Satz 5.5 in [44], en(z) has eigenvalue λn ∈ [1/4,∞) under −∆′1. If

λn = 1/4, then y−1/2en(z) is in S1(M,χ). Since each eigenvalue has finite multiplicity, we

can let N0 ≥ 0 such that {y−1/2en(z) : 1 ≤ n ≤ N0} is an orthonormal basis of S1(M,χ).

Note that S1(M,χ) could be empty, in which case N0 = 0.

For n ∈ N, let tn =√λn − 1/4, sn = 1/2+itn. We can use Eq. (66) in [27], the asymptotic

Mµ,ν(y) = Oµ,ν(ey) as y →∞ and the vanishing property of cusp forms at all cusps to write

en(z) =∞∑

u=−∞,u6=0

cn,uW sgn(u)2

,sn−12(4π|u|y)e2πiux,

where Wµ,ν(z) is the W -Whittaker function and cn,u are constants. Note if tn = 0, i.e.

y−1/2en(z) is a holomorphic cusp form, then Wsgn(u)/2,sn−1/2(4π|u|y) = (4π|u|y)1/2e−2π|u|y

and cn,u = 0 for u ≤ 0. Now we can use the Rankin-Selberg unfolding trick to calculate

〈Qm(·, s), en(·)〉

〈Qm(·, s), en(·)〉 =

∫Γ0(M)\H

y1/2Qm(z, s)en(z)dxdy

y2

=

∫ ∞0

∫ 1

0

y1/2ϕ∗m(z, s)en(z)dxdy

y2

= (4π|m|)1/2

∫ ∞0

e−2π|m|y(4π|m|y)s−1cn,mW sgn(m)2

,sn−12(4π|m|y)

dy

y

= (4π|m|)1/2cn,mΓ(s− 1

2− itn

)Γ(s− 1

2+ itn

)Γ(s− sgn(m)

2

) (2.1.7)

The last step uses the Mellin transform of the W -Whittaker function [5, Eq. (8b)] and

the substitution sn = 1/2 + itn. When Re(s) > 1, the sum defining D(z, s) is absolutely

13

convergent [44, Satz 8.1], since y1/2Qm(z, s) ∈ D1(M,χ) for Re(s) > 1. When 0 < Re(s) ≤ 1,

we can write D(z, s) as

D(z, s) = (4π|m|)1/2∑n∈N

cn,mΓ(s+ 1− 1

2− itn

)Γ(s+ 1− 1

2+ itn

)Γ(s+ 1− sgn(m)

2

) en(z) ·s− sgn(m)

2

(s− 12)2 + t2n

Since t2n = λn − 1/4 and∑

n>M λ−2n converges [44, Satz 8.1], we can apply Cauchy-Schwarz

inequality to see that the sum on the right hand side above converges absolutely on compact

subsets of {s ∈ C : Re(s) > 0, s 6= 1/2}. At s = 1/2, the first N0 terms in the sum produce

a simple pole since tn = 0 for all 1 ≤ n ≤ N0. The rest of the sum still converges absolutely.

So the right hand side above gives the analytic continuation of D(z, s) to Re(s) > 0.

For the continuous spectrum, the contribution from the Eisenstein series on the right

hand side of (2.1.6) can be treated similarly. For any cusp ι, we can write the Fourier

expansion of Eι(z, s) at infinity in the following form (for ι =∞, see [27, eq. (76)’])

Eι(z, s) = ys + ψι(s)y1−s +

∑m 6=0

ψι,m(s)W sgn(m)2

,s−12(4π|m|y)e2πimx, (2.1.8)

where ψι(s) and ψι,m(s) are products of gamma factors and Selberg-Kloosterman zeta func-

tions. It is well-known that The Eisenstein series can be analytically continued in s to the

whole complex plane. When Re(s) > 1/2, the poles of Eι(z, s) are in the interval s ∈ (1/2, 1]

[44, Satz 10.3]. On the line Re(s) = 1/2, Eι(z0, s) is holomorphic in s for any fixed z0 ∈ H

[44, Satz 10.4]. So both ψι(s) and ψι,m(s) admit analytic continuation to Re(s) > 0 and are

holomorphic on Re(s) = 1/2. Using the same unfolding trick above, we can evaluate

〈Qm(·, s), Eι(·, 12

+ ir)〉 = ψι,m(

12

+ ir)(4π|m|)1/2 Γ

(s− 1

2− ir

)Γ(s− 1

2+ ir

)Γ(s− sgn(m)

2

) .

Then C(z, s) can be written as

C(z, s) =∑ι

1

4π

∫ ∞−∞

ψι,m(

12

+ ir)(4π|m|)1/2 Γ

(s− 1

2− ir

)Γ(s− 1

2+ ir

)Γ(s− sgn(m)

2

) Eι(z,12

+ ir)dr

=∑ι

1

4π

∫ ∞−∞

ψι,m(

12

+ ir)(4π|m|)1/2 Γ

(s+ 1− 1

2− ir

)Γ(s+ 1− 1

2+ ir

)Γ(s+ 1− sgn(m)

2

) (2.1.9)

·(s− sgn(m)

2)

(s− 12− ir)(s− 1

2+ ir)

· Eι(z, 12

+ ir)dr.

14

When Re(s) > 0, C(z, s + 1) is absolutely convergent. So when Re(s) 6= 1/2, we can apply

Cauchy-Schwarz to bound expression (2.1.9) by∫ ∞−∞

∣∣∣∣∣ s− sgn(m)2

(s− 12)2 + r2

∣∣∣∣∣ dr. (2.1.10)

When Re(s) > 1/2, let C(z, s) be the expression (2.1.9), which gives the analytic continuation

of C(z, s) in this region. When Re(s) < 1/2, we can define C(z, s) in a similar fashion as

in §6 of [42]. We start with (2.1.9) and Re(s) ∈ (1/2, 1/2 + ε) for some small ε > 0, then

deform the contour so that it goes above 1/2−si

and below s−1/2i

. In the process, the following

residue is picked up

(π|m|)1/2Γ(2s− 1)

Γ(s− sgn(m)

2

) ∑ι

ψι,m(s)Eι(z, s) + ψι,m(1− s)Eι(z, 1− s). (2.1.11)

Finally we can reduce the real part of s to less than 1/2 and change the contour back to

the real line. So when Re(s) < 1/2, let C(z, s) be the sum of (2.1.11) and (2.1.9), which

are holomorphic. The bound (2.1.10) and m ≥ 1 guarantees the existence of the limits of

C(z, s) as Re(s) approaches 1/2 from the left and from the right. The procedure defining

C(z, s) shows that the limits agree. So we define C(z, s) on Re(s) = 1/2 to be this limit.

Then C(z, s) gives the analytic continuation of C(z, s) to Re(s) > 0. Putting these together

with the analytic continuation of D(z, s), we have the analytic continuation of Qm(z, s) to

Re(s) > 0.

Now to analytically continue Pm(z, s), we can simply compare it to Qm(z, s). Applying

the power series expansions of 1F1 (α; β; z) and the exponential map to (2.1.1) and (2.1.2),

we see that for y small and 0 < Re(s) < 2,

|(4π|m|y)−s+1/2(φ∗m(z, s)− ϕ∗m(z, s))| = Om(y).

So the difference Pm(z, s)−Qm(z, s) defined by termwise subtraction is a holomorphic func-

tion in s for Re(s) > 0. That means the analytic continuation of Pm(z, s) to Re(s) > 0

follows from that of Qm(z, s). Furthermore, they have the same poles and residues in the

region Re(s) > 0. Thus, to prove the second half of the proposition, it suffices to analyze

the poles and residues of Qm(z, s) at s = 1/2.

15

Since m > 0, expression (2.1.10) is bounded by an absolute constant for all s ∈ (1/2, 2].

So C(z, s) does not contribute to the pole at s = 1/2. For a Maass cusp form en(z), the

right hand side of (2.1.7) vanishes at s = 1/2 if tn 6= 0. Otherwise, y−1/2en(z) ∈ S1(M,χ)

and 〈Qm(·, s), en(·)〉 has a simple pole of residue (4π|m|)1/2cn,m. Thus, we have

Ress=1/2Pm(z, s) = Ress=1/2Qm(z, s) = (4π|m|)1/2

N0∑n=1

cn,my−1/2en(z). (2.1.12)

Since {y−1/2en(z) : n = 1, . . . , N0} is a basis of S1(M,χ), the matrix

{cn,m : 1 ≤ n ≤ N0, 1 ≤ m ≤ K}

has rank equals to M for K sufficiently large. Therefore the residues at s = 1/2 of Pm(z, s)

generate S1(M,χ).

The following theorem is an immediate consequence of the proposition above.

Theorem 2.1.2. Using the notations above, the following map is a surjection

ξ1 : H1(M,χ)→ S1(M,χ),

i.e. for any cusp form h(z) ∈ S1(M,χ), there exists h(z) ∈M1(M,χ) with shadow h(z).

Proof. When k = 1, Eq. (1.1.1) becomes ∆1 = ξ1◦ξ1. So the Poincare series Pm(z, s) satisfies

∆1(Pm(z, s)) =(s− 1

2

)2Pm(z, s) (2.1.13)

when Re(s) > 1. Since the difference between both sides is holomorphic in s and Pm(z, s)

can be analytically continued to Re(s) > 0 as in the proposition, Eq. (2.1.13) is valid for

Re(s) > 0. At s = 1/2, suppose Pm(z, s) has the following Taylor series expansion in s

Pm(z, s) = g−1(z)(s− 1

2

)−1+ g0(z) + g1(z)

(s− 1

2

)+Oz

((s− 1

2

)2), (2.1.14)

with gj(z) ∈ F1(M,χ) real-analytic for j = −1, 0, 1. Since ξ1 commutes with the slash

operator and ∆1(g1(z)) = ξ21(g1(z)) = g−1(z), we have ξ1(g1(z)) ∈ F1(M,χ) is real-analytic

and a preimage of g−1(z) under ξ1.

16

By considering the Fourier expansion of g1(z), we know that g1(z) has at most linear

exponential growth near the cusps implies ξ1(g1(z)) has the same property. Suppose Qm(z, s)

has the Laurent expansion

g−1(z)(s− 1

2

)−1+ f0(z) + f1(z)

(s− 1

2

)+Oz

((s− 1

2

)2)

near s = 1/2. Then from the spectral expansion of Qm(z, s), i.e. y−1/2(D(z, s) + C(z, s))

as in (2.1.6), it is not hard to see that f1(z) has at most linear exponential growth near the

cusps. The difference Pm(z, s)−Qm(z, s) is a Poincare series defined for Re(s) > 0. So the

coefficient of (s−1/2) of its Laurent series expansion around s = 1/2, say h1(z), has at most

linear exponential growth near the cusps. That means the sum of f1(z) and h1(z), which is

g1(z) by analytic continuation, also has this property.

Thus, ξ1(g1(z)) is in H1(M,χ) and a preimage of g−1(z) ∈ S1(M,χ) under ξ1. Since the

functions {Ress=1/2Pm(z, s) : m ≥ 1} span the space S1(M,χ), the map ξ1 : H1(M,χ) →

S1(M,χ) is surjective.

2.2 Principal Part Coefficients of Harmonic Maass Forms

Let M be an odd, square-free positive integer and χ : (Z/MZ)× −→ C× a character. In

this section, we will relate the regularized inner products between g(z) ∈ S1(M,χ) and

f(z) ∈M !1(M,χ) to linear combinations of coefficients of a mock-modular form g(z), whose

shadow is g(z), via Stokes’ theorem. The regularization technique is standard and has been

used in many places before (see for example [8, 11, 12, 14, 23]).

The usual Petersson inner product 〈f, g〉 can be regularized as follows. Since M is square-

free, Γ0(M) has 2ω(M) inequivalent cusps

{ιd : d |M},

with ι1 equivalent to the cusp infinity. Here, ω(M) is the number of distinct prime divisors

of M . The cusp ιd is related to the cusp infinity by the matrix σd =( √

dαd βd/√d

M/√d√d

)in SL2(R).

Take a fundamental domain of Γ0(M)\H, cut off the portion with Im(z) > Y for a large

17

Y and intersect it with its translate under σd for all d | M . We will call this the truncated

fundamental domain F(Y ). Now, define the regularized inner product by

〈f, g〉reg := limY→∞

∫F(Y )

f(z)g(z)ydxdy

y2. (2.2.1)

If f(z) ∈ M1(M,χ), then this is the usual Petersson inner product. Now let g ∈ H1(M,χ)

be a preimage of g(z) under ξ1 with the following Fourier expansions at each cusp ιd

(g|1σd)(z) =∑n∈Z

c+d (n)qn −

∑n≥1

c(g|1σd, n)β1(n, y)q−n,

The expression for (g|1σd)(z) follows from the commutativity between ξ1 and the slash op-

erator. Note that∑

n∈Z c+d (n)qn is a mock-modular forms with shadows (g|1σd)(z). As a

special case of Prop. 3.5 in [12], we can express 〈f, g〉reg in terms of these Fourier coefficients.

Lemma 2.2.1 (See Prop. 3.5 in [12]). Let f(z) ∈ M !1(M,χ) and g(z) ∈ S1(M,χ). In the

notations above, we have

〈f, g〉reg =∑n∈Z

∑d|M

c+d (n)cd(f,−n). (2.2.2)

Notice that the right hand side of equation (2.2.2) depends on the choice of g, whereas

the left hand side only depends on g(z). So if we replace g with h(z) ∈ M !1(M,χ), then

Lemma 2.2.1 still holds and we obtain

0 =∑n∈Z

∑d|M

cd(h, n)cd(f,−n)

where h(z) has Fourier expansions∑

n∈Z cd(h, n)qn at the cusp ιd. So the right hand side of

Eq. (2.2.2) gives a pairing between f ∈M !1(M,χ) and G ∈ H1(M,χ)/M !

1(M,χ) defined by

{f,G} := 〈f, ξ1(G)〉reg =∑d|M

Const((f |1 σd) · (G |1 σd)). (2.2.3)

This is in fact a perfect pairing when one restricts to f ∈ S1(M,χ) as a consequence of Serre

duality (see [9, §3]). In that case, the first sum is only over n < 0 and we obtain relations

among the principal part coefficients of h(z) at various cusps. We remark that this holds for

other weights as well. So given some Fourier coefficients, we know that they are the principal

part coefficients of a weakly holomorphic modular form in M !k(M,χ) if and only if its pairing

with cusp forms in S2−k(M,χ) vanishes.

18

Proposition 2.2.2. Let {cd(−n) ∈ C : d | M, 1 ≤ n ≤ n0} be a set of complex numbers.

Then there exists f ∈M !1(M,χ) such that

f |1 σd =

n0∑n=1

cd(−n)q−n +O(1)

for each d |M if and only if ∑d|M

n0∑n=1

cd(−n)cd(h, n) = 0

for all h(z) ∈ S1(M,χ) with h |1 σd =∑

n≥1 cd(h, n)qn.

2.3 Transformation Calculations

In this section, we will give some results on transformations of modular forms under different

operators. These will be useful in calculating the Fourier expansion of certain modular forms

in Chapter 3.

2.3.1 Commutation Lemmas

First, we will state some general results about the commutations between the U -operator

and different Atkin-Lehner involutions. In this section, M > 0 will be an odd integer and

Φ(z) will be a real-analytic modular form of level 4M , weight k/2 and character χ for some

k ∈ Z and character χ : (Z/4M)× −→ C×. For p | M a prime, let Up be the U -operator

defined by

Up :=

p−1∑λ=0

[(1 λp

), p1/4

]. (2.3.1)

For d | 4M a positive integer such that gcd(d,M/d) = 1, write χ = χdχ4M/d and let Wd be

the Atkin-Lehner involution defined by

Wd :=[( dr s

4Mt du ) , d−1/4(Mt/du

)√4Mtz + du

]= ˜(

r s4Mt/d du )

[( d 1 ) , d−1/4

(Mt/dd

)εdu

]=[( 1

d ) , d1/4εu] ˜( dr s

4Mt/d u

).

(2.3.2)

The following lemma shows the effect of the Atkin-Lehner involution on the character.

19

Lemma 2.3.1. In the notations above, let

Φ′(z) := (Φ |k/2 Wd)(z).

Then Φ′(z) is a real-analytic modular form of level 4M , weight k/2 and character χ′, where

χ′ : (Z/4MZ)× −→ C×

α 7→(dα

)kχd(α)χ4M/d(α).

Proof. It is not hard to see that

W−1d =

[( 1

d ) , d1/4(Mt/dd

)ε−1du

] ˜(du −s

−4Mt/d r

).

Suppose γ = ( A B4MC D ) ∈ Γ0(4M). Then we have

Φ |k/2 WdγW−1d = Φ |k/2 ˜(

r s4Mt/d du )

[( d 1 ) , d−1/4

] ˜( A B4MC D )

[( 1

d ) , d1/4] ˜(

du −s−4Mt/d r

)=(dD

)kΦ |k/2 ˜(

r s4Mt/d du ) ˜(

A Bd4MC/d D

) ˜(du −s

−4Mt/d r

)=(dD

)kΦ |k/2 ˜( ∗ ∗

4M∗ δ )

=(dD

)kχ(δ)Φ |k/2,

where δ ≡ −4MtsA/d + durD (mod 4M). Since dur ≡ 1 (mod 4M/d) and −4Mts/d ≡ 1

(mod d), we have

χ(δ) = χ4M/d(durD)χd(−4MtsA/d) = χ4M/d(D)χd(A).

This implies the lemma as AD ≡ 1 (mod 4M).

Lemma 2.3.2. The following quantity is independent of the choice of r, s, t, u for Wd

εkdu(td

)kχd(t)χ4M/d(u)Φ |k/2 Wd. (2.3.3)

Proof. For a different choice

W ′d = ˜(

r′ s′

4Mt′/d du′) [

( d 1 ) , d−1/4(Mt′/dd

)εdu′],

20

we have the following calculations

Φ |k/2 W ′d(Wd)

−1 =(εduεdu′

)k ((Mt′/dd

)(Mt/dd

))kΦ |k/2 ˜(

r′ s′

4Mt′/d du′) ˜(

du −s−4Mt/d r

)=(εduεdu′

)k (t′td

)kχ4M/d(du

′r)χd(−4Mt′s/d)Φ

=(εduεdu′

)k (t′td

)kχ4M/d(u)χd(t)χ4M/d(u

′)χd(t′)Φ,

which implies Eq. (2.3.3).

For simplicity, we will take the Atkin-Lehner involution with t = u = 1 and r = α, s = β.

So from now on,

Wd :=[(

dα β4M d

), d−1/4

√4Mz + d

]. (2.3.4)

Lemma 2.3.3. In the notations above, suppose d, d′ |M and gcd(d, d′) = 1. Then

Φ |k/2 WdWd′ =(dd′

)kχd′(d)Φ |k/2 Wdd′ . (2.3.5)

Proof. Since gcd(d, d′) = 1, some matrix calculations tell us that

(dα β4M d

) (d′α′ β′

4M d′

)=( ∗ ∗

4M/d d

) (α′ β′d

4M/(dd′) d′

)( dd

′1 )

= ( 1dd′ )

( ∗ ∗4M/(dd′) 1

) (dα′ β′

4M/d′ 1

).

Since ε−1d ε−1

d′

(M/dd

)(M/(dd′)dd′

)= ε−1

dd′

(4Mα′/d+4M/d′

4Mβ′+dd′

), we could set

W ′dd′ := ˜( ∗ ∗

4M/d d

) ˜(α′ β′d

4M/(dd′) d′

) [( dd

′1 ) , (dd′)−1/4

(εdd′εdεd′

)(M/dd

)(M/(dd′)dd′

)ε−1dd′

],

and obtain Eq. (2.3.5) as follows

Φ |k/2 WdWd′ =Φ |k/2 W ′dd′

=(d′α′+ddd′

)kχdd′(d

′α′ + d)Φ |k/2 Wdd′

=(dd′

)kχd′(d)Φ |k/2 Wdd′ .

This also implies that

Φ |k/2 WdWd′ =(dd′

)k (d′d

)kχd′(d)χd(d

′)Φ |k/2 Wd′Wd. (2.3.6)

21

Lemma 2.3.4. In the notations above, if gcd(p, d) = 1, then

Φ |k/2 UpWd =(pd

)kχd(p)Φ |k/2 WdUp. (2.3.7)

Proof. This again follows from the straightforward calculation(1 λp

) (dα β4M d

)=(dα+4Mλ ∗

4Mp dδλ

) (1 (β+dλ)β′

p

),

with dαβ′ ≡ 1 (mod p) and

δλ = 1− 4M(β+dλ)β′

d≡ 1 (mod 4M/d)

for all 0 ≤ λ ≤ p− 1. Set

Wd,λ = [(dα+4Mλ ∗

4Mp dδλ

), d−1/4

(Mp/dδλ

)√4Mpz + dδλ].

Then by Lemma 2.3.2, we have

Φ |k/2 UpWd =

p−1∑λ=0

(Mp/dδλ

)kΦ |k/2 Wd,λ

[(1 (β+dλ

β′

), p1/4

]=

p−1∑λ=0

(pd

)kχd(p)χ4M/d(δλ)Φ |k/2 Wd

[(1 (β+dλ

β′

), p1/4

]=(pd

)kχd(p)Φ |k/2 WdUp.

2.3.2 Trace Down Lemmas

Now, we will prove two level-lowering lemmas for modular forms.

Lemma 2.3.5. Let Φ(z) be a real-analytic function on H such that it has at most linear

exponential growth at the cusps and

(Φ |k/2 γ)(z) = χ(d)Φ(z)

for all γ ∈ Γ0(4Npr+1) with r ≥ 1, gcd(N, p) = 1 and χ : (Z/4NprZ)∗ −→ C× a character.

Define Φ(z) to be

Φ(z) := Φ |k/2 Up.22

Then Φ(z) satisfies (Φ |k/2 γ

)(z) = χ(d)

(pd

)kΦ(z)

for all γ ∈ Γ0(4Npr).

Proof. This is a purely group theoretic lemma and quite similar to Lemma 7 in [2], its the

integral weight counterpart. For completeness, we will include the argument here.

Let γ =(

a b4Nprc d

)∈ Γ0(4Npr). Suppose r ≥ 1, then

Φ |k/2 γ =

p−1∑λ=0

Φ |k/2[(

1 λp

), p1/4

] [(a b

4Nprc d

), j(γ, z)

]=

p−1∑λ=0

Φ |k/2[(

a+4Nprcλ βλ4Npr+1c δλ

) (1 a(b+dλ)

p

), p1/4

(Nprcd

)ε−1d

√4Nprcz + d

]=

p−1∑λ=0

Φ |k/2˜(

a+4Nprcλ βλ4Npr+1c δλ

) [(1 a(b+dλ)

p

), p1/4

(pd

)]=

p−1∑λ=0

χ(δλ)Φ |k/2[(

1 a(b+dλ)p

), p1/4

]=

p−1∑λ=0

χ(d)(pd

)kΦ |k/2

[(1 a(b+dλ)

p

), p1/4

]= χ(d)

(pd

)kΦ.

When r = 0 in the above lemma, one needs both the U -operator and the Atkin-Lehner

involution to lower the level.

Lemma 2.3.6. Let Φ(z) be a real-analytic function on H such that it has at most linear

exponential growth at the cusps and

(Φ |k/2 γ)(z) = χ(d)(pd

)kΦ(z)

for all γ ∈ Γ0(4Np) with gcd(N, p) = 1 and χ : (Z/4N)∗ −→ C× a character. Define Φ(z)

to be

Φ(z) := Φ |k/2 Up + Φ |k/2 Wp (2.3.8)

where Wp is defined as in Eq. (2.3.4). Then Φ(z) satisfies(Φ |k/2 γ

)(z) = χ(d)Φ(z)

23

for all γ ∈ Γ0(4N).

Remark 2.3.7. For different choices

W ′p :=

[(pa′ b′

4Npc′ pd′

), p−1/4

(Nc′

d′

)√4Npc′z + pd′

],

W ′′p :=

[(pa′′ b′′

4Npc′′ pd′′

), p−1/4

(Nc′′

d′′

)√4Npc′′z + pd′′

],

Lemma 2.3.2 implies that

εkpd′χ(d′)Φ |k/2 W ′p = εkpd′′χ(d′′)Φ |k/2 W ′′

p .

So for general W ′p, one has Φ = Φ |k/2 Up +

(εpd′

εp

)kχ(d′)Φ |k/2 W ′

p.

Proof. The proof is almost the same as the proof of Lemma 2.3.5. Let γ = ( a b4Nc d ) ∈ Γ0(4N).

Notice that the condition satisfied by Φ(z) is equivalent to(Φ |k/2

[γ,(pd

)j(γ, z)

])(z) = χ(d)Φ(z).

By quadratic reciprocity, the character of Φ can be written as

χ(·)(

(−1)(p−1)/2

·

)k (·p

)k.

The only difference now in evaluating Φ |k/2 γ is that one of the λ in the sum of Up will

switch with Wp. For completeness, we record the calculations here. When p | a+ 4Ncλ, we

have by the remark above

Φ |k/2[(

1 λp

), p1/4

]γ = εkdΦ |k/2 W ′

p,

=(εdεpεpd

)k (cp

)k (cp

)kχ(d)

((−1)(p−1)/2

d

)kΦ |k/2 Wp

= χ(d)Φ |k/2 Wp.

where W ′p =

[(pa+4Ncλ

pb+dλ

4Npc pd

), p−1/4

(Ncd

)√4Npcz + pd

]. On the other hand, we have

Φ |k/2Wpγ = Φ |k/2[(

pα β4Np p

), p−1/4

√4Npz + p

][( a b

4Nc d ) , j(γ, z)]

= Φ |k/2[(

1p

), p1/4

] (pα β4N 1

) ˜( a b′

4Nc pd′) [

( p 1 ) , p−1/4] [(

1 4Ncp

), p1/4

]= Φ |k/2

[γ,(

p4Nb′+pd′

)j(γ, z)

] [(1 4Nc

p

), p1/4

]= χ(d)Φ |k/2

[(1 4Nc

p

), p1/4

],

where γ =(pαa+4Ncβ αb′+βd′

4Np(a+c) 4Nb′+pd′

)and pd′ ≡ d (mod 4Nc).

24

2.4 Weight One Space Decomposition

2.4.1 Projection Operators

Let M be a positive, odd integer and Φ be a real-analytic function on H with at most linear

exponential growth at all cusps and k ∈ Z. Suppose it satisfies the transformation property

(Φ |k γ)(z) = χM(d)Φ(z),

for all γ ∈ Γ0(M) and χM : (Z/MZ)∗ −→ C× a character satisfying χM(−1) = (−1)k. Recall

that the space of such functions is denoted by Fk(M,χM). Let ` | M be a prime such that

gcd(`,M/`) = 1 and ` > 0. Write

M = M ′`, χM = χM ′χ`

with χ∗ a character having conductor ∗ = M ′, `.

Notice that any such Φ ∈ Fk(M,χM) has a Fourier expansion at the cusp 1/M ′ with

variable q. If χ` is non-trivial, then we cannot trace Φ down to level M ′. Fortunately,

it is still possible to obtain results similar to Corollary 2.3.8 when χ`(·) = χ`(·) =( ·`

)is

quadratic.

For ε = ±1, define the operator prεχ` by

prεχ`(Φ) := 12

(εχM′ (`)χ`(−M ′)

G(χ`)Φ |k U`W` + Φ

)(2.4.1)

where G(χ`) is the Gauss sum and

U` =`−1∑λ=0

( 1 λ` ) ,W` =

(`α βM `

).

It is not difficult to check that prεχ`(Φ) is in Fk(M,χM ′χ`). The following lemma describes

the kernel of these operators in this case.

Proposition 2.4.1. Let ε ∈ {±1}, Φ(z) ∈ Fk(M,χM ′χ`) with χ` =( ·`

)and Fourier expan-

sion

Φ(z) =∑n∈Z

a(Φ, n, y)qn

at infinity. Then the followings are equivalent.

26

(1) For all n ∈ Z relatively prime to `,

χ`(n)a(Φ, n, y) = −εa(Φ, n, y). (2.4.2)

(2) prεχ`(Φ) = 0.

(2)′ pr−εχ` (Φ) = Φ.

(3)

Φ |k W` = −εχ`(M ′)G(χ`)

Φ |k U`. (2.4.3)

Proof. For any Ψ ∈ Fk(M,χM ′χp), write its Fourier expansions at infinity as

Ψ(z) =∑n∈Z

a(Ψ, n, y)qn.

Some calculations show that

Ψ |k U`W` =`−1∑λ=0

Ψ |k(α+M ′λ β+`λM `2

)( ` 1 )

=`−1∑

λ=0,`-α+M ′λ

Ψ |k(α+M ′λ ∗M ∗

) (` (α+M ′λ)β

`

)+ Ψ |k W ′

` ( ` 1 )

= χM ′(`)χ`(−M ′)`−1∑µ=1

χ`(µ)Ψ |k(` µ`

)+ Ψ |k W ′

` ( ` 1 )

= χM ′(`)χ`(−M ′)G(χ`)∑n∈Z

χ`(n)a(Ψ, n, y)qn + χM ′(`)Ψ |k W` ( ` 1 ) ,

where W ′` =

(`α+M ′λ0

`β+`λ

M `2

)with ` | α +M ′λ0. That means

prεχ`(Ψ) =1

2

(∑n∈Z

(εχ`(n) + 1)a(Ψ, n, y)qn +εχ`(−M ′)

G(χ`)Ψ |k W` ( ` 1 )

). (2.4.4)

(1)⇔ (2)

Taking Ψ = Φ in Eq. (2.4.4) shows that ⇐ is clear. For ⇒, we know that there exists

Φε(z) such that

Φε(z + 1) = Φε(z),

Φε |k ( ` 1 ) = prεχ`(Φ).

27

Since prεχ`(Φ) ∈ Fk(M,χM ′χ`), the second condition implies that

Φε |k(

a `bM ′c d

)= χM ′(d)χ`(d)Φε

for all(

a `bM ′c d

)∈ SL2(Z) with a, b, c, d ∈ Z. Given any ( ∗ ∗

M ′∗ d ) ∈ Γ0(M ′) with gcd(`, d) = 1,

we can write it as ( 1 t1 )(∗ `∗

M ′∗ d

). Thus,

Φε |k ( ∗ ∗M ′∗ d ) = χM ′(d)χ`(d)Φε.

On the other hand, since χ` is non-trivial, one can find b′, c, d, d′ ∈ Z such that gcd(d,Mc′) =

gcd(d′,Mb′) = 1 and χ`(dd′) 6= χ`(M

′b′c+ dd′). That means

χM ′(dd′)χ`(dd

′)Φε = Φε |k ( ∗ ∗M ′c d )

(∗ b′

M ′∗ d′)

= Φε |k ( ∗ ∗M ′∗ M ′b′c+dd′ )

= χM ′(dd′)χ`(M

′b′c+ dd′)Φε.

By the choice of b′, c, d and d′, we know that χM ′(dd′) 6= 0. Thus, Φε must vanish and

prεχ`(Φ) = 0.

(2)⇔ (2)′

This follows easily from

Φ = prεχ`(Φ) + pr−εχ` (Φ) = pr−εχ` (Φ). (2.4.5)

(2)⇔ (3)

Notice that (2) and Eq. (2.4.1) is the same as

Φ = −εχM′ (`)χ`(−M′)

G(χ`)Φ |k U`W`

The equivalence then from applying W` to both sides and using the relations

Φ |k U`W 2` = χ`(−1)χM ′(`)Φ |k U`.

28

If χM ′ is quadratic and χ` is an arbitrary character, then the proposition above can be

modified to yield similar results. In this case, set

M ′ = N, ` = p, χM = χNχp, χN(·) =( ·N

).

Let χp be an arbitrary character of conductor p. Define the operator prεχp for ε = ±1 by

prεχp(Φ) := 12

(εχN (p)χp(−N)

G(χp)Φ |k UpWp + Φc

)(2.4.6)

where Φc(z) = Φ(z). Since Wp sends F(M,χNχp) to F(M,χNχp) and Φc ∈ F(M,χNχp),

we use Φc instead of Φ in defining the projection operator here. The following proposition

is the analogue of Prop. 2.4.1.

Proposition 2.4.2. Let ε ∈ {±1}, Φ(z) ∈ Fk(Np, χNχp) with χN =( ·N

)quadratic, χp

non-trivial and the Fourier expansion

Φ(z) =∑n∈Z

a(Φ, n, y)qn

at infinity. Then the followings are equivalent

(1) For all n ∈ Z relatively prime to p,

χp(n)a(Φ, n, y) = −εa(Φ, n, y) (2.4.7)

(2) prεχp(Φ) = 0.

(2)′ pr−εχp (Φ) = Φc.

(2)′′ pr−εχp (iΦ) = 0.

(3)

Φc |k Wp = −εχp(N)

G(χp)Φ |k Up. (2.4.8)

Remark 2.4.3. The subspace of Fk(Np, χNχp) satisfying any of the four conditions above

is a real vector subspace of Fk(Np, χNχp).

29

Proof. Let Ψ ∈ Fk(Np, χNχp) with Fourier expansions

Ψ(z) =∑n∈Z

a(Ψ, n, y)qn

at infinity. Then Ψc(z) =∑

n∈Z a(Ψ, n, y)qn and the same calculations in Prop. 2.4.1 gives

us the following analogue of Eq. (2.4.4).

prεχp(Ψ) =1

2

(∑n∈Z

(εχp(n)a(Ψ, n, y) + a(Ψ, n, y))qn +εχp(−M ′)

G(χp)Ψ |k Wp ( p 1 )

).

The rest of the proof follows, mutatis mutandis, from that of Prop. 2.4.1. The equivalence

to (2)′′ follows from substituting iΦ for Φ in condition (1) and replacing ε with −ε.

Proposition 2.4.4. For distinct `, `′ | M and ε, ε′ ∈ {±1}, the projection operators satisfy

the following properties

pr−εχ` ◦ prεχ` = 0,

prεχ` ◦ prε′

χ`′= prε

′

χ`′◦ prεχ` ,

prεχ` ◦ ξk = ξk ◦ prεχ`(−1)χ`

.

(2.4.9)

Proof. For Φ ∈ F(M,χM), let a(Φ, n, y), a(Φc, n, y) and b`(Φ, n, y) be the Fourier coefficients

of Φ,Φc, prε`(Φ) and prεχ`(Φ) respectively. For ` 6= p in the first equation, the calculations in

Prop. 2.4.1 tells us that whenever gcd(`, n) = 1,

b`(Φ, n, y) = 12

(εχ`(n)a(Φ, n, y) + a(Φ, n, y)

).

When ` = p, we could write

bp(Φ, n, y) = 12

(εχp(n)a(Φc, n, y) + a(Φ, n, y)

).

Since ε2 = 1, we have in both cases.

χ`(n)b`(Φ, n, y) = εb`(Φ, n, y) whenever gcd(`, n) = 1.

By Prop. 2.4.1, the first two equations hold.

30

For the second equation, notice that we can write

prε` ◦ prε′

`′ (Φ) = A((ε, `), (ε′, `′))Φ |k U`WÙ`′W`′ +B((ε, `), (ε′, `′)),

where A(ε, ε′, `, `′) and B(ε, ε′, `, `′) are stable under switching (ε, `) and (ε′, `′). By Lemma

2.3.4 and 2.3.3, we have

Φ |k U`WÙ`′W`′ = χ`′(`)Φ |k UÙ`′W`W`′ = Φ |k UÙ`′W``′ ,

which is also stable under switching (ε, `) and (ε′, `′). So the third equation holds. By the

same procedure, one could verify that the fourth equation holds as well.

The last equation follows from the definitions of the projection operators and the fact

that ξk commutes with slash operator by changing the weight from k to 2 − k, character

from χM to χM , and the coefficients to their complex conjugates.

2.4.2 Applications of the Projection Operators

Now, we will apply the projection operator to the space H1(M,χM), where M = Np is odd,

square-free and χM = χNχp with χN(·) =( ·N

)quadratic and χp an arbitrary, non-trivial

character. All the results in §2.4.1 still holds since the subspace H1(M,χM) ⊂ F1(M,χM)

is defined by the differential operator ξ1, which commutes with the slash operator. For

ε ∈ {±1} and each prime ` |M , define the space Hε1,`(M,χM) by

Hε1,`(M,χM) :=

{f ∈ H1(M,χM) : prεχ`(f) = 0

}. (2.4.10)

The space Hε1,`(M,χM) is a complex vector space unless ` = p, in which case it is a real

vector space. For each d |M , we could define the real vector space H1,d(M,χM) by

H1,d(M,χM) :=⋂

`|d positive prime

H+1,`(M,χM)

⋂`|Md

positive prime

H−1,`(M,χM). (2.4.11)

By the definitions of the projeciton operators and Props. 2.4.4, we could write

H1(M,χM) = H+1,`(M,χM)⊕H−1,`(M,χM)

31

for all ` | M . By Props. 2.4.1 and 2.4.2, we know that f ∈ H1,d(M,χM) if and only if the

following conditions are satisfied for all ` | N

f |1 W` = −εχ`(Np/`)ε`√`

f |1 U`,

f |1 Wp = −εχp(N)

G(χp)f c |1 Up.

Furthermore, Prop. 2.4.4 tells us that projection operators of different ` commute with each

other. This implies that

H1(M,χM) =⊕d|M

H1,d(M,χM) (2.4.12)

as real vector spaces. Remark 2.4.3 also implies that for any d | N , f ∈ H1,d(M,χM) if and

only if if ∈ H1,dp(M,χM). Since the decomposition 2.4.12 is defined entirely using the slash

operator, it could be naturally defined for subspaces S1(M,χM) ⊂M1(M,χM) ⊂M !1(M,χM)

of H1(M,χM) and M1(M,χM) ∼= H1(M,χM).

For any positive ` |M , define `∗ |M to be

`∗ := χ`(−1)`. (2.4.13)

Then we could write M = M+ ·M−, where

Mε :=∏`|M`∗=ε`

` (2.4.14)

for ε ∈ {±1}. Since χ`(·) =( ·`

)when ` | N is prime, Mε contains all the primes dividing N

which are congruent to ε modulo 4. Define the quantity d |M by

d := gcd(d,M+) · gcd(M/d,M−). (2.4.15)

The next proposition shows that decomposition in Eq. (2.4.12) behaves nicely with respect

to the differential operator ξ1.

Proposition 2.4.5. The following sequence is exact

M !1,d

(M,χM) ↪→ H1,d(M,χM)ξ1−→ S1,d(M,χM). (2.4.16)

32

Proof. By the last two equations in (2.4.9), the image of H1,d(M,χM) lies in S1,d(M,χM).

For any f ∈ S1,d(M,χM), let f ∈ H1(M,χM) be its preimage under ξ, whose existence is

given by Theorem 2.1.2. Define fd by

fd :=

∏`|d

pr−χ`

∏`|(M/d)

pr+χ`

f .

Then the Eqs. (2.4.9) tell us that

ξ1(fd) = ξ1

∏`|d

pr−χ`

∏`|(M/d)

pr+χ`

f =

∏`|d

pr−χ`

∏`|Md

pr+χ`

ξ1f

=

∏`|d

pr−χ`

∏`|Md

pr+χ`

f = f.

The last step follows from (2) ⇔ (2)′ in Props. 2.4.1 and 2.4.2. By the first two equations

in Eq. (2.4.9), we know that

pr+χ`

(fd) = 0 for all ` | d,

pr−χ`(fd) = 0 for all ` | Md.

So fd ∈ H1,d(M,χM) by definition and ξ1 : H1,d(M,χM) −→ S1,d(M,χM) is surjective. The

kernel is the holomorphic subspace of H1,d(M,χM), which is exactly M !1,d

(M,χM).

Combining the proposition above with Lemma 2.2.1, we could deduce the following result

on regularized inner product.

Proposition 2.4.6. Let d1, d2 | N , h ∈ M !1,d1

(M,χM), f ∈ S1,d2(M,χM) with Fourier ex-

pansions

h =∑n∈Z

c(h, n)qn, f =∑n≥1

c(f, n)qn

at infinity. Let f ∈ H1,d2(M,χM) be any preimage of f under ξ1 as in Prop. 2.4.5 with

Fourier expansion

f =∑n∈Z

c(f , n)qn −∑n≥1

c(f, n)β1(n, y)q−n

33

at infinity. If d1 = d2, then

〈h, f〉reg =∑n∈Z

(c(f ,−n)c(h, n) + c(f ,−pn)c(h, pn)

)δN(n), (2.4.17)

where δN(n) := 2ω(gcd(N,n)) is the number of divisors of gcd(N, n). Otherwise, the regularized

inner product is 0.

Proof. By Lemma 2.2.1, we could rewrite 〈h, f〉reg as

〈h, f〉reg =∑d|M

Const(

(h |1 Wd)(f |1 Wd)).

Applying Lemmas 2.3.3, 2.3.4 and Props. 2.4.1 and 2.4.2 to the right hand side above gives

us

〈h, f〉reg =∑d|N

δd1,d2(d)

dConst

((h |1 Ud)(f |1 Ud) +

1

p(hc |1 Udp)(f c |1 Udp)

),

=∑n∈Z

(c(h, n)c(f ,−n) + c(h, pn)c(f ,−pn)

) ∏`|gcd(n,N) prime

(δd1,d2(`) + 1) (2.4.18)

where δd1,d2(`) is defined by

δd1,d2(`) :=

1, ` | gcd(d1, d2) · gcd(M/d1,M/d2),

−1, ` | gcd(d1,M/d2) · gcd(M/d1, d2).

and δd1,d2(d) :=∏

`|d prime δd1,d2(`).

If ` | d2, then Prop. 2.4.4 tells us that

χ`(n)c(f ,−n) = −c(f ,−n),

for all n ∈ Z not divisible by `. So if d1 6= d2, then there exists `′ such that

χ`′(n)c(f ,−n) = −εc(f ,−n), χ`′(n)c(h, n) = εc(h, n)

with ε ∈ {±1} for all n ∈ Z satisfying `′ - n by Prop. 2.4.1. That means for these n, we have

c(f ,−n)c(h, n) =(χ`′(n)c(f ,−n)

)(χ`′(n)c(h, n)) = −c(f ,−n)c(h, n),

34

and c(f ,−n)c(h, n) = 0. Furthermore, δd1,d2(`′) = −1 and the factor

∏`|gcd(n,N)(δd1,d2(`)+1)

vanishes. Thus, the sum in Eq. (2.4.18) vanishes identically and 〈h, f〉reg = 0.

If d1 = d2, then δd1,d2(`) = 1 for all ` | N and the factor

∏`|gcd(n,N) prime

(δd1,d2(`) + 1) = 2ω(gcd(n,N))

is the number of divisors of gcd(n,N). Substituting this into Eq. (2.4.18) gives us Eq.

(2.4.17).

For each d | D, let nd, rd ∈ N be the quantities

nd := max{ord∞f : f ∈ S1,d(Np, χDχp)},

rd := dimR S1,d(Np, χDχp).(2.4.19)

Props. 2.2.2 and 2.4.6 together gives us the following result about the order at infinity of the

preimage in 2.4.5.

Proposition 2.4.7. For any d | N and f ∈ S1,d(Np, χNχp), there exists f ∈ H1,d(Np, χNχp)

such that ξ1(f) = f and

ord∞(f) ≥ −nd.

Proof. Let f1 ∈ H1,(Np, χNχp) be a preimage of f under ξ1 as in Prop. 2.4.5 with ord∞(f1) =

−n0 and principal partn0∑n=1

c(f1,−n)q−n.

If n0 > nd, then one could find complex numbers {c(−n) : 1 ≤ n ≤ n0} such that c(−n0) 6= 0

and

for all ` | d, χ`(−n)c(−n) = −c(−n), for all n ∈ Z relatively prime to `,

for all ` | Nd

, χ`(−n)c(−n) = c(−n), for all n ∈ Z relatively prime to `,

χp(−n)c(−n) = −c(−n), for all n ∈ Z relatively prime to p,

n0∑n=1

(c(−n)c(h, n) + c(−pn)c(h, pn)

)δN(n) = 0, for all h ∈ S1,d(Np, χDχp).

35

By Prop. 2.2.2, there exists f2 ∈M !1,d

(Np, χDχp) with the principal part

c(−n0)q−n0 +

n0−1∑n=1

c(−n)q−n

if p - n0. Since χp(−n0)c(f ,−n0) = −c(f ,−n0) and c(−n0) 6= 0, the ratio c(f ,−n0)/c(−n0)

is a real number and f1 − c(f ,−n0)/c(−n0)f2 ∈ H1,d(Np, χDχp) is a harmonic Maass form

with image fφ,A under ξ1 and smaller order of pole at infinity.

If p | n0, then Prop. 2.2.2 also implies that there exist f3, f4 ∈ M !1,d

(Np, χDφ) with the

following principal parts at infinity respectively

q−n0 +

n0−1∑n=1

c(−n)q−n, iq−n0 +

n0−1∑n=1

c(−n)q−n.

By the same idea as before, we could still subtract R multiples of f3 and f4 from f1 to reduce

n0. An induction on the size of n0 shows that we could take it to be nd.

36

CHAPTER 3

Weight One Newforms of Imaginary Dihedral Type

Let D < 0 be an odd, fundamental discriminant and K = Q(√D) an imaginary quadratic

field with ring of integers OK . The weight one newforms we want to study arise from non-

trivial characters of certain ray class groups of OK , which becomes the newform associated

to characters of Pic(O) for some order O ⊂ Ok after suitable twisting. The structure of

the chapter is as follows. In §3.1, we will give some background information on weight one

newforms and Galois representations following [48]. Then in §3.2, we will gather some facts

about Pic(O) and use its characters to construct newforms. In §3.3, we will look at two

dimensional, odd, complex Galois representations arising from ray class group characters of

imaginary quadratic fields, and the weight one newforms associated to them via Theorem

3.1.1. Finally in §3.4.1, we will look at the inner product between newforms constructed in

§3.3.

3.1 Newforms of Weight One

In this section, we will describe the results connecting complex, odd, two-dimensional, irre-

ducible representations of GQ and weight one newforms.

Let Q/Q be an algebraic closure of Q and GQ = Gal(Q/Q) be the absolute Galois group

of Q, with the profinite topology. Because of the difference in topologies, a continuous

representation ρ : GQ −→ GL2(C) will have finite image. Its projective image under the

projection GL2(C) −→ PGL2(C) is a finite subgroup of PGL2(C). The finite subgroups

of GLn(C) is well-known to be classified as either cyclic, dihedral, tetrahedral, octahedral,

or icosahedral, when the projective image is Cn, D2n, A4, S4 or A5. If the representation

37

is irreducible, then the projective image can only be one of the last four types. We say a

continuous representations ρ : GQ −→ GL2(C) is odd if ρ(c) = −1, where c ∈ GQ is complex

conjugation. The conductor of ρ can be defined using the data ρ|Ip with Ip ⊂ GQ the inertia

subgroup at rational prime p.

In [18], Deligne and Serre attached odd, irreducible Galois representations ρf : GQ →

GL2(C) to weight one newforms f =∑

n≥1 a(f, n)qn of level N and nebentypus χ such that

the conductor and determinant of ρf are N and χ respectively, and for all primes p - N

trρf (Frobp) = a(f, p).

In the other direction, Langlands used solvable base change and proved the modularity

of Galois representations with solvable image [39]. In particular, he attached weight one

newforms to odd, continuous, irreducible representation ρ : GQ −→ GL2(C) with projective

image A4 and Tunnell did the case with projective image S4 [54]. Finally, the A5 case,

which is not solvable, was proved as a consequence of the resolution of Serre’s Conjecture

by Khare-Wintenberger [34, 35]. As a result, we have the following bijection between Galois

representations and weight one newforms.

Theorem 3.1.1 (Deligne-Serre, Langlands-Tunnell, Khare-Wintenberger). There is bijec-

tion between the set of odd, irreducible, continuous representations ρ : GQ −→ GL2(C) of

conductor N and det(ρ) = χ and weight one newforms of level N and nebentypus χ.

From now on, we will say a weight one newform is of dihedral, resp. tetrahedral, octahe-

dral, or icosahedral type if its associated Galois representation has dihedral, resp. tetrahedral,

octahedral, icosahedral projective images. In the dihedral case, the Galois representation is

induced from a Hecke character of either a real or imaginary quadratic field. We will say

that it is of imaginary or real dihedral type depending on the nature of the quadratic field.

38

3.2 Structure of Pic(O)

Let O ⊂ OK ⊂ K be a subring. Its index in OK is f 2 for some positive integer f . We call

f the conductor of O and denote O by Of . Explicitly, one can write

Of = {m+nf√D

2: m,n ∈ Z,m ≡ nfD (mod 2)}.

When f > 1, the order Of is not a Dedekind domain. So we do not expect every fractional

ideal to factor uniquely into prime ideals. On the other hand, every fractional ideal relatively

prime to the conductor f has unique factorization into primes ideals in Of . Let I(Of ), resp.

P (Of ), be the group of invertible, resp. principal ideals of Of . Define the Picard group of

Of to be

Pic(Of ) := I(Of )/P (Of ).

When f = 1, Pic(Of ) is just the class group of K, which is canonically isomorphic to C(D),

the group of equivalence classes of primitive binary quadratic forms of discriminant D < 0

(see §4.1.1 for details). Via the following map, this relationship holds more generally between

Pic(Of ) and C(Df 2) (Theorem 7.7 [17]).

C(Df 2)∼=−→ Pic(Of )

[A,B,C] 7−→ [A, −B+f√D

2]Of

(3.2.1)

In the notation of §4.1.1, we will denote Pic0(Of ) and Pic2(Of ) the kernel and image of

Pic(Of ) under the squaring map.

By Proposition 7.20 in [17], there is an isomorphism between I(Of ) and IK(f), the group

of fractional ideals of OK relatively prime to f . So there is another way to describe Pic(Of )

(Prop 7.22 [17])

Pic(Of ) ∼= IK(f)/PK,Z(f), (3.2.2)

where PK,Z(f) ≤ IK(f) is the subgroup of principal ideals (α) of K with α ≡ a (mod f)

for some a ∈ Z, gcd(a, f) = 1. Let IK = I(OK) and PK = P (OK). By Chebotarev density

theorem, the natural map IK(f)→ IK/PK ∼= Pic(OK) is surjective with kernel IK(f) ∩ PK

containing PK,Z(f). So we have the following exact sequence (Eq. (7.25) in [17])

1 −→ IK(f) ∩ PK/PK,Z(f) −→ IK(f)/PK,Z(f) −→ IK/PK −→ 1 (3.2.3)

39

The kernel is well-understood from the following exact sequence (Eq. (7.27) & Ex. 7.30 in

[17])

1 −→ {±1} −→ (Z/fZ)××O×K −→ (OK/fOK)×

−→ IK(f) ∩ PK/PK,Z(f) −→ 1.(3.2.4)

From this description, we can determine h(Of ), the size of Pic(Of ). It is given by the

following formula (Theorem 7.24 [17])

h(Of ) =h(OK)f

[O×K : O×f ]

∏p|f

(1−

(D

p

)1

p

). (3.2.5)

Here(Dp

)is the Kronecker symbol and O× denotes the units in the ring O. In particular,

h(Of ) is always an integral multiple of h(OK). When f | f ′, we have Of ′ ⊂ Of and hence a

canonical map

π : Pic(Of ′) � Pic(Of )

aOf ′ 7→ aOf .(3.2.6)

For a class A ∈ Pic(Of ), define the theta series ϑA(z) by

ϑA(z) :=1

#O×f+

∑a∈IK(f),[a]=A

qNm(a) =∑n≥0

rA(n)qn. (3.2.7)

Let ψ : Pic(Of )→ C× be a character and define

gψ(z) :=∑

A∈Pic(Of )

ψ(A)ϑA(z) =∑n≥1

cψ(n)qn. (3.2.8)

By Hecke’s work [31], we know that gψ ∈ M1(Γ0(|D|f 2), χD1f ) is an eigenform, where

χD =(D·

)is the Dirichlet character associated to K and 1f is the trivial character of

conductor f . Furthermore, if ψ is not a genus character of C(Df 2) ∼= Pic(Of ), then gψ(z) is

a cuspform.

When f = p is a prime that splits into pp in OK , the ring OK/pOK is isomorphic to

Z/pZ× Z/pZ via

OK/pOK −→ Z/pZ× Z/pZ

α 7→ (α (mod p), α (mod p)).(3.2.9)

40

So the group (OK/pO)×/(Z/pZ)×, which surjects onto IK(p) ∩ PK/PK,Z(p) with kernel

O×K/{±1}, is isomorphic to (Z/pZ)× via

(Z/pZ)× × (Z/pZ)× −→ (Z/pZ)×

(u, v) 7→ uv−1.

Then we could restrict ψ to (OK/pO)×/(Z/pZ)× and view it as a character of (Z/pZ)×.

3.3 Modular Forms of Imaginary Dihedral Type

This section will describe the weight one newforms we are interested in. First, we will

study the structure of the ray class group and its characters. They give rise to holomorphic

modular forms of imaginary dihedral type. Then we will analyze the relationship between

the characters of ray class group and ring class group. Finally, we will discuss mock-modular

forms with imaginary dihedral shadows.

3.3.1 Ray Class Group and Its Characters

Let K = Q(√D) as before and p ≥ 3 a rational prime ideal that splits into pp in OK . Let Ip,

resp. Pp, be the group of fractional, resp. principal fractional, ideals of OK relatively prime

to p and Pp,1 be the group of principal fractional ideals a of K such that there exists α ∈ K

satisfying a = (α) and vp(α − 1) > 0. Suppose Hp is the ray class field of modulus p with

M/Q the Galois closure over Q, then Hp/K is abelian and contains H, the Hilbert class field

of K. Furthermore, Gal(Hp/K) is canonically isomorphic to Ip/Pp,1, which fits into exact

sequences similar to (3.2.3) and (3.2.4)

1 −→ Pp/Pp,1 −→ Ip/Pp,1 −→ Pic(OK) −→ 1

1 −→ O×K −→ (Z/pZ)× −→ Pp/Pp,1 −→ 1.(3.3.1)

The map (Z/pZ)× −→ Pp/Pp,1 comes from the embedding Z ↪→ OK .

Let φ : Ip/Pp,1 −→ C× be a non-trivial character and φ1 the composition

φ1 : (Z/pZ)×/O×K −→ Ip/Pp,1φ−→ C×

41

Since−1 ∈ O×K , φ1 as a character of (Z/pZ)× has even order and there exists φ2 : (Z/pZ)× −→

C× satisfying

φ1φ22 = 1p.

Let φ′2(·) := φ2(·)(·p

)be another character of conductor p. Then it also satisfies φ1(φ′2)2 =

1p. For another φ′ : Ip/Pp,1 −→ C× giving rise to the same φ1, we know that φφ′ factors

through (Z/pZ)×/O×K and is a character of Pic(OK).

Let ρφ : GQ −→ GL2(C) be the induced representation from φ. Then the kernel of ρφ is

contained in Gal(M/M) and the projective image of ρ(φ) is dihedral. Since φ is non-trivial,

ρφ is irreducible and

det(ρφ) = χDφ1,

where χD =(D·

). Let c ∈ GQ be complex conjugation, then

det(ρφ(c)) = (χDφ1)(−1) = −1.

So one can attach a weight one newform fφ(z) ∈ S1(|D|p, χDφ1) of imaginary dihedral type

to ρφ. It can be written out explicitly as

fφ(z) =∑

a∈Ip,a⊂OK

φ(a)qNm(a) =∑n≥1

cφ(n)qn. (3.3.2)

From this expression, one can express the Fourier coefficients cφ(n) explicitly in terms of the

Hecke character φ.

Proposition 3.3.1. The Fourier coefficients cφ(n) is multiplicative with respect to n. For `

prime and r ∈ N, we have

cφ(`r) =

φ(l)r+1−φ(l)r+1

φ(l)−φ(l)(`) = ll in OK

φ(`)r/2 (`) inert in OK and 2 | r

0 (`) inert in OK and 2 - r

(r + 1)φ(l)r (`) = l2 in OK

φ(p)r (`) = p in OK .

(3.3.3)

Let T` be the Hecke operator defined by

T`(fφ)(z) :=∑n≥1

cφ(`n)qn +∑n≥1

(χDφ1)(`)cφ(n)q`n. (3.3.4)

42

Then T`(fφ) = cφ(`)fφ for all prime `.

Proof. The multiplicativity of cφ(n) follows from the Euler product of the L-series associated

to fφ, which is the same L-series associated to ρφ. Eq. (3.3.3) and T`(fφ) = cφ(`)fφ are both

direct consequences of Eq. (3.3.2).

Proposition 3.3.2. In the notation above, suppose fφ ∈ S1(|D|p, χDφ1). Then for all

characters ϕ : Pic(OK) −→ C×,

fφϕ(z) =∑

a∈Ip,a⊂OK

φ(a)ϕ(a)qNm(a) ∈ S1(|D|p, χDφ1).

Proof. By Eq. (3.3.1), φϕ is a character of Ip/Pp,1 and its composition with (Z/pZ)× −→

Ip/Pp,1 is the same as φ1 : (Z/pZ)× −→ Ip/Pp,1φ−→ C× since (Z/pZ)× is in the kernel of

ϕ.

3.3.2 Genus Theory

In this section, we will decompose the form fφ into the sum of forms from different genera.

Define the set

ΣD := {d : d ≡ 1 (mod 4), d | D}, ΣD := Σ/ ∼, (3.3.5)

where d ∼ d′ if dd′ = D. For each d ∈ ΣD, write dOK = d2 and let ϕd be the genus

character of Pic(OK) determined by d. Notice that this is well-defined with respect to ∼.

Since D is an odd, fundamental discriminant, ΣD has size 2ω(D)−1 and is isomorphic as sets

to the subgroup of Pic(OK) consisting of elements of order at most 2, which gives it a group

structure.

For an arbitrary class A ∈ Pic(OK), we could define the form fφ,A by

fφ,A(z) :=1

H(D)

∑ϕ:Pic(OK)→C×

ϕ(A−1)fφϕ(z) =∑n≥1

cφ,A(n)qn, (3.3.6)

43

where H(D) is the class number of K = Q(√D). By the orthogonality of the characters ϕ,

we could write the Fourier coefficient cφ,A(n) in the following form

cφ,A(n) =1

H(D)

∑ϕ:Pic(OK)→C×

ϕ(A−1)cφϕ(n) =∑

a⊂OK ,a∈IpNm(a)=n

[a]=A∈Pic(OK)

φ(a). (3.3.7)

Then we have the following decomposition

fφ(z) =∑

A∈Pic(OK)

fφ,A(z), (3.3.8)

which follows from the orthogonality of the characters.

Let (`) = ll be a split prime in OK and L = [l] ∈ Pic(OK). As an immediate consequence

of Prop. 3.3.1, we have

T`(fφ,A) =1

H(D)

∑ϕ:Pic(OK)→C×

ϕ(A−1)T`(fφϕ)

=1

H(D)

∑ϕ

ϕ(A−1)((φϕ)(l) + (φϕ)(l)

)fφϕ

= φ(l)fφ,AL + φ(l)fφ,AL−1 .

(3.3.9)

Now, define dA ∈ ΣD by

dA :=∏

`|D, ϕ`∗ (A)=−1

`∗, (3.3.10)

where `∗ is given in Eq. (2.4.13). The following result tells us about the vanishing of the

coefficients cφ,A(n).

Proposition 3.3.3. The Fourier coefficient cφ,A(n) vanishes whenever(dn

)6= ϕd(A) for

some d ∈ ΣD. Equivalently, fφ,A ∈ S1,dA(|D|p, χDφ1).

Proof. In the last sum in Eq. (3.3.7), the ideal a ⊂ OK is in the same genus class as A,

which implies that ϕd(A) = ϕd(a) =(

dNm(a)

)for all d ∈ ΣD. Thus, this sum is empty, hence

zero, if(dn

)6= ϕd(A) for some d ∈ ΣD. This is equivalent to

(dn

)cφ,A(n) = ϕd(A)cφ,A(n) whenever gcd(n, d) = 1. (3.3.11)

44

By taking d = `∗ for ` | D a positive prime and applying Prop. 2.4.1, we see that

pr+χ`fφ,A = 0 for all ` | dA,

pr−χ`fφ,A = 0 for all ` | |D|dA

.

Using Prop. 3.3.1, we could deduce that

φ1(n)cφ,A(n) = cφ,A(n)

whenever gcd(n, p) = 1 for all A ∈ Pic(OK). Prop. 2.4.2 then implies that pr−φ1(fφ,A) = 0.

By Definition 2.4.11, fφ,A ∈ S1,dA(|D|p, χDφ1).

For each d ∈ ΣD, we could now define fφ,d ∈ S1,d(|D|p, χDφ1) by

fφ,d(z) :=∑

A∈Pic(OK), dA=d

fφ,A(z) =∑n≥1

cφ,d(n)qn. (3.3.12)

Then fφ =∑

d∈ΣDfφ,d with each summand containing the information in different genera of

Pic(OK). For example, dA = 1 if and only if A is in the principal genus of Pic(OK), also

a subgroup of Pic(OK). Since there are 2ω(D)−1 genera and every genus is represented by

a class in Pic(OK), the size of the principal genus is H(D)

2ω(D)−1 . The principal genus contains

the subgroup Pic2(OK) of Pic(OK), consisting of squared classes. By Lemma 4.1.5, the size

of Pic0(OK) is the same as the number of genera. Thus, Pic2(OK) has size H(D)

2ω(D)−1 and is

exactly the principal genus. For each d ∈ ΣD, pick any A ∈ Pic(OK) such that dA = d, if it

exists. Then we could write

fφ,d =∑

C∈Pic2(OK)

fφ,AC =1

2ω(D)−1

∑B∈Pic(OK)

fφ,AB2 . (3.3.13)

Let (`) = ll be a split prime ideal in OK and L = [l] ∈ Pic(OK). Applying Eq. (3.3.9)

produces

T`fφ,d =1

2ω(D)−1

∑B∈Pic(OK)

(φ(l)fφ,AB2L + φ(l)fφ,AB2L−1

)=(φ(l) + φ(l)

)fφ,d′ ,

(3.3.14)

where d′ = dAL = dAL−1 .

45

3.3.3 Relationship to Characters of Pic(Op)

After understanding the relationship between fφ and Pic(OK), we will now consider its

relationship to Pic(Op). The following result tells us that the twist of fφ =∑

n≥1 cφ(n)qn by

φ2, defined as

fφ ⊗ φ2 :=1

G(φ2)

p∑µ=1

φ2(µ)fφ |1 ( p µp ) =∑n≥1

φ2(n)cφ(n)qn, (3.3.15)

is a newform of the shape (3.2.8) for some character ψ of Pic(Op).

Proposition 3.3.4. In the notation above, let ψ : Pic(Op) −→ C× be a character defined by

ψ : Pic(Op) −→ C×

a 7→ φ(a)φ2(Nm(a)).

Suppose φ : Ip/Pp,1 −→ C× is non-trivial. Then fφ ⊗ φ2 satisfies

(fφ ⊗ φ2)(z) =∑

A′∈Pic(Op)

ψ(A′)ϑA′(z) ∈ S1(|D|p2, χDp2).

Proof. The level and nebentypus of fφ⊗φ2 is given by Prop. 3.1 in [3]. Note that χDp2 = χD1p

since p - D. To prove the rest of the proposition, we first need to check that ψ : IK(p) −→ C×

is a well-defined character of Pic(Op) ∼= IK(p)/PK,Z(p). This is clear from the following

calculations

ψ(a) =φ(a)φ2

(a2+b2p2D

r2

)= φ1

(a+bp

√D

r

)φ2

(a2

r2

)=(φ1φ

22)(a2/r2) = 1

for all a =(a+bp

√D

r

)∈ PK,Z(p).

Now since fφ⊗ φ2 and gψ(z) :=∑A′∈Pic(Op) ψ(A′)ϑA′(z) =

∑n≥1 cψ(n)qn are both eigen-

forms in S1(|D|p2, χDp2), it is enough to compare their `th coefficients, for almost all primes

`, to show that they are the same. Suppose (`) = ll splits in OK , then

cφ(n)φ2(n) = (φ(l) + φ(l))φ2(`)

= φ(l)φ2(Nm(l)) + φ(l)φ2(Nm(l))

= ψ(l) + ψ(l) = cψ(`).

46

When ` is inert in OK , cφ(`)φ2(`) = cψ(`) = 0. Finally the (pn)th coefficient can be written

as

cψ(pn) =∑

A′∈Pic(Op)

ψ(A′)rA′(pn) =∑

A′∈Pic(Op)

ψ(A′)rπ(A′)(n/p)

=∑

B∈Pic(OK)

ψ(B)rB(n/p)∑

A′∈Pic(Op)A′∈ker(π)

ψ(A′).

Here π : Pic(Op) −→ Pic(OK) is the projection in Eq. (3.2.6) with f ′ = p and f = 1. Since

φ1 is non-trivial, ψ does not factor through π : Pic(Op) −→ Pic(OK) and cψ(pn) = 0 =

cφ(pn)φ2(pn).

Thus, the difference fφ ⊗ φ2 − gψ is of the form h(dz) for some d | D and modular form

h(z) ∈ QJqK of level dividing D. Since this difference has level Dp2 and D is square-free, we

must have h(z) = 0.

Let ψ be defined as above. Consider m+n√D ∈ OK in the following subgroup of Pic(Op)

ker(π : Pic(Op) −→ Pic(OK)) = (OK/pOK)×/((Z/pZ)×O×K/{±1}

)∼= (Z/pZ)×/(O×K/{±1}),

which is defined in the exact sequence (3.2.4). If vp(m+n√D− a) > 0 with a ∈ Z, then the

value of ψ on m+ n√D becomes

ψ(m+ n√D) = φ1(m+ n

√D)φ2(m2 − n2D) = φ1(a)φ2(2am− a2) = φ2

(2ma− 1).

By changing m+ n√D to m+ λ+ n

√D for λ ∈ Z, the quantity 2(m+λ)

a+λ− 1 runs through all

residue classes modulo p. Thus, the restriction of ψ to the kernel of π : Pic(Op) −→ Pic(OK)

is φ2.

For each A ∈ Pic(OK), define

gψ,A := fφ,A ⊗ φ2 =∑n≥1

cψ,A(n), (3.3.16)

cψ,A(n) =∑

a⊂OK ,a∈IK(p),Nm(a)=n,

π([a])=A∈Pic(OK)

ψ(a) =∑

A′∈Pic(Op)π(A′)=A

ψ(A′)rA′(n), (3.3.17)

47

where π : Pic(Op) −→ Pic(OK) is the projection maps as in (3.2.6). Note that if n = pn′,

then rA′(pn′) = rA

(n′

p

)and∑A′∈Pic(Op)π(A′)=A

ψ(A′) = ψ(A′′)∑

B′∈ker(π)

φ2(B′) = 0 (3.3.18)

since φ2 is non-trivial. Here A′′ ∈ Pic(Op) is any class whose image under π is A ∈ Pic(OK).

Thus, cψ,A(pn′) = 0.

As immediate consequences of Props. 3.3.3 and 3.3.4, we have

gψ(z) =∑

A∈Pic(OK)

gψ,A(z), (3.3.19)

(dn

)cψ,A(n) = ϕd(A)cψ,A(n) whenever gcd(n, d) = 1, (3.3.20)

for all d ∈ ΣD. When A = A0 ∈ Pic(OK) is the principal class, the Fourier coefficients

cφ,A0(n) have the following properties.

Proposition 3.3.5. When n < |D|4

, the Fourier coefficient cψ,A0(n) has the form

cψ,A0(n) =

1, when n = k2, p - k

0, otherwise.

(3.3.21)

Proof. Let 1 ≤ n < |D|4

be a positive integer and a ⊂ OK an ideal in the set

IK,p(A0, n) := {a ⊂ O : a ∈ IK(p),Nm(a) = n, π([a]) = A0}.

By the exact sequences (3.2.3) and (3.2.4), we know that ker(π) consists of principal ideals

a = (α), α ∈ K, such that

ordp(α) = ordp(α) = 0.

When α ∈ OK , we could write α = a+b√D

2with a, b ∈ Z. Since Nm(a) = Nm(α) = n < |D|

4,

we have b = 0. In that case, the ideal a = (α) is in PK,Z(p), hence principal in Pic(Op).

So when n < |D|4

, the set IK,p(A0, n) is non-empty only for n a square not divisible by p, in

which case it has only one element. So whenever n < |D|4

is a perfect square not divisible by

p, Eq. (3.3.17) tells us that

cψ,A0(n) =∑

a∈IK,p(A0,n)

ψ(a) = 1.

48

Corollary 3.3.6. When n < |D|4

, the Fourier coefficient cφ,A0(n) has the form

cφ,A0(n) =

φ1(k), when n = k2, p - k,

φ(p)cφ,[p]−1(m), when n = pm,

0, otherwise.

(3.3.22)

Proof. Using the relation cψ,A(n) = cφ,A(n)φ2(n), it is easy to deduce Eq. (3.3.22) from Prop.

3.3.5 when p - n. Otherwise for n = pm, Prop. 3.3.1 and Eq. (3.3.7) implies that

cφ,A0(pm) =1

H(D)

∑ϕ

cφϕ(pm) =1

H(D)

∑ϕ

(φϕ)(p)cφϕ(m) = φ(p)cφ,[p]−1(m).

3.4 Mock-Modular Forms with Imaginary Dihedral Shadow

3.4.1 Petersson Inner Product of Newforms

For an integer M , let EM(z, s) be the non-holomorphic Eisenstein series of weight zero, level

M defined by

EM(z, s) :=∑

γ∈Γ∞\Γ0(M)

(Im(γz))s, (3.4.1)

where Γ∞ = {( ∗ ∗0 ∗ ) ∈ SL2(Z)}. In particular, we can write

E1(z, s) =1

2

∑c,d∈Z

gcd(c,d)=1,(c,d)6=(0,0)

ys

|cz + d|2s=

1

2ζ(2s)

∑c,d∈Z

(c,d) 6=(0,0)

ys

|cz + d|2s.

For convenience, we denote E1(z, s) by E(z, s). It has a simple pole at s = 1 and the

well-known expansion

E(z, s) = ys + ϕ(s)y1−s +O(e−y)

as y →∞, where z = x+ iy and

ϕ(s) =Γ(

12

)Γ(s− 1

2

)Γ(s)

ζ(2s− 1)

ζ(2s).

49

Kronecker’s first limit formula states that

2ζ(2s)E(z, s) =π

s− 1+ 2π

(γ − log(2)− log(

√y|η(z)|2)

)+O(s− 1), (3.4.2)

where γ is the Euler constant. The factor of 2 comes from ±I ∈ Γ∞.

By Eq. (II 2.16) in [29], the Eisenstein series EM(z, s) can be expressed in terms of E(z, s)

as

EM(z, s) = M−s∏`|M

(1− `−2s)−1∑d|M

µ(d)

dsE

(M

dz, s

), (3.4.3)

where µ(d) is the Mobius function. Using (3.4.2) and Rankin-Selberg unfolding trick, we can

relate the inner product between dihedral newforms to values of modular functions.

Proposition 3.4.1. Let φ, φ′ : Ip/Pp,1 −→ C× be ray class group characters and φ1, φ′1 :

(Z/pZ)×/O×K −→ C× their associated characters via (3.3.1). Suppose that φ1 = φ′1 are not

quadratic. Then inner product 〈fφ, fφ′〉 vanishes unless φ′ = φ, in which cases we have

〈fφ, fφ〉 = − 4H(D)

#O×K#O×pIψ2 ,

Iψ2 :=∑

[Q]∈C(Dp2)

ψ2(Q) log(√yQ|η(τQ)|2),

(3.4.4)

where ψ is the character of Pic(Op) ∼= C(Dp2) associated to φ by Prop. 3.3.4 and τQ =

xQ + iyQ = B+p√D

2is the CM point associated to the binary quadratic form Q = [A,B,C].

Proof. Let φ2 : (Z/pZ)× −→ C× be a character such that φ1φ22 = 1p. Since φ1 = φ′1, the

map (φφ′) ◦ Nm is trivial and we could write

φ = φ′ϕ

for some character ϕ : Pic(OK) −→ C×. By Prop. 3.3.2, we could associate ring class group

character ψ and ψ′ to φ and φ′ respectively. Defined a character ψ2 of Pic(Op) by

ψ2(a) := φ(a)φ(a) = ψ(a)ψ′(a). (3.4.5)

When φ = φ′, we have

ψ2(a) = φ(a)φ(a) = φ2(a)φ(Nm(a)) = ψ2(a).

50

Since φ1 = φ′1 is not quadratic, we have

φ(a) = φ1(a) 6= φ′1(a) = φ′(a)

for some a ∈ (Z/pZ)×/O×K and ψ2 is not the trivial character. By class field theory, the

characters ϕ, ψ, ψ2 are also characters of Gal(K/K). Denote their induced two-dimensional

representations of Gal(Q/Q) by ρϕ, ρψ and ρψ2 respectively.

Let M = Np be a square-free integer. Since the residue of E(z, s) at s = 1 is 3π

and

independent of z, the residue of EM(z, s) at s = 1 is

Ress=1EM(z, s) =3

πM−1

∏`|M

(1− `−2)−1∑d|M

µ(d)

d

=3

π

∏`|M

(1 + `)−1.

This gives us the relationship

3

π

∏`|M

(1 + `)−1 · 〈fφ, fφ′〉 = Ress=1

∫Γ0(M)\H

fφ(z)fφ′(z)EM(z, s)ydxdy

y2. (3.4.6)

Now, we can use the Rankin-Selberg method to unfold the right hand side and obtain∫Γ0(M)\H

fφ(z)fφ′(z)EM(z, s)ydxdy

y2=

Γ(s)

(4π)sL(s, φ, φ′),

L(s, φ, φ′) :=cφ(n)cφ′(n)

ns.

(3.4.7)

Up to Euler factors at primes dividing M , the function L(s, φ, φ′) equals to L(s, ρφ ⊗ ρφ′),

the L-function of the tensor product of the representations ρφ and ρφ′ . Alternatively, we

could explicitly compute the Euler factors at all the places using Prop. 3.3.1. For a prime `,

we want to evaluate the sum ∑r≥0

cφ(`r)cφ′(`r)

(`s)r. (3.4.8)

51

If (`) splits into ll in OK and ` 6= p, then Eq. (3.4.8) becomes

∑r≥0

(∑rt=0 φ(l)tφ(l)r−t

) (∑rt′=0 φ

′(l)t′φ′(l)r−t′)

(`s)r=∑r≥0

∑rt,t′=0 φ(l)tφ(l)r−tφ′(l)t′φ′(l)r−t′

(`s)r

= (1− `−2s)(

1− φ(l)φ′(l)`−s)−1 (

1− φ(l)φ′(l)`−s)−1

(1− φ(l)φ′(l)`−s

)−1 (1− φ(l)φ′(l)`−s

)−1

= (1− `−2s)(1− ϕ(l)`−s

)−1 (1− ϕ(l)`−s

)−1 (1− ψ2(l)`−s

)−1 (1− ψ2(l)`−s

)−1.

If (`) is inert in OK , then ψ2(`) = 1 and Eq. (3.4.8) becomes∑r≥0

φ(`)rφ′(`)r

(`s)2r= (1− φ(`)φ′(`)`−2s)−1 = (1− `−2s)−1.

If (`) = (p) = pp or (`) = l2 is ramified in OK , then ψ2(l) = 0 or

ψ2(l) = φ(l)φ′(l)φ′(l)φ′(l) = φ(l2)ϕ(l)φ′(`) = ϕ(l).

respectively and Eq. (3.4.8) becomes∑r≥0

φ(l)rφ′(l)r

(`s)r= (1− ϕ(l)`−s)−1.

Multiplying these together, we find that

L(s, φ, φ′) =L(s, ρψ2)L(s, ρϕ)(1− ϕ(p)p−s)

ζ(2s)(1− p−2s)

∏`|N

(1 + ϕ(l)`−s)−1, (3.4.9)

By definition, we could write

L(s, ρψ2) =∑

A∈Pic(Op)

ψ2(A)L(s,A),

L(s,A) :=∑a∈Op[a]=A

Nm(a)−s =1

#O×p

∑α∈b

[b]=A−1

(Nm(α)

Nm(b)

)−s.

where b = [A, B+p√D

2] ⊂ Op be an ideal such that [b] = A−1 and Nm(b) = A. Set C =

(B2 − p2D)/4A and we have the following bijections

{(m,n) ∈ Z2} ↔ {Nm(α) : α ∈ b}

(m,n)↔ Nm

(Am+

B + p√D

2n

)= Nm(b)(Am2 +Bmn+ Cn2).

52

Then we have

L(s,A) =1

#O×p

∑′

m,n∈Z

(Am2 +Bmn+ Cn2

)−s=

(p√|D|/2)−s

#O×p

∑′

m,n∈Z

ysQ

|m+ nτQ|2s

=21+sζ(2s)

(p√|D|)s#O×p

E(τQ, s)

where Q = [A,B,C] and τQ = xQ+iyQ = −B+p√D

2A. By the isomorphism (3.2.1), the function

L(s, ψ2) becomes

L(s, ρψ2) =21+sζ(2s)

(p√|D|)s#O×p

∑[Q]∈C(Dp2)

ψ2(τQ)E(τQ, s). (3.4.10)

Similarly, we could deduce that

L(s, ρϕ) =21+sζ(2s)

(√|D|)s#O×K

∑[Q]∈C(D)

ϕ(τQ)E(τQ, s). (3.4.11)

From Eqs. (3.4.10), (3.4.11), (3.4.9) and the fact that ψ2 is not trivial, we see that L(s, φ, φ′)

has a pole at s = 1 when ϕ is trivial. In that case,

ψ2(l) = φ(l)φ(l).

Now putting together Eqs. (3.4.2), (3.4.6), (3.4.7), (3.4.10) and (3.4.11), we obtain Eq.

(3.4.4).

Corollary 3.4.2. Let A1,A2 ∈ Pic(OK) and φ be a ray class group character such that φ1

is non-quadratic. Then we have

〈fφ,A1 , fφ,A2〉 = − 4

#O×K#O×pIψ2,A1(A2)−1 ,

Iψ2,B2 :=∑

[Q]∈C(Dp2)π(Q)2=B2

ψ2([Q]) log(√yQ|η(τQ)|2).

(3.4.12)

Remark 3.4.3. Notice that if A1(A2)−1 6∈ Pic2(OK), then the inner product above vanishes.

53

Proof. By Eq. (3.3.6) and Prop. 3.4.1, we have

〈fφ,A1 , fφ,A2〉 =1

H(D)2

∑ϕ1,ϕ2

ϕ1(A1−1)ϕ2((A2)−1)〈fφϕ1 , fφϕ2〉

=1

H(D)2

∑ϕ1

ϕ1(A1−1)ϕ1(A2)〈fφϕ1 , fφϕ1〉

= − 4

H(D)#O×K#O×p

∑ϕ1

ϕ1(A1−1A2)

∑[Q]∈C(Dp2)

ψ2(Q)ϕ21(Q) log(

√yQ|η(zQ)|2)

= − 4

H(D)#O×K#O×p

∑[Q]∈C(Dp2)

ψ2(Q) log(√

yQ|η(zQ)|2)∑

ϕ1

ϕ1((A1)−1A2Q2)

= − 4

#O×K#O×p

∑[Q]∈C(Dp2)

π(Q)2=A1(A2)−1

ψ2(Q) log(√yQ|η(zQ)|2).

Corollary 3.4.4. Suppose d ∈ ΣD, A ∈ Pic(OK) and φ is a ray class group character such

that φ1 = φ′1 is non-quadratic. Then we have

〈fφ,d, fφ,A〉 =

− 4

#O×K#O×pIψ2 , dA = d,

0, Otherwise.

(3.4.13)

Proof. Since fφ,d ∈ S1,d(|D|p, χDφ1) and fφ,A ∈ S1,dA(|D|p, χDφ1), Prop. 2.4.6 implies that

〈fφ,d, fφ,A〉 = 0 if dA 6= d. Otherwise, Eqs. (3.3.13) and (3.4.12) give us

〈fφ,d, fφ,A〉 =1

2ω(D)−1

∑B∈Pic(OK)

〈fφ,A−1B2 , fφ,A〉.

= − 4

#OK#Op1

2ω(D)−1

∑B∈Pic(OK)

∑[Q]∈C(Dp2)π(Q)2=B2

ψ2(Q) log(√

yQ|η(τQ)|2)

= − 4

#OK#OpIψ2 .

3.4.2 Principal Part Coefficients of Mock-Modular Forms

For each d ∈ ΣD, recall that nd and rd are defined in (2.4.19). Let {ft : 1 ≤ t ≤ rd} be a

q-echelon basis of S1,d(|D|p, χDφ1) over R and mt := ord∞(ft). Then 1 ≤ mt ≤ nd for all

54

1 ≤ t ≤ rd. Now we could apply Prop. 2.4.7 and Cor. 3.4.2 to fφ,d ∈ S1,d(|D|p, χDφ1) to

obtain the following result about mock-modular forms with special principal part coefficients.

Proposition 3.4.5. There exists a mock-modular form

fφ,d =

rd∑t=1

cφ,d(−mt)q−mt +

∑n≥0

cφ,d(n)qn ∈M1,d(|D|p, χDφ1)

with shadow fφ,d such that when n ≤ 0,

cφ,d(n) = αnIψ2 (3.4.14)

for some αn ∈ Q(φ), where Q(φ) is the number field obtained from Q by adjoining the values

of φ and φ2.

Proof. By Prop. 3.3.3 and Eq. (3.3.12), fφ,d ∈ S1,d(|D|p, χDφ1). So the existence of fφ,d ∈

M1,d(|D|p, χDχp) with ord∞(fφ,d) < nd with such principal part is given by Prop. 2.4.7. This,

along with Prop. 2.4.6 and Cor. 3.4.4, gives us the equationrd∑t=1

cφ,d(−mt)cφ,A(mt)δ|D|(mt)+∑

1≤t≤rdp|mt

cφ,d(−mt)cφ,A(mt)δ|D|(mt) = 〈fφ,d, fφ,A〉 = −4Iψ2

#O×K#O×p

for any A ∈ Pic(OK) satisfying dA = d.

If there exists any subspace of S1,d orthogonal to fφϕ,d for any character ϕ of Pic(OK),

then we could obtain equations similar to the one above with the right hand side replaced

by 0. These rd equations are sufficient to determine cφ,d(−mt) for all 1 ≤ t ≤ rd by Prop.

2.4.7. Since all the entries in this rd × rd matrix are in Q(φ) and the right hand side rd × 1

matrix has entries either 0 or −4#O×K#O×p

Iψ2 , we know that each cφ,d(−mt) can be written in

the form of Eq. (3.4.15). Applying the same procedure to the inner product between fφ,d

and the Eisenstein series proves Eq. (3.4.15) for n = 0.

There is also a refined result regarding the principal part coefficients of the mock-modular

form fφ,A ∈M1,dA, which we state here. The proof is the same as the one for Prop. 2.4.7.

Proposition 3.4.6. For every A ∈ Pic(OK), there exists a mock-modular form

fφ,A =

rdA∑t=1

cφ,A(−mt)q−mt +

∑n≥0

cφ,A(n)qn ∈M1,dA(|D|p, χDφ1)

55

with shadow fφ,A such that when n ≤ 0,

cφ,A(n) =∑

B2∈Pic2(OK)

αn,A,B2Iψ2,B2 (3.4.15)

for some αn,A,B2 ∈ Q(φ).

56

CHAPTER 4

Proof of Main Theorem

4.1 Counting Arguments

In this section, we will prove a counting argument crucial to our results.

4.1.1 Background

In this section, we will go over the background on positive definite binary quadratic forms

following the treatments in [6] and [17]. The proofs of the results directly from these sources

are omitted.

Let D0 = Df 2 < 0 be a discriminant with D < 0 a fundamental discriminant and

f an integer. Denote a positive definite binary quadratic form with discriminant D0 by

[A,B,C], A > 0 and the set of all such forms by C(D0). If gcd(A,B,C) = 1, then the form

is call primitive. From now on, forms [A,B,C] can be taken to be positive definite and

primitive.

The group SL2(Z) has a right action on [A,B,C] via

[A,B,C] · γ := [A′, B′, C ′],

where γ = ( a bc d ) ∈ SL2(Z) and

A′ = Aa2 +Bac+ Cc2,

C ′ = Ab2 +Bbd+ Cd2,

B′ = 2Aab+ 2Ccd+B(ad+ bc).

(4.1.1)

It is not hard to see that the action preserves primitivity and positive definiteness. Notice γ

57

and −γ =( −1

−1

)· γ have the same action. So the group

Γ := PSL2(Z) = SL2(Z)/{± ( 11 )}

acts on Q. Two forms [A,B,C] and [A′, B′, C ′] are equivalent 1 if there exists γ ∈ SL2(Z)

such that Q · γ = Q′. A form [A,B,C] is called reduced (see Eq. (2.7) in [17]) if

|B| ≤ A ≤ C, and B ≥ 0 if either |B| = A or A = C. (4.1.2)

This is a convenient notion due to this theorem.

Theorem 4.1.1 (Theorem 2.8 [17]). Every primitive positve definite form is equivalent to a

unique reduced form.

After Dirichlet, we say two forms [A1, B1, C1] and [A2, B2, C2] of the same discriminants

D0 are united forms if

gcd(A1, A2,B1+B2

2) = 1.

As a consequence of some simple calculations, we can find representatives of united forms

that are convenient for defining a composition law.

Proposition 4.1.2 (Proposition 4.5 [6]). If [A1, B1, C1] and [A2, B2, C2] are united forms,

then there exist forms [A1, B,A2C] and [A2, B,A1C] such that

[A1, B1, C1] ∼ [A1, B,A2C],

[A2, B2, C2] ∼ [A2, B,A1C].

We define the composition of two united forms [A1, B,A2C] and [A2, B,A1C] by

[A1, B,A2C] ◦ [A2, B,A1C] := [A1A2, B, C]. (4.1.3)

The composition law is well-defined with respect to the equivalence relationship (Theorem

4.7 [6]). In fact, it makes the equivalence classes of primitive binary quadratic forms into a

group, which we denote by C(D0). As a corollary, we have

1This is proper equivalence in the work of Gauss. Lagrange used the action of GL2(Z) instead of SL2(Z)in his notion of equivalence.

58

Corollary 4.1.3. Let Q = [A,B,AC] be a binary quadratic form. Then Q2 is represented

by the binary quadratic form [A2, B, C].

Remark 4.1.4. To be more precise, if [A,B,AC] · ( a bc d ) = [A′, B′, A′C ′], then [A2, B, C] ·

(X WY Z ) = [A′2, B′, C ′] with

X = a2 − Cc2, Y = 2Aac+Bc2.

For T ∈ R, let τT be the translation defined by

τT := ( 1 T1 ) . (4.1.4)

If ( a bc d ) = τAt for some t ∈ Z, then Eq. (4.1.1) implies (X WY Z ) = τt, where

It is clear from the definition that C(D0) is abelian. So the image of C(D0) under the

squaring map is again an abelian group, which we denote by C2(D0). The kernel, denoted

by C0(D0), contains forms of order at most 2 and we have the following short exact sequence

0 −→ C0(D0) −→ C(D0)·2−→ C2(D0)→ 0.

The subgroup C0(D0) is well-understood. When D0 is odd, let Qm ∈ C(D0) be the class

represented by the forms Qm, which is defined as

Qm := [Am, Bm, Am],

Am = 14

(m− D0

m

), Bm = 1

2

(m+ D0

m

),

(4.1.5)

where m | D0 and gcd(m, D0

m) = 1. It is clear that Qm ∈ C0(D0) as Q2

m · γm = [A2m, Bm, 1] ·

γm = Q0 where

γm =( −1

1Bm+1

2

).

Thus, Qm ∼ QD0/m and the set

C ′0(D0) := {Qm : m | D0,m2 < −D0, gcd(m, D0

m) = 1} ⊆ C0(D0)

has size 2ω(D0)−1, where for any N ∈ Z

ω(N) := number of distinct prime divisors of N. (4.1.6)

On the other hand, Prop. 3.11 in [17] tells us that there are exactly 2ω(D0)−1 elements in

C0(D0). It is then natural to expect the following lemma to hold.

59

Lemma 4.1.5. In the notations above, C ′0(D0) = C0(D0).

Proof. It suffices to show that for two m,m′ | D0 satisfying

gcd(m,D0/m) = gcd(m′, D0/m′) = 1,

the classes Qm and Qm′ are the same if and only if m = m′ or mm′ = D0. By Theorem 4.1.1,

it is enough to look at the reduced forms equivalent to the Qm’s. Since Qm ∼ QD0/m, it is

enough to cnosider m | D0 such that −D0

m2 > 1. Suppose −D0

m2 ≥ 3, then 0 < m ≤ Am and Qm

is equivalent to the reduced form [m,m,Am]. Otherwise if 1 < −D0

m2 ≤ 3, then |Bm| ≤ Am

and Qm is equivalent to the reduced form [Am, |Bm|, Am]. Thus, for any given m,m′ | D0

such that m2 < −D0, (m′)2 < −D0 and Qm ∼ Qm′ , we know that Am = Am′ , which implies

m = m′.

The composition law on C0(D0) can now be easily described in terms of Qm.

Lemma 4.1.6. For m,m′ | D0 satisfying gcd(m,D0/m) = gcd(m′, D0/m′) = 1, define

M | D0 by

M :=mm′

gcd(m,m′)2. (4.1.7)

Then gcd(M,D0/M) = 1 and Qm ◦ Qm′ = QM .

Proof. Since gcd(m,D0/m) = gcd(m′, D0/m′) = 1, it is not hard to see that

M = gcd(m, D0

m′) · gcd(m′, D0

m),

D0

M= gcd(m,m′) · gcd(D0

m, D0

m′).

Thus, M | D0 and gcd(M,D0/M) = 1.

For the second part of the claim, we will first prove the case m | m′. Let m′ = mn with

n ∈ Z. We have seen before that Qm ∼ [m,m,Am], Qm′ ∼ [m′,m′, Am′ ]. Using Arndt’s

composition algorithm (Theorem 4.10 [6]), it is easy to compute that

[m,m,Am] ◦ [mn,mn,Amn] = [n,mn, ∗] ∼ [n, n,An].

The equivalence step follows from applying τ(1−m)/2 andm | D0 is odd. Thus,Qm◦Qmn = Qn.

60

Next, suppose gcd(m,m′) = 1, then M = mm′, m | D0/m′ and we have

Qm ◦ Qm′ = Qm ◦ QD0/m′

= QD0/(mm′) = Qmm′ = QM .

Finally, suppose gcd(m,m′) = n, then M = mm′

n2 and

Qm ◦ Qm′ = Qm ◦ Qn ◦ QnQm′

= Qm/n ◦ Qm′/n = Qmm′

n2

= QM .

When D0 is even, similar results hold and details can be found in §3.B in [17].

4.1.2 Counting Theorem

In this section, we will prove Lemma 4.1.9, which is crucial to the counting result, Theorem

4.1.10. Throughout, D0 = Df 2 < 0 will be an odd discriminant with D fundamental. For

any primitive, positive definite binary quadratic form P = [A2, B, C] of discriminant D0,

A > 0 and x, y ∈ Z, define the quantity

I(P, [x, y]) := 2A2x+ByA

∈ Q. (4.1.8)

If gcd(A,B) = 1, then Q = [A,B,AC] is primitive and P = Q2 = [A2, B, C] by Corollary

4.1.3. Since Q is primitive, gcd(A,D0) = 1 and Ip(x, y) is well-defined modulo D0.

In general, the function I(Q2, [x, y]) is not well-defined as a function on the class of Q or

Q2. Fortunately, Lemma 4.1.7 below shows that I(Q2, [x, y]) (mod D0) is well-defined as a

function on the class of Q for many choices of x, y ∈ Z.

Lemma 4.1.7. Let Q = [A,B,AC] be primitive and x, y ∈ Z such that

gcd(Q2(x, y), f) = 1.

Suppose Q · ( a bc d ) = Q′ := [A′, B′, A′C ′] and Q2 · γ = (Q′)2 with γ = (X WY Z ) defined as in

Remark 4.1.4, then

I(Q2, [x, y]) ≡ I((Q′)2, [x′, y′]) (mod D0), (4.1.9)

61

where [x′, y′] := [x, y] · (γ−1)t.

Remark 4.1.8. There will be a negative sign in the congruence if we choose −γ instead of

γ.

Proof. For simplicity, write

IQ = I(Q2, [x, y]), IQ′ = I((Q′)2, [x′, y′]).

First, we have the following important observation from completing the square

4A2Q2(x, y) = (2A2x+By)2 +D0y2. (4.1.10)

After unfolding the definitions, it is not hard to see that

Q2(x, y) = (Q′)2(x′, y′),

I2Q = 4Q2(x, y) + D0

A2 y2,

I2Q′ = 4(Q′)2(x′, y′) + D0

(A′)2(y′)2.

This implies that

I2Q ≡ (IQ′)

2 ≡ 4Q2(x, y) (mod D0). (4.1.11)

Let p | D0 be a prime. If r := ordp(D0) ≥ 2, then p | f and p - Q2(x, y) since gcd(f,Q2(x, y)) =

1. That means only one of IQ + IQ′ and IQ − IQ′ is divisible by p, hence

IQ ≡ ±IQ′ (mod pr)⇐⇒ IQ ≡ ±IQ′ (mod p) (4.1.12)

So to prove Eq. (4.1.9), it suffices to show

IQ ≡ IQ′ (mod D′),

where D′ | D0 is square-free and defined by

D′ :=∏

p|D0,p-Q2(x,y)

p. (4.1.13)

To simplify the proof, we will consider two cases depending on γ modulo D′. Bear in mind

that B2 ≡ 4A2C (mod D0) in both cases.

62

Case (1) γ ≡ ( 1 W0 1 ) (mod D′0)

Remark 4.1.4 implies that

a2 − Cc2 = X ≡ 1 (mod D′),

2Aac+Bc2 = Y ≡ 0 (mod D′).

Applying the congruences above and equation (4.1.1) to Q ∼ Q′ gives us

4AA′ = 4A(Aa2 +Bac+ ACc2)

≡ 4A2a2 + 4ABac+B2c2

≡ 4A2(1 + Cc2)− 2B2c2 +B2c2

≡ 4A2 (mod D′)

Since gcd(A,D0) = 1, we conclude that A ≡ A′ (mod D′). Now applying equation (4.1.1) to

Q2 · γ = (Q′)2 gives us

B′ = 2A2XW + 2CY Z +B(XZ +WY ) ≡ 2A2W +B (mod D′),

x′ = Zx−Wy ≡ x−Wy (mod D′),

y′ = −Y x+Xy ≡ y (mod D′).

Putting these together, we see that

I ′ ≡ A′(2(A′)2x′ +B′y′)

≡ A(2A2(x−Wy) + (2A2W +B)y)

≡ I (mod D′)

Case (2) γ ≡(α 0β α

)(mod D′)

63

Similar to before, we have

(A′)2 ≡ A2α2 +Bαβ + Cβ2,

x′ ≡ αx (mod D′),

y′ ≡ −βx+ αy (mod D′),

B′ ≡ B + 2Cβα (mod D′),

a2 − Cc2 = X ≡ α (mod D′),

2Aac+Bc2 = Y ≡ β (mod D′),

A′ = Aa2 +Bac+ ACc2.

Substituting these into I ′ gives

I ′ ≡ A′(2(A2α2 +Bαβ + Cβ2)(αx) + (B + 2Cβα)(−βx+ αy))

≡ 2A′A2(4A4αx+ 2A2Bβx+B2βy + A2Bαy)

≡ 2A′A(2A2α +Bβ)I

≡ 4A′A(4A2(a2 − Cc2) + 2B(2Aac+Bc2))I

≡ 4A′A(2Aa+Bc)2I

≡ (4A2a2 + 4ABac+ 4A2Cc2)(2Aa+Bc)2I

≡ I (mod D′)

Notice the analysis in both cases works fine if D′ is replaced with any of its divisor. Now

in the general case γ = (X WY Z ), let p | D′ be a prime. We say that γ1, γ2 ∈ SL2(Fp) are

translation equivalent if there exists t, t′ ∈ Fp such that

τtγ1τt′ = γ2,

where τT is translation by T as in Eq. (4.1.4). It is not difficult to see that any γ ∈ SL2(Fp)

is translation equivalent to a matrix γ′ of the shape ( ∗ 0∗ ∗ ). 2

2Indeed, τtγτt′ =(∗ t′(X+tY )+W+tZ∗ ∗

)and p - (X + tY ) for some t.

64

Let γ′ = τ−t1γτt2 such that γ′ ≡ ( ∗∗ ∗ ) (mod p). Note the choice of t1, t2 depends on p.

Set Q1 := Q · τAt1 , Q2 := Q′ · τA′t2 . Then Remark 4.1.4 implies

Q1 · (τ−At1 ( a bc d ) τA′t2) = Q2,

(Q1)2 · γ′ = (Q2)2

Then the analysis of the two cases above shows that

IQ ≡ IQ1 ≡ IQ2 ≡ IQ′ (mod p),

where the IQj ’s are defined in the same way as IQ, IQ′ . Since this congruence holds for all

p | D′, which is square-free by definition, we have IQ ≡ IQ′ (mod D0) for any γ.

From this Lemma, we see that the value I(Q2, [x, y]) modulo D0 only depends on the

class of Q if Q2(x, y) is relatively prime to f .

One can now ask whether the congruence in Eq. (4.1.9) could distinguish the classes of Q

and Q′. There will be a minus sign in it since replacing γ by −γ has this effect on Eq. (4.1.9).

If f = 1 and D0 = D is composite and divides k, then the answer is certainly false since Eq.

(4.1.9) will be true for any form Q′′ with (Q′′)2 = Q2. On the other hand, if gcd(k,D0) = 1,

then the answer is positive, as we will see.

The discussion in §4.1.1 then tells us that the ambiguity comes from C2(D0) and should

be related to gcd(D0, k) intuitively. The following lemma gives a precise statement, which

strengthens and completes Lemma 4.1.7.

Lemma 4.1.9. Let Q = [A,B,AC] be primitive of discriminant D0 and x, y ∈ Z such that

gcd(Q2(x, y), f) = 1. Suppose Q′ = [A′, B′, A′C ′] satisfies (Q′)2 = Q2 · γ. Then

I(Q2, [x, y]) ≡ ±I((Q′)2, [x, y](γ−1)t) (mod D0)

if and only if Q′Q−1 ∼ QM for some QM as in Eq. (4.1.5) such that M | Q2(x, y).

Proof. Notice that Lemma 4.1.7 allows us to choose convenient representatives of Q and Q′

here in the proof. We will use the same notation as in Lemma 4.1.7 and consider two cases.

65

Case (i): Q2 ∼ Q0 = [1, 1, 1−D0

4]

Choose Q = Qm and Q′ = Qm′ with m,m′ | D0 as in Eq. (4.1.5). Then Q2 ·γ = (Q′)2 where

γ = γmγ−1m′ =

(1

Bm′−Bm2

1

).

Substitute these and B2m ≡ 4A2

m (mod D0) into IQm gives us

IQm ≡ 2AmBm(Bmx+ 2y) (mod D0).

Since gcd(m, D0

m) = 1, we know that gcd(Bm, D0) = 1. This implies

gcd(Q2(x, y), D0) = gcd(IQ, D0) = gcd((2A2mx+Bmy), D0)

= gcd((B2mx+ 2Bmy), D0) = gcd((Bmx+ 2y), D0).

Now substitute the [x′, y′] = [x, y](γ−1)t and B2m′ − 4A2

m′ ≡ 0 (mod D0) into IQm′ yields

IQm′ ≡ 2Am′Bm′(Bmx+ 2y) (mod D0).

Let p | D0 be a prime. Suppose p | Q2(x, y), then p | gcd(Q2(x, y), D0) = gcd((Bmx+2y), D0)

and IQm ≡ IQm′ ≡ 0 (mod p). Otherwise, if p - Q2(x, y) and p | m, then

IQm ≡ −(Bmx+ 2y) ≡

IQm′ (mod p) p | m′

−IQm′ (mod p) p - m′.

Similarly, when p - Q2(x, y) and p | D0

m, we have

IQm ≡ (Bmx+ 2y) ≡

IQm′ (mod p) p | D0

m′

−IQm′ (mod p) p - D0

m′.

Let M be defined as in Eq. (4.1.7). We can then summarize the results as follows

IQm ≡

±IQm′ (mod p), p | Q2

m(x, y)

IQm′ (mod p), p - Q2m(x, y) and p | D0

M

−IQm′ (mod p), p - Q2m(x, y) and p |M.

(4.1.14)

Since gcd(Q2(x, y), f) = 1, the same argument in Lemma 4.1.7 implies

IQ ≡ ±IQ′ (mod D0)⇐⇒ IQ ≡ ±IQ′ (mod σ(D0)), (4.1.15)

66

where σ(N) is the largest square-free integer dividing N ∈ Z and defined by

σ(N) :=∏p|N

p.

Combining this general fact with the analysis above in the specific case Q2 ∼ Q0, we have

IQm ≡ ±IQm′ (mod D0)⇐⇒ IQm ≡ ±IQm′ (mod σ(D0))

⇐⇒ σ(M) | Q2m(x, y) or σ(D0

M) | Q2

m(x, y)

⇐⇒M | Q2m(x, y) or D0

M| Q2

m(x, y).

If σ(M) | Q2m(x, y), then gcd(M, f) = 1 since gcd(Q2

m(x, y), f) = 1. Since M | D0 = Df 2,

we know that M | D is square-free and the third equivalence above follows. Now Lemma

4.1.6, we have QmQ−1m′ ∼ QM ∼ QD0/M and finished proving this case. Notice that since

gcd(M, D0

M) = 1, M | Q2

m(x, y) and D0

M| Q2

m(x, y) happen simultaneously only when f = 1

and D0 | Q2m(x, y).

Case (ii): Q2 6∼ Q0

Let ` be a prime represented by QM ∼ Q−1Q′ such that gcd(`,D0) = 1. This is equivalent

to finding a prime ideal l of the ring of integers of the imaginary quadratic field Q(√D0)

with norm ` and contained in the ideal class corresponding to QM , which is possible by

Chebotarev density theorem.

For the representative of Q, choose [A,B,AC] such that ` | A. Since ` is prime and

represented by QM , any form representing ` is equivalent to QM or Q−1M = QM , in particular

Q′M := [`, B, A2C`

] ∼ QM .

We can then choose Q′ = [A′, B′, A′C ′], where

A′ = A`, B′ = B,C ′ = `2C.

By the definition of composition in Eq. (4.1.3), it is easy to check that Q′ ◦Q′M = Q. Also,

we have

Q2 = [A2, B, C], (Q′)2 = [(A′)2, B, C ′].

67

Since [`2, B, (A′)2C] = (Q′M)2 ∼ Q2M ∼ [1, 1, 1−D0

4], there exist a, c ∈ Z such that

`2a2 +Bac+ (A′)2Cc2 = 1,

which also implies gcd(a, (A′)2c) = 1 and

A2(a)2 +Ba((A′)2c) + C((A′)2c)2 = (A′)2.

That means Q2 · γ = (Q′)2 with

γ =(

a b(A′)2c d

)∈ SL2(Z).

Set γ :=(a (A′)2bc d

), X := A′x,X ′ := A′x′. It is easy to check that

[X ′, y′] = [X, y](γ−1)t, (Q′M)2 · γ = [1, B,A2C].

Substituting these into Eq. (4.1.9) gives us

`(2`2X +By) ≡ (2X ′ +By′) (mod D0),

and equivalence (4.1.15) becomes

IQ ≡ ±IQ′ (mod D0)⇐⇒ IQ ≡ ±IQ′ (mod σ(D0))

⇐⇒ `(2`2X +By) ≡ ±(2X ′ +By′) (mod σ(D0))

⇐⇒ I((Q′M)2, [X, y]) ≡ ±I([1, B,A2C]2, [X ′, y′]) (mod σ(D0))

⇐⇒M | gcd((Q′M)2(X, y), D0) or D0

M| gcd((Q′M)2(X, y), D0)

⇐⇒M | gcd((2`2X +By), D0) or D0

M| gcd((2`2X +By), D0)

⇐⇒M | gcd((2A2x+By), D0) or D0

M| gcd((2A2x+By), D0)

⇐⇒M | Q2(x, y) or D0

M| Q2(x, y).

Here, the fourth “iff” follows from case (i) proved above. The fifth and seventh “iff” both

follows from Eq. (4.1.10)

68

Now, we are ready to prove the main counting theorem. For a binary quadratic for

Q = [A,B,C] of discriminant D0, let τQ := B+√D0

2A∈ H be the CM point associated to it.

Let d < 0 be another discriminant, q = [a, b, c] a binary quadratic form with discriminant

d, and d(τQ, τq) the hyperbolic distance between τQ and τq Its hyperbolic cosine has the

following convenient expression

cosh(d(τQ, τq)) = 2Ac+2Ca−Bb√D0d

. (4.1.16)

The group SL2(Z) acts isometrically on H via linear fractional transformation. It is easy to

check that for γ ∈ SL2(Z),

γ · τQ = τQ·(( 1

−1 )γ−1( 1−1 )). (4.1.17)

Let Q ∈ C(D0) and k ∈ Z. Define the sets and their sizes by

SQ(k, d) :={q = [a, b, c] ∈ Z3 : a > 0, disc(q) = d, cosh(d(τQ, τq)) = k√D0d}, (4.1.18)

RQ(n) :={(Q,±(x, y)) : x, y ∈ Z, [Q] = Q, Q(x, y) = n}/ ∼, (4.1.19)

ρQ(k, d) = #SQ(k, d), (4.1.20)

rQ(n) = #RQ(n). (4.1.21)

Here (Q,±(x, y)) ∼ (Q′,±(x′, y′)) if there exists γ ∈ SL2(Z) such that Q′ = Q′ · γ and

[x′, y′] = [x, y](γ−1)t.

The set SQ(k, d) counts the number of CM points of a fixed discriminant and at a fixed

hyperbolic distance from a given CM point τQ. By Eq. (4.1.17), the quantity ρQ(k, d) depends

only on the class of Q. The main counting theorem will tell us that this number is closely

related to rQ2(n).

Theorem 4.1.10. Let P ∈ C2(D0) and k ∈ Z such that gcd(k, f) = 1. Then

∑Q∈C(D0),Q2=P

ρQ(k, d) = rP

(k2−D0d

4

)· 2ω(gcd(D0,k)). (4.1.22)

Proof. Let Q = [A,B,AC] such that [Q2] = P . By Eq. (4.1.16), a form q ∈ SQ(k, d) is the

69

same as a triple [a, b, c] satisfying

b2 − 4ac = −d,

2Ac+ 2ACa−Bb = k.

Now there is a map φQ between SQ(k, d) and RQ2

(k2−D0d

4

)defined by

φQ : SQ(k, d) −→ RQ2

(k2−D0d

4

)[a, b, c] 7→ (Q2,±(c− Ca,Ba− Ab)).

It is more convenient to describe φQ by the linear map2AC −B 2A

B −A 0

−C 0 1

·a

b

c

=

k

±y

±x

.

Here the ± sign in front of x and y are the same. The determinant of the 3×3 matrix above

is D0, so is an injective linear map, and φQ is a function.

Now given (Q2,±(x, y)) ∈ RQ2

(k2−D0d

4

), it is in the image of φQ if and only if

1

D0

−A B 2A2

−B 4AC 2AB

AC BC B2 − 2A2C

·k

±y

±x

=

a

b

c

∈ Z3.

If a = (−Ak+By+2A2x)/D0 ∈ Z, then c = Ca+x ∈ Z and b = (Ba−y)/A =√d+ 4ac ∈ Z.

Thus, we have

(Q2,±(x, y)) ∈ im(φQ)⇐⇒ (2A2x+By) ≡ ±Ak (mod D0),

⇐⇒ I(Q2, [x, y]) ≡ ±k (mod D0) (4.1.23)

Now only one of ± holds unless D0 | k, in which case both signs can occur. That means φQ

is an injective map when D0 - k and a 2 to 1 map when D0 | k.

By Eq. (4.1.10), we have

I(Q2, [x, y])2 ≡ k2 (mod D0).

70

That means there exists ε(p,Q, [x, y]) ∈ {±1} for each p | D0 such that

I(Q2, [x, y]) ≡ ε(p,Q, [x, y])k (mod p).

So Eq. (4.1.23) is satisfied if and only if ε(p,Q, [x, y]) = ±1 is independent of p | D0.

Using the congruence summary (4.1.14), we can find ((Q′)2,±(x′, y′)) ∼ (Q2,±(x, y))

such that

I(Q2, [x, y]) ≡ ε(p,Q, [x, y])I((Q′)2, [x′, y′]) (mod p).

So equivalence (4.1.23) implies that ((Q′)2,±(x′, y′)) ∈ im(φQ′). Furthermore, Lemma 4.1.9

also tell us that

im(φQ′) = im(φQ)⇐⇒ Q−1Q′ ∼ QM for some M | k.

Now if D0 - k, then every element in RQ2

(k2−D0d

4

)lies im(φQQm) for exactly 2ω(gcd(k,D0)) of

m | D0. If D0 | k, then φQQm is a 2 to 1 surjection for every m | D0, gcd(m,D0/m) = 1. By

Lemma 4.1.5, the set {Q ◦Qm : m | D0,m2 < −D0} contains exactly all the representatives

of the classes Q ∈ C(D0) satisfying Q2 = P . By counting ∪m|D0,m2<−D0im(φQQm) with

repetition, we obtain Eq. (4.1.22).

When gcd(k, f) > 1, the function ρQ(k, d) is more difficult to evaluate. In the special

case when gcd(k, f) = f1 and d is a non-square residue modulo some prime dividing f1, we

have the following result.

Proposition 4.1.11. Write f = f1f2 and let k ∈ Z be an integer such that gcd(k, f2) = 1.

If d < 0 is a discriminant such that(d`

)= −1 for some ` | f1, then ρQ(f1k, d) = 0 for all

Q ∈ C(D0).

Proof. First, we could choose a representative Q = [A, fB, f 2AC] of Q. Then gcd(A, f) = 1

since Q is primitive. Using this Q, we see that if q = [a, b, c] ∈ SQ(f1k, d), then

2Ac+ 2f 2ACa− fBb =√D0d cosh(d(τQ, τq)) = f1k.

71

Since gcd(2A, f) = 1, there exists c′ ∈ Z such that c = f1c′. So the set SQ(f1k, d) becomes

SQ(f1k, d) = {q = [a, b, f1c′] ∈ Z3 : a > 0, b2 − 4af1c

′ = d, 2Ac′ + 2f1f22ACa− f2Bb = k}

= {q = [f1a, b, c′] ∈ Z3 : f1a > 0, b2 − 4(f1a)c′ = d, 2Ac′ + 2f 2

2AC(f1a)− f2Bb = k}

=

{q = [a′, b, c′] ∈ Z3 : a′ > 0, f1 | a′, disc(q) = d, cosh(τQ′ , τq) = k√

Df22 d

},

where a′ = f1a,Q′ = [A,B′, AC ′] = [A, f2B,ACf

22 ] ∈ C(Df 2

2 ). For any [a′, b, c′] ∈ SQ(f1k, d),

define x, y ∈ Z by 2AC ′ B′ 2A

B′ −A 0

−C ′ 0 1

·a′

b

c′

=

k

y

x

.

It is easy to check that 4(A2x2 + B′xy + Cy2) = (k2 − Df 22d). After multiplying on both

sides by the inverses of the two 3× 3 matrices, we obtaina′

b

c′

=1

Df 22

−A B′ 2A2

−B′ 4AC ′ 2AB′

AC ′ B′C ′ (B′)2 − 2A2C ′

·k

y

x

.

Since f1 | a′, we have −Ak+B′y+2A2xDf22

∈ Z and

−Ak +B′y + 2A2x

Df 22

≡ 0 (mod f1).

Some calculations show that

(B′y + 2A2x− Ak)(B′y + 2A2x+ Ak) = 4A2(A2x2 +B′xy + C ′y2) + ((B′)2 − 4A2C ′)y2 − A2k2

= A2(k2 −Df 22d) +Df 2

2 y2 − A2k2

= Df 22 (y2 − A2d).

So A2d ≡ y2 (mod f1), which implies that(d`

)6= −1 for all ` | f1 since gcd(A, f) = 1. This

contradicts our condition on d in the statement of the proposition. So the set SQ(f1k, d) is

empty for all Q ∈ C(D0).

72

4.1.2.1 Character Sum Identities

In this section, we will give some character sum identities necessary for our results.

Let p > 2 be a prime, χp =(·p

)the Dirichlet character of conductor p and χ :

(Z/pZ)× −→ C×. For λ ∈ Z/pZ, define the character sum S(λ, χ) by

S1(λ, χ) := G(χ)G(χχp)G(χp)

p−1∑r=0

χ(r + 1)χ(r + λ)χ(r)χp(r), (4.1.24)

where G(ψ) =∑

k∈(Z/pZ)× ψ(k)ep(k) is the Gauss sum associated to ψ for any character ψ.

Via a substitution, S1(λ, χ) is closely related to another character sum defined for a, b ∈ Z/pZ

by

S2(a, b, χ) := 1G(χ)G(χp)

p−1∑u,v=0

χ (uv) (χχp) (u+ v) ep(au+ bv), (4.1.25)

where ep(z) := e2πiz/p. The following lemma gives an explicit evaluation of S1(λ, χ).

Lemma 4.1.12. In the notations above, if χ 6= χp, then

S1(λ, χ) = 12

(1 + χp(λ)) ·(χ(1 +

√λ)2 + χ(1−

√λ)2). (4.1.26)

Remark 4.1.13. When χp(λ) 6= −1, the quantity√λ makes sense in Fp. Otherwise, the

factor (1 + χp(λ)) vanishes and it is not necessary to evaluate the second factor.

Proof. When χp(λ) = −1, the substitution r 7→ λr

yields

S1(λ, χ) = χp(λ)S1(λ, χ).

So S1(λ, χ) = 0 in this case.

When χp(λ) = 0, Eq. (4.1.24) becomes

G(χχp)G(χp)

G(χ)S1(λ, χ) =

p−1∑r=0

χ(r + 1)ep(r)

= 1G(χ)

p−1∑r,s=0

χ(s)χp(r)ep(s(r + 1))

= 1G(χ)

p−1∑s=0

(χχp)(s)ep(s)

p−1∑rs=0

χp(rs)ep(rs)

= G(χχp)G(χp)

G(χ).

73

Here in the second step, we used the Gauss sum substitution for

χ(r + 1)G(χ) =∑

s∈Z/pZ

χ(s)ep(s(r + 1)).

When χp(λ) = 1, apply Gauss sum substitution for χ((r + 1)(r + λ)) gives us

S1(λ, χ) = G(χ)G(χχp)G(χp)

p−1∑s,r=0

χ(rs)χp(r)ep(s(r + 1)(r + λ))

= 1G(χχp)G(χp)

p−1∑s,r=0

(χχp)(rs)χp(s)ep(s(r +√λ)2 + sr(1−

√λ)2).

Define a map (u, v) : (Z/pZ)2 −→ (Z/pZ)2 by

u(s, r) = sr, v(s, r) = s(r +√λ)2.

For each (u, v) ∈ (Z/pZ)2, the number of preimages under this map depends on the number

of solutions in (r, s) ∈ (Z/pZ)2 to

rs = u, rv = u(r +√λ)2.

When v 6= 0, this quantity is either 0, 1, or 2 and can be expressed as

δ(u, v) := (1 + χp(1− 4√λuv)).

Thus, we have

S1(λ, χ) = 1G(χχp)G(χp)

p−1∑u=0

(χχp)(u)χp

(−u√λ)ep(u(1−

√λ)2)

+ 1G(χχp)G(χp)

p−1∑u,v=0

(χχp)(u)χp(v)ep(v + u(1−√λ)2)δ(u, v)

= 1G(χχp)G(χp)

p−1∑u=0

(χχp)(u)χp

(−u√λ)ep(u(1−

√λ)2)

+ 1G(χχp)G(χp)

p−1∑u,v=0

(χχp)(u)χp(v)ep(v + u(1−√λ)2)

+ 1G(χχp)G(χp)

p−1∑u=0

p−1∑v=1

(χχp)(u)ep(v + u(1−√λ)2)χp(v − 4

√λu).

74

Notice that χp(−4√λu) = χp(−u

√λ) for any u ∈ Z/pZ since χp is a quadratic character. So

the first term and third term combines to change the index of v to v ∈ Z/pZ. The summation

in the second term can also be evaluated easily as the product of two Gauss sums. This gives

us

S1(λ, χ) = 1G(χχp)G(χp)

p−1∑u,v=0

(χχp)(u)χp(v − 4√λu)ep(v − 4

√λu+ (1 +

√λ)2u)

+ (χχp)(1−√λ)2

=χ(1 +√λ)2 + χ(1−

√λ)2.

Corollary 4.1.14. Suppose χ 6= χp, then

S2(a, b, χ) = 12

(χp(a) + χp(b)) ·(χ(a+ 2

√ab+ b) + χ(a− 2

√ab+ b)

)(4.1.27)

Proof. The Gauss sum substitution for (χχp)(u+ v) gives us

S2(a, b, χ) = 1G(χ)G(χp)G(χχp)

p−1∑u,v,r=0

χ(uvr)χp(r)ep((r + a)u+ (r + b)v)

= G(χ)G(χp)G(χχp)

p−1∑r=0

χ(r + a)χ(r + b)χ(r)χp(r).

This is clearly 0 when a = b = 0 since χ 6= χp. Otherwise, since it is easy to see that

S2(a, b, χ) = S2(b, a, χ) from its definition, we can suppose a 6= 0 without loss of generality.

Then

S2(a, b, χ) = χ(a)χp(a)S1( ba, χ).

So Eq. (4.1.27) follows directly from Eq. (4.1.26).

4.2 Fourier Expansions

The notation in this section will be consistent with those in §3.3. Let D < 0 be an odd

fundamental discriminant and p - D be an odd prime that splits into pp in OK , where

K = Q(√D). Let φ : Ip/Pp,1 −→ C× be a ray class group character with φ1 : (Z/pZ)∗ −→

75

Ip/Pp,1φ−→ C× non-quadratic and φ2 : (Z/pZ)∗ −→ C× a character satisfying φ1φ

22 = 1p.

Denote the absolute value of D by

N := |D|. (4.2.1)

Let fφ =∑

n≥1 a(φ, n)qn ∈ S1(Np, χDφ1) be the weight one newform of imaginary di-

hedral type associated to φ and gψ = fφ ⊗ φ2 ∈ S1(Np2, χDp2) as in Prop. 3.3.4. We could

write

fφ =∑

A∈Pic(OK)

fφ,A, gψ =∑

A∈Pic(OK)

gψ,A

as in Eq. (3.3.8) and (3.3.19). Then we could find mock-modular forms with shadow gψ,A by

twisting mock-modular forms fφ,A, whose shadow is fφ,A.

Proposition 4.2.1. Let A ∈ Pic(OK) be any class and fφ,A ∈ M1,dA(|D|p, χDφ1) be any

mock-modular form with shadow fφ,A ∈ S1,dA(|D|p, χDφ1) as in Prop. 2.4.5. Then the mock-

modular form

gψ,A(z) := φ2(−1)(fφ,A ⊗ φ2)(z) ∈M1(|D|p2, χD) (4.2.2)

has shadow gψ,A(z) and the associated harmonic Maass form gψ,A(z) satisfies

gψ,A |1 W` =ϕ`∗(A)

(D/``

)ε`√`

gψ,A |1 U`,

gψ,A |1 Wp2 = φ2(−1)gψ,A

(4.2.3)

for all primes ` | D, ` > 0, where `∗ = (−1)(`−1)/2`, Wp =(p2α β|D|p p

),Wp2 =

(p2α β|D|p2 p2

), W`

and U` are defined as in §2.4.1.

Proof. Let fφ,A(z) ∈ H1,dAbe harmonic Maass form associated to fφ,A(z). Then the function

gψ,A(z) is the holomorphic part of the harmonic Maass form φ2(−1)(fφ,A ⊗ φ2

), which is

sent to gψ,A under ξ1.

Write the Fourier expansion of fφ,A and gψ,A at infinity as

fφ,A =∑n∈Z

cφ,A(n, y)qn, gψ,A =∑n∈Z

cψ,A(n, y)qn.

76

Since fφ,A ∈ H1,dA(|D|p, χDφ1) and cψ,A(n, y) = φ2(n)cφ,A(n, y), the Fourier coefficients

cψ,A(n, y) satisfy

χ`(n)cψ,A(n, y) = ϕ`∗(A)cψ,A(n, y) for all n ∈ Z relatively prime to `

for all ` | D. The first equation in (4.2.3) is then implied by Prop. 2.4.1.

The second equation in (4.2.3) follows from Eq. (3.3.15) and the calculations below.

φ2(−1)gψ,d |1 Wp2 =1

G(φ2)

p−1∑µ=1

φ2(µ)fφ,d |1 ( p µp )Wp2

=1

G(φ2)

p−1∑µ=1

φ2(µ)fφ,d |1(pα+|D|µ ∗|D|p dµ

)(p β(|D|µ)

p

)=

1

G(φ2)

p−1∑µ=1

φ2(µ)χD(p)φ1(|D|µ)fφ,d |1(p β(|D|µ)

p

)=φ2(|D|β)

G(φ2)

p−1∑µ′=1

φ2(µ′)fφ,d |1(p µ′p

)= gψ,d.

Here, (pα + |D|µ)dµ ≡ 1 (mod |D|p), |D|β ≡ −1 (mod p) and χD(p) = 1.

Since gψ,A is obtained from fφ,A by twisting, its transformation under Γ0(Np) can be

described by the following lemma.

Lemma 4.2.2. For any γ =(

r bNpc u

)∈ Γ0(Np), we have

φ2(−1)gψ,A |1 γ =φ2(−Nrc)χN(u)

G(φ2)G(φ1)f cφ,A |1 Up ( p 1 )

+χN(u)

G(φ2)

p−1∑µ=1

p-r+Ncµ

φ2

(µ(r +Ncµ)

2)fφ,A |1

(p (r+Ncµ)uµ

p

).

(4.2.4)

Proof. Notice that if p | c, Eq. (4.2.4) is just gψ,A |1 γ = χN(u)gψ,A. So we suppose p - c.

77

The rest of the proof follows from Eq. (3.3.15) and (4.2.2) and the calculations below.

φ2(−1)gψ,A |1 γ =1

G(φ2)

p∑µ=1

φ2(µ)fφ,A |1 ( p µp )(

r bNpc u

)=

1

G(φ2)

p∑µ=1

p-r+Ncµ

φ2(µ)fφ,A |1(r+Ncµ bµNpc uµ

)(p (r+Ncµ)uµ

p

)

+φ2(µ0)

G(φ2)fφ,A |1

(pr+Ncµ0

pbp+uµ0

Npc pu

)( p 1 )

=1

G(φ2)

p∑µ=1

p-r+Ncµ

φ2(µ)χN(uµ)φ1(uµ)fφ,A |1(p (r+Ncµ)uµ

p

)

+φ2(−rNc)χN(u)φ1(c)φ1(N)

G(φ2)G(φ1)f cφ,A |1 Up ( p 1 )

=χN(u)

G(φ2)

p∑µ=1

p-r+Ncµ

φ2(µ)φ1(r +Ncµ)fφ,A |1(r+Ncµ bµNpc uµ

)(p (r+Ncµ)uµ

p

)

+φ2(−Nrc)χN(u)

G(φ2)G(φ1)f cφ,A |1 Up ( p 1 )

Let fφ,A(z) and gψ,A(z) be the harmonic Maass forms as in Prop. 4.2.1 with the following

Fourier expansions at infinity

fφ,A(z) =∑n∈Z

cφ,A(n)qn, gψ,A(z) =∑n∈Z

cψ,A(n)qn.

For each δ | N , δ > 0 and p, let Wδ and Wp be the Atkin-Lehner involutions as in Eq. (2.3.4)

Wδ =[(

δαδ βδ4Np2 δ

), δ−1/4

√4Np2z + δ

], Wp =

[(p2α β4Np p

), p−1/4

√4Npz + p

],

and Uδ, Up be the U -operator as in Eq. (2.3.1). For each A ∈ Pic(OK), define the following

functions

ΦA(z) := gψ,A |1 ( 41 ) θ(z),

ΨA(z) := (ΦA(z)) |3/2

∏`|N,` prime

(U` + W`)

Up(Up + Wp),(4.2.5)

78

where θ =∑

n∈Z qn2

is the weight 12

theta function. It is easy to check that for any γ =(a b

4Np2 d

)∈ Γ0(Np2), we have

(ΦA |3/2 γ)(z) =(Nd

)ΦA(z).

Then by Lemma 2.3.5 and 2.3.6, the function ΨA(z) satisfies

(ΨA |3/2 γ)(z) = ΨA(z)

for all γ ∈ Γ0(4). After subtracting off appropriate poles from ΦA(z), we could apply

holomorphic projection to it and obtain an identity between finite linear combinations of

c+ψ (n) and an infinite sum from the Fourier expansion of ΨA(z). This infinite sum will become

the special values of modular function after applying the appropriate counting argument.

Before calculating the Fourier coefficients of ΨA, we need the following lemma.

Lemma 4.2.3. For any δ | N, δ > 0, we have

ΦA |3/2 Wδ =ϕδ∗(A)

δgψ,A |1 Uδ ( 4

1 ) · θ |1/2 Uδ. (4.2.6)

Proof. This lemma follows from an induction on the number of prime divisors of δ. When

δ = ` is prime, we have

ΦA |3/2 W` = gψ,A |1 ( 41 )(

`α` β`4Np2 `

)· θ |1/2

˜(α` β`

4Np2/` `

) [( ` 1 ) , `−1/4

(N/``

)ε`

]= gψ,A |1

(`α` 4β`Np2 `

)( 4

1 ) · θ |1/2[( ` 1 ) , `−1/4

(N/``

)ε`

]=ϕ`∗(A)

(−N/``

)ε`√`

gψ,A |1 U` ( 41 ) ·

(N/``

)ε−1` θ |1/2

[( ` 1 ) , `−1/4

]=ϕ`∗(A)√

`gψ,A |1 U` ( 4

1 ) · θ |1/2[( ` 1 ) , `−1/4

]=ϕ`∗(A)

`gψ,A |1 U` ( 4

1 ) · θ |1/2 U`.

When δ = δ′`, we could use Lemma 2.3.3 and 2.3.4 to find that

ΦA |3/2 Wδ = ΦA |3/2 Wδ′W`

gψ,A |1 U`Wδ′ =(`δ′

)gψ,A |1 Wδ′U`,

θ |1/2 U`Wδ′ =(`δ′

)θ |1/2 Wδ′U`.

79

Using this, we could obtain

ΦA |3/2 Wδ =ΦA |3/2 Wδ′W`

=ϕ(δ′)∗(A)

δgψ,A |1 Uδ′ ( 4

1 )(

`α` β`4Np2 `

)· θ |1/2 Uδ′W`

=ϕ(δ′)∗(A)

δ

(`δ′

)gψ,A |1

(`α` 4β`Np2 `

)Uδ′ ( 4

1 ) ·(`δ′

)θ |1/2 WÙδ′

=ϕ(δ′)∗(A)ϕ`(A)

δ′`gψ,A |1 UÙδ′ ( 4

1 ) · θ |1/2 UÙδ′

=ϕδ∗(A)

δgψ,A |1 Uδ ( 4

1 ) · θ |1/2 Uδ.

By induction, the proof is complete.

Now, we are ready to calculate the Fourier expansion of ΨA at infinity.

Proposition 4.2.4. The mth Fourier coefficient of ΨA(z) at infinity, c(ΨA,m, y), has the

form

c(ΨA,m, y) = 2√p∑

δ|N,δ>0

ϕδ∗(A)(SA,δ(m, y) + S ′A,δ(m, y)), (4.2.7)

where

SA,δ(m, y) =∑k∈Z

cψ,A

(Np2m− δ2k2

4,

4y

p2N

),

S ′A,δ(m, y) =φ2(−1)∑k∈Z

cψ,A

(Nm− p2δ2k2

4,4y

N

)

+φ2(−4Nm)εp

√pG(φ′2)

G(φ1)G(φ2)

∑k∈Z

δ2k2≡Nm (mod p)

cφ,A

(Nm− δ2k2

4,

4y

pN

)

+ φ2(4)∑k∈Z

S2(Nm, δ2k2, φ2)cφ,A

(Nm− δ2k2

4,4y

N

),

φ′2(·) = φ2(·)(·p

).

When m = pm′ is divisible by p, the expression S ′A,δ(pm) above simplifies to∑k∈Z

cψ,A

(Npm′ − δ2k2

4,4y

N

).

When(−mp

)= −1, the expression S ′A,δ(m) vanishes identically.

80

Proof. By Lemma 2.3.3 and 2.3.4, we know that for any positive `, δ | N satisfying gcd(`, δ) =

1

ΦA |3/2 U`Wδ = ΦA |3/2 WδU`,

ΦA |3/2 WδW` = ΦA |3/2 Wδ`.

So we could write

ΦA |3/2∏

`|N,` prime

(U` + W`) =∑

δ|N,δ>0

ΦA |3/2 WδUN/δ.

Applying Lemma 2.3.4 yields

ΨA =∑

δ|N,δ>0

ΦA |3/2 WδUN/δUp(Up + Wp)

=∑

δ|N,δ>0

ΦA |3/2 WδUNp2/δ + ΦA |3/2 WδUN/δUpWp

=∑

δ|N,δ>0

ΦA |3/2 WδUNp2/δ + ΦA |3/2 WδUpWpUN/δ.

The first term in the summand can be handled easily using Lemma 4.2.3. The main

technical complications arise in the calculations of the second term, which we will carry out.

Here, we will choose

Wp =[(

p2α β4Np p

), p−1/4

√4Npz + p

]This is possible since gcd(N, p) = 1. It is not essential to the result, but simplifies the

calculations. For each δ | N, δ > 0, we denote

ΦA,δ := ΦA |3/2 Wδ.

Then we have

ΦA,δ |3/2 UpWp =ΦA,δ |3/2 Wp2 +

p−1∑λ=1

(Npdλ

)ε3dλΦA,δ |3/2 γλ ·

[(p β(4Nλ)

p

), 1],

Wp2 =[(

p2α β4Np2 p2

), p−1/2

√4Np2z + p2

],

γλ =(pα+4Nλ bλ

4Np dλ

)∈ Γ0(4Np),

dλ ≡pα + 4Nλ (mod 4p), dλ ≡ p (mod 4N).

81

Substituting these terms into Eq. (4.2.8), we have

ΦA |3/2 WδUpWpUN/δ = CA,δ(PA,δ,1 + PA,δ,2 + PA,δ,3) |3/2 UN/δ, (4.2.10)

where

PA,δ,1 =φ2(−1) (gψ,A |1 Uδ ( 41 ))(θ |1/2

[(p2

1

), p−1

]Uδ

),

PA,δ,2 =εpχN(p)φ2(−4)G(φ′2)

G(φ1)G(φ2)f cφ,A |1 Upδ

(4p

1

)· (θ ⊗ φ′2) |1/2 Uδ,

PA,δ,3 =εpχN(p)φ2(−δ)

G(φ2)

p−1∑λ,µ′=1p-4λ+µ′

(λδp

)φ2(µ′)φ2

2(δN(4λ+ µ′))fφ,A |1(p −δN2(4λ+µ′)

p

)Uδ ( 4

1 )

· θ |1/2[(

p β(4δNλ)p

), 1]Uδ,

φ′2(·) = φ2(·)(·p

),

µ′ =µδ.

The mth Fourier coefficient of 12δ−5/4PA,δ,1 and 1

2δ−5/4PA,δ,2 are

√pφ2(−1)

∑k∈Z

cψ,A

(δm− p4δk2

4,4y

δ

)(4.2.11)

pεpχN(p)φ2(−4)G(φ′2)

G(φ1)G(φ2)

∑k∈Z

δk2≡ m (mod p)

cφ,A

(δm− δk2

4,4y

pδ

)φ1(δk) (4.2.12)

The mth Fourier coefficient of 12δ−5/4PA,δ,3 is

εpχN(p)φ2(−δ)G(φ2)

p−1∑λ,µ′=1p-4λ+µ′

k∈Z

(λδp

)φ2(µ′)φ2

2(δN(4λ+ µ′))cφ,A

(δm− δk2

4,y

δ

)

· ep(βN(4λ+ µ′)(m− δk2) + β(4Nλ)δk2

),

where ep(z) = e2πiz/p. Set

u = N(4λ+ µ′), v = 4Nλ−N(4λ+ µ′),

then µ′ = v(N(u+ v)u) and the change of variable (u, v) is injective. So we could rewrite

the mth Fourier coefficient of PA,δ,3 as

χN(p)φ2(−Nδ)(−Nδp

)√p∑k∈Z

S2(βm, βδk2, φ2)cφ,A

(δm− δk2

4,y

δ

), (4.2.13)

83

where S2(a, b, χ) is defined in Eq. (4.1.25) and given by

S2(βm, βδk2, φ2) =1

εp√pG(φ2)

p−1∑u,v=1

(u+vp

)φ2(u+ v)φ2(uv) · ep

(β(mu+ δk2v)

).

After Substituting Using the fact that χN(p) =(−Np

)=(pN

)= 1 and adding together Eq.

(4.2.11), (4.2.12) and (4.2.13), we could write themth Fourier coefficient of (2CA,δδ√p)−1N1/4ΦA |3/2

WδUpWpUN/δ as follows.

φ2(−1)∑k∈Z

cψ,A

(Nm− p2δ2k2

4,y

N

)

+φ2(−4Nm)εp

√pG(φ′2)

G(φ1)G(φ2)

∑k∈Z

δ2k2≡Nm (mod p)

cφ,A

(Nm− δ2k2

4,y

pN

)

+ φ2(−Nδ)(δp

)∑k∈Z

S2(βNm/δ, βδk2, φ2)cφ,A

(Nm− δ2k2

4,y

N

),

(4.2.14)

The sum S2(a, b, φ2) can be evaluated using Eq. (4.1.27). After simplifying the expression,

we get S ′A,δ(m, y).

Whenm = pm′, the first two terms in expression (4.2.14) both vanishes since cψ,A(pn, y) =

0, and the third term becomes∑k∈Z

cψ,A

(Npm′ − δ2k2

4,4y

N

).

Thus, the (pm′)th Fourier coefficient of ΦA |3/2 WδUpWpUN/δ is

2√pϕδ∗(A)

∑k∈Z

cψ,A

(Npm′ − δ2k2

4,4y

N

). (4.2.15)

By Lemma 4.2.3, we could find the (pm′)th Fourier coefficient of ΦA |3/2 WδUNp2/δ to be

2√pϕδ∗(A)

∑k∈Z

cψ,A

(Np3m′ − δ2k2

4,

4y

p2N

). (4.2.16)

Summing these together over δ | N , we find that the (pm′)th Fourier coefficient of ΨA to be

the expression

2√p∑

δ|N,δ>0k∈Z

ϕδ∗(A)

(cψ,A

(Npm′ − δ2k2

4,4y

N

)+ cψ,A

(Np3m′ − δ2k2

4,

4y

p2N

)). (4.2.17)

84

When(−mp

)= −1, the second term in (4.2.14) vanishes since the summation is empty.

The third term has contribution only from k = pk′ since S2(βNm/δ, βδk2, φ2) = 0 by Eq.

(4.1.27) otherwise. So this term becomes

φ2(−Nδ)(δp

)∑k∈Z

S2(βNm/δ, βδk2, φ2)cφ,A

(Nm− δ2k2

4,4y

N

)= φ2(4Nδβ/δ)

(δβNm/δ

p

)∑k′∈Z

cφ,A

(Nm− p2δ2(k′)2

4,4y

N

)φ2(Nm/4)φ2(−1)

= − φ2(−1)∑k′∈Z

cψ,A

(Nm− p2δ2(k′)2

4,4y

N

),

which cancels the first term exactly. So S ′A,δ(m, y) = 0 in this case.

A special case of this proposition is when A = B2 ∈ Pic2(OK) and(−mp

)= −1. Here,

ϕδ∗(B2) = 1 for all δ | N and the mth Fourier coefficient of ΨB2(z) is simply

2√p∑k∈Z

2ω(gcd(D,k))

cφ,B2

(Np2m− k2

4

)φ1

(k

2

)−

cψ,B2

(k2 −Np2m

4

)β1

((k2 −Np2m)

4,

4y

p2N

) (4.2.18)

where ω(·) is the function defined in Eq. (4.1.6). Summing over such classes A ∈ Pic2(OK)

gives us the mth Fourier coefficient of Ψ1 :=∑A∈Pic2(OK) ΨA, which is

2√p∑k∈Z

2ω(gcd(D,k))

(cψ,1

(Np2m− k2

4

)− cψ,1

(k2 −Np2m

4

)β1

((k2 −Np2m)

4,

4y

p2N

) )(4.2.19)

when(mp

)= −1.

4.3 Fourier Coefficients and Values of Modular Functions

4.3.1 Borcherds Lift

Let M !1/2 be the space of weakly holomorphic modular forms of weight 1/2 and level 4

satisfying Kohnen’s plus space condition. It has a canonical basis {f−d}d≤0 with d ≡ 0, 1

(mod 4) and Fourier expansions

f−d(z) = qd +∑n≥1

c(f−d, n)qn.

85

Let f(z) ∈ M !1/2 be a weakly holomorphic form with integral Fourier coefficients c(f, n). In

[7], Borcherds constructed an infinite product Ψf (z) using c(f, n) as exponents, and showed

that it is a modular form of weight c(f, 0) and some character. The divisors of Ψf (z) are

supported on cusps and imaginary quadratic irrationals. In particular, if τ is a quadratic

irrational of discriminant D < 0, then its multiplicity in Ψf (z) is

ordτ (Ψf ) =∑k>0

c(f,Dk2).

For example, when f(z) = f−d(z) with d < 0, the Borcherds product Ψ−d(z) := Ψfd(z)

equals to ∏q∈C(d)/Γ

(j(z)− j(τq))1/wq , (4.3.1)

where C(d) is the set of all positive definite binary quadratic forms [a, b, c] of discriminant

d satisfying a > 0. Note that when d is fundamental, wd, the number of roots of unity in

Q(√d), is equal to 2wq for all q ∈ C(d).

4.3.2 Automorphic Green’s Function

In this section, we will follow the construction in [28, §5] to express the automorphic Green’s

function as an infinite sum. For two distinct points zj = xj+iyj ∈ H, the invariant hyperbolic

distance d(z1, z2) between them is defined by

cosh d(z1, z2) :=|z1 − z2|2

2y1y2

+ 1

=(x1 − x2)2 + y2

1 + y22

2y1y2

.

(4.3.2)

Note d(z1, z2) = d(γz1, γz2) for all γ ∈ PSL2(R). The Legendre function of the second kind

Qs−1(t) is defined by

Qs−1(t) =

∫ ∞0

(t+√t2 − 1 coshu)−sdu, Re(s) > 1, t > 1,

Q0(t) = 12

log(1 + 2

t−1

).

(4.3.3)

Let Γ = PSL2(Z). For two distinct points z1, z2 ∈ Γ\H, the following convergent series

defines the automorphic Green’s function

Gs(z1, z2) :=∑γ∈Γ

gs(z1, γz2), Re(s) > 1, (4.3.4)

86

where

gs(z1, z2) := −2Qs−1(cosh d(z1, z2)). (4.3.5)

Recall that E(τ, s) is defined in (3.4.1) and ϕ1(s) is the coefficient of y1−s in the Fourier

expansion of E(τ, s). Proposition 5.1 in [28] tells us that for distinct z1, z2 ∈ Γ\H, the values

of the j-function are related to the values of the automorphic Green’s function by

log |j(z1)− j(z2)|2 = lims→1

(Gs(z1, z2) + 4πE(z1, s) + 4πE(z2, s)− 4πϕ1(s))− 24. (4.3.6)

For a fixed z1 ∈ H, one could evaluate z2 at CM points arising from binary quadratic forms

in C(d). The number of such CM points is give by the Hurwitz class number H(−d). Adding

up these values gives us the following proposition.

Proposition 4.3.1. Let d,D0 < 0 be congruent to 0 or 1 modulo 4 and Q ∈ C(D0). If

τQ 6= τq for any q ∈ C(d), then

log |Ψ−d(τQ)|2 = lims→1

∑k>√dD0

ρQ(k, d)(−2)Qs−1

(k√dD0

)+H(−d)4πE(τQ, s) +R(d, s)

,

(4.3.7)

where R(d, s) =∑

q∈C(d)/Γ(4πE(τq, s)− 4πϕ1(s)− 24) and ρQ(k, d) is the counting function

defined by Eq. (4.1.20).

Proof. Let Q = [A,B,C] ∈ C(D0), q = [a, b, c] ∈ C(d), then τQ = −B+√D0

2A, τq = −b+

√d

2a. Some

computations verify that

k :=√dD0 cosh d(τQ, τq) = 2Ac+ 2Ca−Bb ∈ Z.

Thus, the set SQ(k, d) defined by Eq. (4.1.18) can be rewritten as

SQ(k, d) ={q ∈ C(d) : cosh d(τq, τQ) = k√

dD0

}.

Now let z1 = τQ, z2 = τq in Eq. (4.3.6) and sum over q ∈ C(d)/Γ. With the following

87

observation

∑q∈C(d)/Γ

1

wqGs(τQ, τq) =

∑q∈C(d)/Γ

∑γ∈Γ

1

wq(−2)Qs−1(cosh d(τQ, γτq))

=∑q∈C(d)

(−2)Qs−1(cosh d(τQ, τq))

=∑

k>√dD0

ρQ(k, d)(−2)Qs−1

(k√dD0

),

we have Eq. (4.3.7). The sum is over k >√dD0 since cosh d(τQ, τq) = 1 precisely when

τQ = τq and τQ 6= τq for any q ∈ C(d).

4.3.3 Holomorphic Projection

In this section, we will use holomorphic projection to express a finite linear combination of

the Fourier coefficients of a mock-modular form as an infinite sum similar to the right hand

side of Eq. (4.3.7).

Recall that ΨA(z) is defined by Eq. (4.2.5) for each A ∈ Pic(OK) and has Fourier

expansion

ΨA(z) =∑m∈Z

c(ΨA,m, y)qm.

We have calculated the Fourier coefficients c(ΨA,m, y) explicitly in Prop. 4.2.4. Using the

following facts

cφ,A(n) = cφ,A(n)− cφ,A(−n)β1(−4πn, y),

cψ,A(n) = φ2(−n)cφ,A(n),

β1(α1, α2y) = β1(α1α2, y), α1, α2 > 0

we could write c(ΨA,m, y)qm into the sum of a holomorphic part,(2√p)aφ,A(m), and non-

88

holomorphic part(2√p)bφ,A(m, y), where for all m ∈ Z

aφ,A(m) =∑

δ|N, k∈Z

ϕδ∗(A)

(cψ,A

(Np2m− δ2k2

4

)+ φ2(−1)cψ,A

(Nm− p2δ2k2

4

))

+φ2(−4Nm)εp

√pG(φ′2)

G(φ1)G(φ2)

∑δ|N, k∈Z

δ2k2≡Nm (mod p)

cφ,A

(Nm− δ2k2

4

)

+ φ2(4)∑

δ|N, k∈Z

S2(Nm, δ2k2, φ2)cφ,A

(Nm− δ2k2

4

),

(4.3.8)

and when(−mp

)= −1

bφ,A(m, y) = −∑

δ|N, k∈Z

ϕδ∗(A)cψ,A

(δ2k2 −Np2m

4

)β1

(δ2k2 −Np2m

Np2, y

). (4.3.9)

Because β1(4πn, y)q−n decays exponentially when n ≥ 1, the pole and constant term of ΨA

at infinity has the form

2√p∑m≥0

aφ,A(−m)q−m.

To apply holomorphic projection to ΨA, one needs to first subtract the pole and constant

term.

For an integer n ≥ 1 congruent to 0, 3 modulo 4, let

gn(z) = q−n +∑m≥1

c(gn,m)qm

be the unique weakly holomorphic modular form of level 4, weight 3/2 in the Kohnen plus

space. They have integral Fourier coefficients and could be constructed explicitly (see [60]).

Let F(z) be the weight 3/2 Eisenstein series studied in [33], which has the following Fourier

expansion

F(z) =∞∑m=0

H(m)qm + y−1/2

∞∑m=−∞

1

16πβ3/2(m2, y)q−m

2

,

and satisfies Kohnen’s plus space condition. Here H(m) is the Hurwitz class number when

m ≥ 1 and H(0) = − 112

. Define the function Ψ∗A(z) to be

Ψ∗A(z) :=1

2√p

ΨA(z)− aφ,A(0)

H(0)F(z)−

∑n≥1

aφ,A(−n)gn(z). (4.3.10)

89

Denote its mth Fourier coefficient by c(Ψ∗A,m, y). Then its holomorphic part, denoted by

a∗φ,A(m), is

a∗φ,A(m) = aφ,A(m)− H(m)aφ,A(0)

H(0)−∑n≥1

aφ,A(−n)c(gn,m). (4.3.11)

The function Ψ∗A(z) has order O(y−1/2) at the cusp infinity. The same decaying prop-

erty holds at the other two cusps of Γ0(4) as well, since Ψ∗A(z) satisfies the Kohnen plus

space condition. So we can consider its holomorphic projection to the Kohnen plus space

S+3/2(Γ0(4)). This will produce the following identities between a∗φ,A(m) and an infinite sum

similar to the one on the right hand side of Eq. (4.3.7).

Proposition 4.3.2. Let m ≥ 1 be a positive integer such that(−mp

)= −1. Then

a∗φ,A(m) = 2 lims→1

∑δ|N

ϕδ∗(A)∑

k>p√Nm/δ

cψ,A

(δ2k2 −Np2m

4

)2Qs−1

(δk

p√Nm

) . (4.3.12)

Proof. To execute the holomorphic projection, we first need to define the weight 3/2 Poincare

series for m ≥ 1 by

Pm(z, s) :=∑

γ∈Γ∞\Γ0(4)

j(γ, z)−3e2mπiγzIm(γz)s/2,

where for γ ∈ Γ0(4)

j(γ, z) :=θ(γz)

θ(z).

This series converges absolutely for Re(s) > 1 and can be analytically continued to Re(s) ≥ 0.

As s → 0, the inner product 〈Pm(z, s),Ψ∗A(z)〉 is the mth Fourier coefficient of a cusp form

in S+3/2(Γ0(4)), since Ψ∗A(z) is already in the plus space. Given S+

3/2(Γ0(4)) = {0}, we know

the limit is zero and obtain the following equation after applying Rankin-Selberg unfolding,

lims→0

(Γ(1+s

2)

(4πm)1/2+s/2a∗φ,A(m) +

∫ ∞0

bφ,A(m, y)e−4πmyy1/2+s/2dy

y

)= 0. (4.3.13)

After some manipulations, we have∫ ∞0

β1(m,µy)e−4πmyy1/2+s/2dy

y=

Γ(1+s2

)

(4πm)1/2+s/2%s (µ) ,

90

where the function %s(µ) is defined by

%s(µ) :=

∫ ∞1

du

(µu+ 1)1+s2 u

, µ > 0. (4.3.14)

After substituting Eq. (4.3.9) and µ = δ2k2

Np2m− 1 into Eq. (4.3.13), we arrive at the following

equation

−∫ ∞

0

bφ,A(m, y)e−4πmyy1/2+s/2dy

y=

Γ(1+s2

)

(4πm)1+s2

2∑δ|N

ϕδ∗(A)∑

k>p√Nm/δ

cψ,A

(δ2k2 −Np2m

4

)%s

(δ2k2

Np2m− 1

).

(4.3.15)

Since cψ,A(n) = 0 whenever n ≤ 0, the sum changed from k ∈ Z to k > p√N and

produced a factor of 2. Now substituting (4.3.15) into (4.3.13) gives us


∑δ|N

ϕδ∗(A)∑

k>p√Nm/δ

cψ,A

(δ2k2 −Np2m

4

)%s

(δ2k2

Np2m− 1

) .

With the following comparisons (see [28, §7] for similar arguments).

%0(µ) = 2Q0(√µ+ 1),

Qs−1(√µ+ 1)− sΓ(s)2

22−sΓ(2s)%s−1(µ) = O(µ−1/2−s/2),

we could substitute %s

(δ2k2

Np2m− 1)

with 2Qs−1

(δk

p√Nm

)in the limit and obtain Eq. (4.3.12).

4.3.4 Proof of Main Theorem

In this section, we will prove Theorem 1.1.1 stated in the introduction by proving a more

general equality. As before, D < 0 is an odd, fundamental discriminant, (p) = pp a prime

that splits in K = Q(√D) and φ is a non-trivial ray class group character modulo p such

that φ1 : (Z/pZ)× −→ C× is non-quadratic.

91

Theorem 4.3.3. Let m ≥ 1 be a positive integer such that(−mp

)= −1 and A = B2 ∈

Pic(OK). Then we have

a∗φ,A(m) = −2∑

[Q]∈C(Dp2)π([Q])2=[QA−1 ]

ψ2(Q)(

log |Ψm(τQ)|2 − 2H(m)H(0)

log(yQ|η(τQ)|2

)). (4.3.16)

Proof. The right hand of Eq. (4.3.16) can be rewritten as∑[Q]∈C(Dp2)π([Q])2=[QA]

ψ2(Q) log |Ψm(τQ)|2 =∑

[P ]∈C2(Dp2)π([P ])=[QA]

ψ(P )∑

[Q]∈C(Dp2)[Q]2=[P ]

log |Ψm(τQ)|2. (4.3.17)

Applying Theorem 4.1.10 and Props. 4.1.11 and 4.3.1 with d = −m,D0 = −Np2 then gives

us

∑Q∈C(Dp2)[Q2]=[P ]

log |Ψm(τQ)|2 = lims→1

∑

k>p√Nm

2ω(gcd(Np,k))rP

(k2−Np2m

4

)(−2)Qs−1

(k

p√Nm

)+

∑Q∈C(Dp2)[Q2]=[P ]

(H(m)4πE(τQ, s) +R(−m, s))

.

Notice that this substitution is valid even when p | k by Prop. 4.1.11. Substituting this into

the right hand side of Eq. (4.3.17) and applying Eqs. (3.3.17) and (3.3.18) then gives us∑[Q]∈C(Dp2)π([Q])2=[QA]

ψ2(Q) log |Ψm(τQ)|2 = lims→1

∑k>p√Nm

2ω(gcd(N,k))cψ,A

(k2−Np2m

4

)(−2)Qs−1

(k

p√Nm

)

+ 4πH(m) lims→1

∑Q∈C(Dp2)π([Q])2=[QA]

ψ2(Q)E(τQ, s).

The p disappears from gcd(Np, k) in the exponent since cψ,A(n) = 0 whenever p | n. Eq.

(3.3.18) implies that the term R(−m, s) also vanishes since it is independent of [Q] ∈ C(Dp2).

By Kronecker’s first limit formula (Eq. (3.4.2)) and Eq. (3.3.18), we have

4π lims→1

∑Q∈C(Dp2)π([Q])2=[QA]

ψ2(Q)E(τQ, s) =2

H(0)

∑Q∈C(Dp2)π([Q])2=[QA]

ψ2(Q) log(yQ|η(τQ)|2

).

Since A ∈ Pic2(OK), Eq. (4.3.12) becomes


∑k>p√Nm

2ω(gcd(N,k))cψ,A

(k2 −Np2m

4

)2Qs−1

(δk

p√Nm

) .

92

Putting together the last three equations, we obtain

a∗φ,A(m) = −2∑

[Q]∈C(Dp2)π([Q])2=[QA]

ψ2(Q)(

log |Ψm(τQ)|2 − 2H(m)H(0)

log(yQ|η(τQ)|2

)).

Conjugating both sides and using the fact that ψ2(Q) = ψ2(Q−1), τQ−1 = −τQ give us Eq.

(4.3.16).

From Eq. (3.3.13), we know that the shadow of

fφ,1(z) =∑n≥−n1

cφ,1(n)qn :=∑

A∈Pic2(OK)

fφ,A(z)

is fφ,1. Here n1 ∈ Z is defined in Eq. (2.4.19). In the same notations as before, we could now

state the theorem relating finite linear combinations of cφ,1(n) with the values of Borcherds

lift.

Theorem 4.3.4. Let φ : Ip/Pp,1 −→ C× be a non-trivial character such that φ1 : (Z/pZ)× −→

C× is non-quadratic and m ≥ 1 be a positive integer such that(−mp

)= −1. Then for any

fφ,1 =∑

n≥−n1cφ,1(n)qn ∈M1,1(Np, χDφ1), we have

∑k∈Z cφ,1

(Np2m−k2

4

)φ1

(k2

)δN(k) + 4

∑Q∈C(Dp2) ψ

2(Q) log (|Ψm(τQ)|)

Iψ2

∈ Q(φ). (4.3.18)

Furthermore, if n1 < min{N4, p}

, then we have the equality

∑k∈Z

cφ,1

(Np2m− k2

4

)φ1

(k

2

)δN(k) = −4

∑Q∈C(Dp2)

ψ2(Q) log (yQ|Ψm(τQ)|) . (4.3.19)

Proof. So we could define

aφ,1(m) :=∑

A∈Pic2(OK)

aφ,A(m), a∗φ,1(m) :=∑

A∈Pic2(OK)

a∗φ,A(m).

Summing Eq. (4.3.16) over A ∈ Pic2(OK) gives us

a∗φ,1(m) + 4∑

Q∈C(Dp2)

ψ2(Q) log |Ψm(τQ)| = 4H(m)

H(0)Iψ2 .

93

Combining with Eq. (4.3.11), we could write

aφ,1(m) + 4∑

Q∈C(Dp2)

ψ2(Q) log |Ψm(τQ)| = H(m)

H(0)(4Iψ2 + aφ,1(0)) +

∑n′≥1

aφ,1(−n′)c(gn,m).

(4.3.20)

Since(−mp

)= −1, we could simplify Eq. (4.3.8) and use cψ,A(n) = φ2(−n)cφ,A(n), φ1φ

22 = 1p

to obtain

aφ,1(m) =∑k∈Z

cφ,1

(Np2m− k2

4

)φ1

(k

2

)δN(k).

Now on the right hand side of Eq. (4.3.20), c(gn,m) ∈ Z and aφ,1(−n′) is some linear

combination of cφ,1(n) with n ≤ 0 with coefficients in Q(φ) by Eq. (4.3.8). Prop. 3.4.5 then

gives us (4.3.18).

If n1 <N4

, then the term aφ,1(−n′) vanishes for all n′ ≥ 1 since the sums in Eq. (4.3.8)

are all empty. Eq. (4.3.20) then becomes

aφ,1(m) + 4∑

Q∈C(Dp2)

ψ2(Q) log |Ψm(τQ)| = H(m)

H(0)(4Iψ2 + aφ,1(0)) . (4.3.21)

By Prop. 4.2.4, the term aφ,1(0) can be written as

aφ,1(0) = 2∑k∈Z

cφ,1(−k2)φ1(k)δN(k) = 4

b√n1c∑k=1

cφ,1(−k2)φ1(k)δN(k). (4.3.22)

Combinig Prop. 2.4.6 and Cor. 3.4.4 gives us

n1∑n=1

(cφ,1(−n)cφ,A0(n) + cφ,1(−pn)cφ,A0(pn)

)δN(n) = 〈fφ,1, fφ,A0〉 = − 4

#O×K#O×pIψ2 .

If n1 < min{N4, p}

, #O×K = #O×p = 2 and the sum above simplifies

n1∑n=1

cφ,1(−n)cφ,A0(n)δN(n) = −Iψ2 .

Cor. 3.3.6 reduces the equation above further to

b√n1c∑k=1

cφ,1(−k2)φ1(k)δN(k) = −Iψ2 .

Substituting this into Eq. (4.3.22) yields aφ,1(0) = −4Iψ2 . Then Eq. (4.3.21) becomes Eq.

(4.3.19).

94

Theorem 1.1.1 is now a consequence of Theorem 4.3.4. Since S1(|D|p, χDφ1) is one

dimensional and |D| > 5, n1 = 1 and the condition n1 < min{N4, p}

is satisfied. Since |D|

is prime, we know that fφ,1 = fφ and there are two Eisenstein series in M1(|D|p, χDφ1).

Because of the vanishing conditions we imposed on the Fourier coefficients c(n), the mock-

modular form fφ(z) in Eq. (1.1.2) is unique. When D′ = −m is a fundamental discriminant,

the function Ψm(z) becomes

Ψm(z) =∏

Q′∈C(D′)

(j(z)− j(τQ′))2/wQ′ .

So Eq. (4.3.19) becomes Eq. (1.1.3).

95

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Mock-Modular Forms of Weight One

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