Introduction to Applications of Modular Forms : Computational Aspects 9783031326288, 9783031326295

Table of contents :
Preface
Acknowledgements
Contents
1 Dirichlet Characters
1.1 The Definition and Basic Properties
1.2 Primitive Dirichlet Characters
1.3 Orthogonal Relations
1.4 Gauss Sums Associated with Dirichlet Characters
1.5 Dirichlet L-Functions and Bernoulli Numbers
2 Modular Forms: Definition and Some Properties
2.1 The Definition of Modular Forms
2.1.1 Modular Groups Γ0(N) and the Transformation Property
2.1.2 Definition of Modular Forms
2.2 Some Properties of the Modular Forms
2.3 Newforms
2.4 Eisenstein Series
2.4.1 Definition of Eisenstein Series
2.4.2 Constant Terms of Eisenstein Series
2.5 The Sum of Divisors Function
2.6 Projections of Modular Forms on the Eisenstein Series
2.7 The Size of the q-Series Coefficients of the Eisenstein Part of a Modular Form
3 Application: Quadratic Forms
3.1 The Modularity and the Constant Terms
3.2 Gauss Sums Associated with Quadratic Forms and the Eisenstein Part of θ(Q;z)
3.2.1 Separable Quadratic Forms
3.3 A Family of Binary Quadratic Forms
3.4 Diagonal Quadratic Forms
3.4.1 Generalized Kronecker Symbol
3.4.2 Constant Terms of Theta Functions Corresponding to the Diagonal Quadratic Forms
3.4.3 An Explicit Formula for Representation Numbers of Diagonal Forms
3.4.4 Diagonal Quadratic Forms Examples
4 Application: Eta Quotients
4.1 The Definition, the Modularity and the Constant Terms
4.2 Finding the Complete Sets of Eta Quotients in Mk(Γ0(N),χ)
4.3 The Eisenstein Part of Eta Quotients
4.4 Eta Quotients in Ek(Γ0(N),χ)
4.4.1 The Level 12 Examples
4.4.2 The Level 16 Examples
4.4.3 The Level 25 Examples
4.5 Eight Families of Eta Quotients Whose Eisenstein Part is a Single Eisenstein Series
4.6 Eta Quotients and Newforms
4.7 Bases for Cusp Form Spaces
5 Various Applications
5.1 The Convolution Sums
5.2 Klein Forms
5.3 The Character Analog of Dedekind's Eta Function
5.4 Cusp Forms and the Character Analog of Dedekind's Eta Function
5.5 Elliptic Curves
References
Index
Index of Notation
Index of SAGE Functions

Citation preview

Synthesis Lectures on Mathematics & Statistics

Zafer Selcuk Aygin

Introduction to Applications of Modular Forms Computational Aspects

Synthesis Lectures on Mathematics & Statistics Series Editor Steven G. Krantz, Department of Mathematics, Washington University, Saint Louis, MO, USA

This series includes titles in applied mathematics and statistics for cross-disciplinary STEM professionals, educators, researchers, and students. The series focuses on new and traditional techniques to develop mathematical knowledge and skills, an understanding of core mathematical reasoning, and the ability to utilize data in specific applications.

Zafer Selcuk Aygin

Introduction to Applications of Modular Forms Computational Aspects

Zafer Selcuk Aygin Northwestern Polytechnic Grande Prairie, AB, Canada Carleton University Ottawa, ON, Canada

ISSN 1938-1743 ISSN 1938-1751 (electronic) Synthesis Lectures on Mathematics & Statistics ISBN 978-3-031-32628-8 ISBN 978-3-031-32629-5 (eBook) https://doi.org/10.1007/978-3-031-32629-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book aims to communicate modular forms to a wider audience. The emphasis is on the computational aspects of modular forms. One should read this book as a supplement to the existing literature on modular forms. In some proofs the author only finesses the parts necessary to develop the computational foundation aimed to be built and refers some crucial details to other books such as [26] by H. Cohen and F. Strömberg, [27] by F. Diamond and J. Shurman, [36] by G. Köhler, [43] by T. Miyake and [53] by W. A. Stein. All of these books are excellent books explaining the theory of modular forms in depth. There are two main strengths of this book. First, it communicates the applications with a minimal theoretic background. Therefore this can be an encouraging first read for those interested in modular forms. Second, the applicability of results is far and reaching to many areas of mathematics that are connected by the theory of modular forms. This book should have something to offer to everyone interested in modular forms ranging from undergraduate students to seasoned researchers. It is assumed that the reader has a basic understanding of generating functions, complex numbers, calculus, linear algebra, and some complex analysis as well as the basics of programming with Python. The computational prowess of this book is the algorithmic approach and the SAGE functions given that can handle complicated calculations regarding applications of modular forms. The best way to use the SAGE functions provided is to use Jupyter notebooks, which is natively supported in the current distribution of SAGE. Simply download the file containing all the codes given in this book through the link https://doi.org/10.1007/9783-031-32629-5_5 and run the following code in the first line of a Jupyter notebook: %run file_path/file_name.ipynb After this, all the SAGE functions provided in the book are ready to be used in that Jupyter notebook. We note that these functions are written using SAGE version 9.6.

v

vi

Preface

This book provides refined versions of some previously known results as well as some new results. These results are used in giving the following types of applications: • Finding Eisenstein part of a given modular form and asymptotical analysis of the results; • Giving examples of modular form spaces generated by special functions like Eisenstein series, eta quotients, and theta functions; • Giving tools for finding modular identities; • Writing modular forms with multiplicative q-series coefficients, such as newforms in terms of special functions like Eisenstein series, eta quotients, and theta functions; • Giving tools to study counting problems related to modular forms; • Providing the SAGE functions that automate the applications. Throughout the book, Z denotes the domain of all integers, N the set of positive integers, N0 the set of all non-negative integers, Q the field of rational numbers, C the field of complex numbers and H the upper half plane of complex numbers. All the divisors considered in the book are positive integer divisors of an integer. The variable z always denotes a complex number with a positive imaginary part, and q always stands for e2πi z . The book starts with a brief introduction to modular forms. Then gives results to determine the Eisenstein part of any given modular form of integer weight. As these results are explicit a SAGE function that can handle the necessary computations to use this result is provided. In certain cases, it was known that the Eisenstein part of a modular form is the main part of it. The results of this book generalize these results to a wide variety of modular forms to determine their asymptotical behavior. In plain words, the results of this book provide tools that can be used to show if the q-series coefficients of Eisenstein part of a modular form is asymptotically equal to the q-series coefficients of that modular form for a wide variety of modular forms. Additionally, this book provides useful information on modular form spaces that can be generated by special functions like the Eisenstein series, eta quotients. As an immediate application, modular identities for some newforms are given, as well as prescribing ways to derive similar identities for others. Some of those results are used to determine algebraic points of elliptic curves modulo p by using the Modularity Theorem. The other main application of modular forms provided in this book is on the representation numbers of quadratic forms. The results and the SAGE functions provided can easily be used to determine formulas for any 2k-ary quadratic form. The author hopes that this book will be helpful to anyone who is interested in the computational aspects of modular forms. Grande Prairie, AB, Canada February 2023

Zafer Selcuk Aygin

Acknowledgements

I would like to thank Professor Emeritus Kenneth S. Williams of Carleton University for helping me through all the steps of writing this book. His encouragement helped me to conceive the idea of this book, later at each step Professor Williams provided invaluable feedback and guidance. Some portion of this book was written when I was visiting the University of Calgary in the Summer of 2022. I am extremely grateful to the Department of Mathematics and Statistics, specifically to Prof. Khoa D. Nguyen, for inviting me as a visiting researcher. He is always generous in sharing his wealth of mathematical knowledge and in providing insightful comments on the results of this book. Professor Amir Akbary of the University of Lethbridge has always been an inspiration to me in terms of attention to detail, and I owe a great deal of gratitude to him for encouraging me to think more critically about the results of this book. I would like to thank my dear friends Dr. Adam Legacy, Dr. Ashley Oostvogels, and Matthew Oostvogels for their kindness and their friendship. I am thankful to have worked with Susanne Filler and Prasanna Kumar Narayanasamy of Springer. Their efficiency has made the process of publishing this book extremely pleasant. Last but not least, I thank my parents Emine Aygin and Musa Aygin and my siblings Zuhal Bas, Huseyin Can Aygin, and Alperen Aygin. We are thousands of kilometres and 9 time zones apart, but they are always with me in every way possible.

vii

Contents

1 Dirichlet Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Primitive Dirichlet Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Orthogonal Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Gauss Sums Associated with Dirichlet Characters . . . . . . . . . . . . . . . . . . . . 1.5 Dirichlet L-Functions and Bernoulli Numbers . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 7 9 11

2 Modular Forms: Definition and Some Properties . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Definition of Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Modular Groups 0 (N ) and the Transformation Property . . . . . . . 2.1.2 Definition of Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Some Properties of the Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Definition of Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Constant Terms of Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Sum of Divisors Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Projections of Modular Forms on the Eisenstein Series . . . . . . . . . . . . . . . . 2.7 The Size of the q-Series Coefficients of the Eisenstein Part of a Modular Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 13 15 24 29 31 31 35 38 43

3 Application: Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Modularity and the Constant Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Gauss Sums Associated with Quadratic Forms and the Eisenstein Part of θ (Q; z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Separable Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 A Family of Binary Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Diagonal Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Generalized Kronecker Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Constant Terms of Theta Functions Corresponding to the Diagonal Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65

58

68 71 73 77 77 78 ix

x

Contents

3.4.3 An Explicit Formula for Representation Numbers of Diagonal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Diagonal Quadratic Forms Examples . . . . . . . . . . . . . . . . . . . . . . . . .

84 90

4 Application: Eta Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Definition, the Modularity and the Constant Terms . . . . . . . . . . . . . . . 4.2 Finding the Complete Sets of Eta Quotients in Mk (0 (N ), χ ) . . . . . . . . . 4.3 The Eisenstein Part of Eta Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Eta Quotients in E k (0 (N ), χ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Level 12 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 The Level 16 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 The Level 25 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Eight Families of Eta Quotients Whose Eisenstein Part is a Single Eisenstein Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Eta Quotients and Newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Bases for Cusp Form Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105 105 114 117 119 120 122 124

5 Various Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Convolution Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Klein Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Character Analog of Dedekind’s Eta Function . . . . . . . . . . . . . . . . . . . 5.4 Cusp Forms and the Character Analog of Dedekind’s Eta Function . . . . . 5.5 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 139 146 149 154 157

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165

Index of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

Index of SAGE Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

125 131 135

1

Dirichlet Characters

1.1

The Definition and Basic Properties

We start with the definition of a Dirichlet character, give some examples and discuss some properties of them that are fundamental to our purposes. Definition 1.1.1 Let N be a positive integer. A Dirichlet character of modulus N is a map χ : Z → C which satisfies the following conditions: (a) For all integers a we have χ(a + N ) = χ(a), that is, χ is N periodic. (b) For all integers a with gcd(a, N )  = 1 we have χ (a) = 0. (c) For all integers a and b we have χ(ab) = χ(a)χ (b), that is, χ is totally multiplicative. We denote the set of all Dirichlet characters of modulus N by D(N ). Example 1.1.1 (Kronecker symbols) Let d be a quadratic discriminant, that is, d ∈ Z satisfies one of the following: d ≡ 1 (mod 4) and d is squarefree, d ≡ 0 (mod 4), d/4 is squarefree and d/4 ≡ 2 or 3 (mod 4).

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-32629-5_1.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 Z. S. Aygin, Introduction to Applications of Modular Forms, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-32629-5_1

1

2

1 Dirichlet Characters

The Kronecker symbol χd (a) is defined to be a totally multiplicative function, that is, for integers a and b we have χd (a)χd (b) = χd (ab), and therefore can be determined by the values at primes and −1, which are as follows:  1 if d > 0, χd (−1) := −1 if d < 0, ⎧ ⎪ if d ≡ 1 (mod 8), ⎪ ⎨1 χd (2) :=

−1 if d ≡ 5 (mod 8), ⎪ ⎪ ⎩0 if 2 | d,

and, when p > 2 is a prime, χd ( p) is the Legendre symbol

χd ( p) :=

⎧ ⎪ ⎪0 ⎨

 

d p L,

that is,

if p | d,

1 if d ≡ a 2 (mod p) for some nonzero integer a, ⎪ ⎪ ⎩−1 if there is no integer a such that d ≡ a 2 (mod p).

The Kronecker symbol χd is a Dirichlet character of modulus |d|. In Chap. 3 we define Kronecker symbols in a more general setting. For our purposes in this chapter and Chap. 2 this definition is sufficient. For further reading on the subject see [44, Sect. 9.3]. For example, χ1 is a Dirichlet character of modulus 1. This is usually referred to as the principal character. A less trivial example is ⎧ ⎪ if a ≡ 0 (mod 3), ⎪ ⎨0 χ−3 (a) =

1 if a ≡ 1 (mod 3), ⎪ ⎪ ⎩−1 if a ≡ 2 (mod 3),

which is a Dirichlet character of modulus 3. The following SAGE command returns D(N ) as a list. SAGE command to obtain Dirichlet characters Input: • N: The positive integer N . DirichletGroup(N)

Output: The elements of D(N ) as a list of elements of DirichletGroup.

1.1 The Definition and Basic Properties

3

Let N be a positive integer and χ be a Dirichlet character of modulus N . If a is an integer coprime to N , then by Euler’s theorem there exists a divisor e of φ(N ) which satisfies a e ≡ 1 (mod N ). Here φ denotes the Euler phi function. Therefore we have χ (a e ) = χ (a)e = 1. Then for all integers, a with gcd(a, N ) = 1, χ(a) is a root of unity. In particular, we have χ(−1) = ±1. Definition 1.1.2 If χ(−1) = 1 holds, then we call χ an even character, and if χ (−1) = −1 holds, then we call χ an odd character. Next, we define real and nonreal Dirichlet characters. Definition 1.1.3 If a Dirichlet character χ takes only real values, that is, χ (a) = −1, 0 or 1, then we call χ a real Dirichlet character, otherwise we call it a nonreal Dirichlet character. All Kronecker symbols are clearly real characters. Conversely, by [44, Theorem 9.13], all real Dirichlet characters can uniquely be given in terms of Kronecker symbols. Next, we fix some notations that we use throughout the book. Notation 1.1.1 Let  and ψ be Dirichlet characters of modulus L and M, respectively. For an integer a their product is denoted by ψ(a) = (a)ψ(a). We note that ψ is a Dirichlet character of modulus lcm(L, M). Now we discuss what is meant by the equality of Dirichlet characters modulo N . Notation 1.1.2 Let  and ψ be Dirichlet characters of modulus L and M, respectively. Let N be a multiple of lcm(L, M). If (a) = ψ(a) for each a coprime to N , then we write  = ψ (mod N ), otherwise, we write   = ψ (mod N ). Finally, we introduce a notation that helps us declutter some equations that appear in upcoming chapters. Notation 1.1.3 Let d be a quadratic discriminant where |d| = t satisfies 8  t. Throughout the book, we often see that t is accompanied by a negative or a positive sign depending on the sign of d. In general settings, this creates notational mayhem. For example, if M

4

1 Dirichlet Characters

is a positive, squarefree and odd integer, then the Kronecker symbols with modulus t | M are χ(−1)(t−1)/2 t . If we are to give a formula that involves these Dirichlet characters, it will not look nice. When 8  t there is a unique quadratic discriminant d with |d| = t. Using this, to avoid complicated expressions to indicate signs, we adopt the notation tq to denote the quadratic discriminant with |tq | = t. With this notation we have χ(−1)(t−1)/2 t = χtq for all t | M. If d is a quadratic discriminant where |d| = t and 8 | t then both −t and t are quadratic discriminants, therefore this notation will create ambiguity. In these cases, we use the ± signs as needed.

1.2

Primitive Dirichlet Characters

Definition 1.2.1 Let N be a positive integer and χ be a Dirichlet character of modulus N . If χo is the Dirichlet character of smallest modulus M such that χ = χo (mod N ), then χo (a) is the primitive Dirichlet character corresponding to χ and M is the conductor of χ . If M = N then we say χ is a primitive Dirichlet character of modulus N (or a primitive Dirichlet character of conductor N). For each quadratic discriminant d the Kronecker symbol χd is a primitive Dirichlet character of conductor |d|, see [44, Theorem 9.13]. On the other hand, any primitive Dirichlet character χ of conductor |d| can be extended to a Dirichlet character of modulus positive integer L where |d| | L by defining  χ(a) if gcd(a, L) = 1, χ(L; a) := 0 if gcd(a, L)  = 1. Example 1.2.1 (Primitive Dirichlet characters) Let χ be a Dirichlet character of modulus N defined by  1 if gcd(a, N ) = 1, χ(a) = 0 if gcd(a, N )  = 1. The primitive Dirichlet character corresponding to χ is the Dirichlet character χ1 . Therefore, the conductor of χ is 1 and we have χo = χ1 . The character χ1 is called the primitive principal character. We denote the set of all primitive Dirichlet characters of modulus dividing N by Do (N ). Note that the mapping o : D(N ) → Do (N ) which maps χ to χo is one-to-one and onto.

1.2

Primitive Dirichlet Characters

5

Example 1.2.2 (Determining Do (12)) We have D(12) = {χ1 (12; ∗), χ−3 (12; ∗), χ−4 (12; ∗), χ12 (12; ∗)}, therefore Do (12) = {χ1 , χ−3 , χ−4 , χ12 }. SAGE has built-in methods to obtain the Dirichlet character χo for a given Dirichlet character χ. SAGE command to obtain χo Input: • chi: The element of DirichletGroup(N) which corresponds to the Dirichlet character χ . chi.primitive_character()

Output: The element of DirichletGroup which corresponds to the primitive Dirichlet character χo .

Example 1.2.3 For example, one can create the set Do (N ) by using the following SAGE command: [chi.primitive_character() for chi in DirichletGroup(N)]

Let N be a positive integer and  and ψ be Dirichlet characters of modulus L and M, respectively. Recall that, if L M | N and (a)ψ(a) = χ (a) for all integers a coprime to N then we say ψ = χ (mod N ). For example, we have χ−3 χ−3 = χ1 (mod 3)(= χ1 (3; ∗)). In the context of modular forms, it is very important to determine all pairs of primitive Dirichlet characters whose product is equal to a given Dirichlet character of modulus N , that is, we are interested in the following set of pairs of primitive Dirichlet characters of conductors L and M where L M | N : E(N , χ ) := {(, ψ) ∈ Do (N ) × Do (N ) : L M | N and ψ = χ (mod N )}.

6

1 Dirichlet Characters

Example 1.2.4 Recall that we have Do (12) = {χ1 , χ−3 , χ−4 , χ12 }. From this we obtain E(12, χ1 ) = {(χ1 , χ1 )}, E(12, χ−3 ) = {(χ1 , χ−3 ), (χ−3 , χ1 )}, E(12, χ−4 ) = {(χ1 , χ−4 ), (χ−4 , χ1 )}, and E(12, χ12 ) = {(χ1 , χ12 ), (χ−3 , χ−4 ), (χ−4 , χ−3 ), (χ12 , χ1 )}. In general, determining these pairs is a straightforward, but tedious task. This task can be automated by the following SAGE function. SAGE function to determine E(N , χ ) Inputs: • N: The positive integer N . • chi: The element of DirichletGroup which corresponds to the Dirichlet character χ.

def CharPairsFind(N,chi): G=DirichletGroup(N) chi.extend(N) out=[] for eps in G: L,psi=eps.conductor(),chi/eps M=psi.conductor() if N%(L*M)==0: out.append([eps.primitive_character(),\ psi.primitive_character()]) return out

Output: A list of pairs of primitive Dirichlet characters in E(N , χ ) as elements of DirichletGroup, where each pair is given as a list as well.

1.3

1.3

Orthogonal Relations

7

Orthogonal Relations

In this section we discuss some orthogonal relations satisfied by Dirichlet characters. Proposition 1.3.1 ([26, Proposition 3.4.2]) Let N be a positive integer. If χ is a Dirichlet character of modulus N , then we have  N φ(N ) if χ = χ1 (mod N ), χ(A) = 0 otherwise. A=1 Below and in the rest of the book χ denotes the complex conjugate of χ. The following orthogonal relation satisfied by Dirichlet characters is used in the proof of Theorem 2.4.3. Lemma 1.3.1 Let  and ψ be Dirichlet characters with modulus L and M, respectively. If a positive integer C is a multiple of lcm(L, M), then we have  C φ(C) if  = ψ (mod lcm(L, M)), (A)ψ(A) = 0 if   = ψ (mod lcm(L, M)). A=1, gcd(A,C)=1

Proof The character ψ is of modulus lcm(L, M), see [43, p. 80]. If  = ψ (mod lcm(L, M)), then for all A coprime to lcm(L, M) we have (A)ψ(A) = 1. On the other hand if gcd(A, C) = 1, then gcd(A, lcm(L, M)) = 1. Therefore we have C

C

(A)ψ(A) =

A=1, gcd(A,C)=1

1 = φ(C).

A=1, gcd(A,C)=1

If   = ψ (mod lcm(L, M)), we consider the character χ1 (C; ∗)ψ which is a nonprincipal Dirichlet character of modulus C. Therefore, by Proposition 1.3.1, we have C

χ1 (C; A)(A)ψ(A) = 0.

A=1

The observation that C A=1

completes the proof.

χ1 (C; A)(A)ψ(A) =

C

(A)ψ(A)

A=1, gcd(A,C)=1



8

1 Dirichlet Characters

Using Lemma 1.3.1, we prove the following statement, which is useful in Chap. 3. Corollary 1.3.1 Let C be a positive integer divisible by 4. Let ρ  = χ1 , χ−4 be a Dirichlet character of modulus R, where R divides C. Then we have C

ρ(A) = 0

A=1 gcd(A,C)=1 A≡1 (mod 4)

and C

ρ(A) = 0.

A=1 gcd(A,C)=1 A≡3 (mod 4)

Proof Since 4 | C, R | C and ρ  = χ−4 , by Lemma 1.3.1 we have C

χ−4 (A)ρ(A) = 0.

(1.1)

A=1 gcd(A,C)=1

We also have C

C

χ−4 (A)ρ(A) =

A=1 gcd(A,C)=1

A=1 gcd(A,C)=1 A≡1 (mod 4) C

=

C

χ−4 (A)ρ(A) +

χ−4 (A)ρ(A)

A=1 gcd(A,C)=1 A≡3 (mod 4) C

ρ(A) −

A=1 gcd(A,C)=1 A≡1 (mod 4)

ρ(A).

(1.2)

A=1 gcd(A,C)=1 A≡3 (mod 4)

Combining (1.1) and (1.2) we have C A=1 gcd(A,C)=1 A≡1 (mod 4)

ρ(A) =

C A=1 gcd(A,C)=1 A≡3 (mod 4)

ρ(A).

(1.3)

1.4

Gauss Sums Associated with Dirichlet Characters

9

On the other hand, since R | C and ρ  = χ1 by Lemma 1.3.1 we have C

ρ(A) = 0.

(1.4)

A=1 gcd(A,C)=1

Using (1.3) and (1.4) together we obtain the desired result.

1.4



Gauss Sums Associated with Dirichlet Characters

Next, we define the Gauss sums associated with Dirichlet characters and state some of their properties that are necessary for our purposes, see [22] to read more on the Gauss sums in various settings. The Gauss sums associated with Dirichlet characters appear in the Fourier series expansions of the Eisenstein series defined in Sect. 2.4. Definition 1.4.1 Let N be a positive integer and χ be a Dirichlet character of modulus N . We define the Gauss sum of χ by the summation G(χ ) :=

N −1

χ(a)e2π ia/N .

a=0

Example 1.4.1 We have G(χ−3 ) =

2

√ χ−3 (a)e2π ia/3 = e2π i/3 − e4πi/3 = i 3.

a=0

In general, Gauss has proved that √ G(χ pq ) =

p if p ≡ 1 (mod 4), √ i p if p ≡ 3 (mod 4),

(1.5)

for all odd primes p. As we often work with products of certain pairs of Dirichlet characters, we need the following property of the Gauss sums. Lemma 1.4.1 ([43, Lemma 3.1.2]) Let  and ψ be primitive Dirichlet characters with conductors L and M, respectively. If gcd(L, M) = 1, then we have G(ψ) = (M)ψ(L)G()G(ψ).

10

1 Dirichlet Characters

This also paves the way to consider Gauss’s result for all primitive real Dirichlet characters, that is, for Kronecker symbols. Lemma 1.4.2 ([26, Proposition 3.4.10.(c)]) If χ is a real primitive Dirichlet character, that is, if χ = χd for some quadratic discriminant d, then we have

G(χ ) = G(χd ) = e(d) |d|, where e(d) :=

 1 if d > 0, i if d < 0.

Proof Let |d| = D. Then we have either D = N , or D = 4N , or D = 8N where N is a positive odd squarefree integer. If D = N , then the proof is given in [17] in detail. Now we use Lemma 1.4.1 to prove the statement for D = 4N . We have χd = χ−4 χ Nq , therefore we have  √ 2i N if N ≡ 1 (mod 4), G(χd ) = χ−4 (N )χ Nq (4)G(χ−4 )G(χ Nq ) = √ 2 N if N ≡ 3 (mod 4). We have d < 0 if N ≡ 1 (mod 4) and we have d > 0 if N ≡ 3 (mod 4). This finishes the proof for this case. The proof of the case when D = 8N is similar by observing that χd = χ8 χ Nq or, =  χ−8 χ Nq depending on the sign of d and the value of N modulo 4. In general, whether χ is real or not, the following SAGE command computes the Gauss sum of a given Dirichlet character. We note that, in the interest of saving resources, we favor using Lemma 1.4.2 for the Gauss sums associated with real Dirichlet characters. SAGE command to compute G(χ ) Input: • chi: The element of DirichletGroup(N) which corresponds to the Dirichlet character χ . chi.gauss_sum()

Output: The value of G(χ ), the Gauss sum associated with χ.

1.5

Dirichlet L-Functions and Bernoulli Numbers

11

This finishes our discussion of the Gauss sums associated with Dirichlet characters. In Chap. 3, we consider Gauss sums which are associated with quadratic forms.

1.5

Dirichlet L-Functions and Bernoulli Numbers

In this section, we define and give some properties of the Dirichlet L-function. This function plays an important role in the normalization of the Eisenstein series, see Chap. 2. Definition 1.5.1 Let N and k be positive integers and χ be a Dirichlet character of modulus N . The Dirichlet L-function of χ, denoted by L(χ , k), is defined by L(χ , k) :=

 p

∞

χ (n) 1 = . 1 − χ( p) p −k nk n=1

Sometimes the Dirichlet L-functions of nonprimitive Dirichlet characters appear together with their corresponding primitive Dirichlet characters. In order to unify our formulas we need the relationship between them. From the infinite product definition of L(χ , k) the following statement is clear. Lemma 1.5.1 ([43, Eq. (3.3.14)]) Let k and N be positive integers and χ a Dirichlet character of modulus N and conductor No . If χo is the primitive Dirichlet character corresponding to χ , then we have L(χ , k) = L(χo , k)

 p k − χo ( p) . pk p|N

The following relationship between the Dirichlet L-functions and Bernoulli numbers helps us to write our formulas in terms of finite sums. Theorem 1.5.1 ([43, Theorem 3.3.4]) Let k and N be positive integers and χ a primitive Dirichlet character of modulus N . If χ(−1) = (−1)k , then we have L(χ , k) =

(−1)k−1 (2π i)k G(χ )Bk (χ ), 2k!N k

where Bk (χ) is called the kth generalized Bernoulli number associated with χ and defined by the formula ∞ N Bk (χ) k χ (a)teat t := . k! eN t − 1 k=0

a=1

12

1 Dirichlet Characters

In general, we resort to SAGE to compute Bernoulli numbers by using the following command. SAGE command to compute Bk (χ ) Inputs: • chi: The element of DirichletGroup(N) which corresponds to the Dirichlet character χ . • k: The positive integer k. chi.bernoulli(k)

Output: The value of Bk (χ ), the kth Bernoulli number associated with χ .

This concludes the discussion of preliminary tools. In the next chapter, we define modular forms and give some properties of them.

2

Modular Forms: Definition and Some Properties

2.1

The Definition of Modular Forms

2.1.1

Modular Groups 0 (N) and the Transformation Property

We first discuss the full modular group SL 2 (Z) and its subgroups. Definition 2.1.1 The full modular group is defined by    ab SL 2 (Z) := : a, b, c, d ∈ Z; ad − bc = 1 . cd   ab If M = ∈ SL 2 (Z), then with the action defined by cd M(z) :=

az + b , cz + d

the matrix M acts on z ∈ H ∪ Q ∪ {i∞}. This action is referred to as linear fractional transformation. Note that if z is in the upper half plane of the complex numbers, then M(z) is in the upper half plane of the complex numbers; and if z ∈ Q ∪ {i∞}, then M(z) ∈ Q ∪ {i∞}. Next, we define the modular subgroups.

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-32629-5_2.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 Z. S. Aygin, Introduction to Applications of Modular Forms, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-32629-5_2

13

14

2 Modular Forms: Definition and Some Properties

Definition 2.1.2 Let N be a positive integer. The modular subgroups are defined by    ab 0 (N ) := : a, b, c, d ∈ Z; ad − bc = 1; c ≡ 0 (mod N ) . cd We note that 0 (1) is equal to the full modular group, that is, we have 0 (1) = SL 2 (Z). From now on we shall use 0 (1) rather than SL 2 (Z). Definition 2.1.3 The elements of the set Q ∪ {i∞} are called the cusps. Two cusps r1 and r2 are said to be equivalent under 0 (N ) if there exists a matrix M ∈ 0 (N ) such that M(r1 ) = r2 . This defines an equivalence relation and we denote by R(0 (N )) a set of cusps formed by taking exactly one cusp from each equivalence class. Whenever we write a/c ∈ Q ∪ {i∞}, it is always assumed that the ratio is in reduced form, that is, gcd(a, c) = 1. Additionally, unless otherwise stated, 1/0 denotes the cusp i∞. Example 2.1.1 (The set R(0 (1))) Let r = a/c be a cusp  with  gcd(a, c) = 1. Then there ab exist integers b and d such that ad − bc = 1, that is, M = ∈ 0 (1). Thus for any cusp cd a/c ∈ Q ∪ {i∞} we have M(i∞) = a/c. Therefore all the cusps of 0 (1) are equivalent, that is, R(0 (1)) consists of only one element. Classically, the choice R(0 (1)) = {i∞} is used. In some applications, for unification purposes, we prefer using R(0 (1)) = {1/1}. Lemma 2.1.1 ([26, Corollary 6.3.23]) For any positive integer N we can choose  A N (c), R(0 (N )) = c|N

where A N (c) = {a/c : a (mod gcd(c, N /c)) and gcd(a, c) = 1}, meaning we pick exactly one a from each congruence class modulo gcd(c, N /c) such that gcd(a, c) = 1. Example 2.1.2 (The set R(0 (N )) for a squarefree N ) Let N be a positive squarefree integer. Then for each c | N we have gcd(c, N /c) = 1. Therefore we can choose R(0 (N )) = {1/c : c | N }.   ab Definition 2.1.4 If there exists an M = ∈ 0 (N ) such that M(z) = z for some z ∈ cd H ∪ Q ∪ {i∞}, then we say z is an elliptic point of the modular group 0 (N ).

2.1 The Definition of Modular Forms

15

Example 2.1.3 The complex unit i is an elliptic point of 0 (1), because we have   1 0 1 (i) = = i. −1 0 −i To read more on fixed points of 0 (N ) and their full characterization, see [27, Sect. 3.7].

2.1.2

Definition of Modular Forms

In this subsection, we give the definition of a modular form and develop a basic understanding of modular forms. In plain words, modular forms are functions that satisfy a certain transformation property for a given modular subgroup and that are holomorphic on the upper half plane of complex numbers and holomorphic at each cusp in Q ∪ {i∞}. There are various approaches to defining modular forms. We start with the transformation property and build up the meaning of holomorphicity at cusps using the transformation property. This approach follows Köhler’s ideas from [36] and serves our purposes the best. Definition 2.1.5 Let k and N be positive integers and χ a Dirichlet character of modulo N . Then f : H → C is a function which satisfies the transformation property for 0 (N ) and χ with weight k, if f (M(z)) = χ(d)(cz + d)k f (z) holds for all M =

  ab ∈ 0 (N ). cd

Below and in the rest of the book q = q(z) denotes the complex number e2π iz where z ∈ H. Lemma 2.1.2 ([36, pp. 15–19]) Let k and N be positive integers and χ a Dirichlet character modulo N . If f : H → C is holomorphic on the upper half plane of complex numbers which satisfies the transformation property for 0 (N ) and χ , then at the cusp i∞ the function f (z) has a Fourier series expansion of the form f (z) =

∞ 

a( f ; n)q n ,

n=−∞

and at the cusp r = a/c ∈ Q the function f (z) has a Fourier series expansion of the form (cz + d)−k f (M(z)) =

∞  n=−∞

n+κ ar ( f ; n)qc,N ,

16

2 Modular Forms: Definition and Some Properties

where b and d are some integers satisfying ad − bc = 1, qc,N := e2π i gcd(c

2 ,N )z/N

,

and 0 ≤ κ < 1 satisfies e2π iκ = χ(1 + acN / gcd(N , c2 )).

(2.1)

Proof Suppose that f (z) is holomorphic on the upper half plane of complex numbers and satisfies the transformation formula f (M(z)) = χ(d)(cz + d)k f (z)     ab 11 for all M = ∈ 0 (N ). Since ∈ 0 (N ), we have cd 01 f (z + 1) = f (z). Therefore f (z) is a holomorphic function on the upper half plane of complex numbers with period 1. Hence it has a Fourier series expansion of the form f (z) =

∞ 

a( f ; n)e2π inz =

n=−∞

∞ 

a( f ; n)q n

n=−∞

at the cusp i∞. We now consider cusps a/c  = i∞. Let M =

  ab ∈ 0 (1). If we let h := cd

N / gcd(N , c2 ), then we have     1h 1 − ach a 2 h ∈ 0 (N ). L := M M −1 = −c2 h 1 + ach 01 Therefore, by the transformation property, we have f (L(z)) = χ(1 + ach)(−c2 hz + 1 + ach)k f (z). Next, we define g(z) := χ(1 + ach)(−cM(hz) + a)k f (M(hz)) and show that g(z + 1) = g(z). First we use the equation M(hz + h) = L M(hz) and the transformation property to obtain g(z + 1) = (−cM(hz + h) + a)k (−c2 h M(hz) + 1 + ach)k f (M(hz)).

(2.2)

2.1 The Definition of Modular Forms

17

On the other hand, we have   ahz + b (−cM(hz + h) + a)(−c2 h M(hz) + 1 + ach) = −c +a . chz + d

(2.3)

Putting (2.3) in (2.2) we obtain g(z + 1) = g(z), that is, g(z) is a holomorphic function on the upper half plane of complex numbers with period 1. Therefore g has a Fourier series expansion of the form g(z) =

∞ 

ar ( f ; n)e2π inz .

n=−∞

Hence we have χ(1 + ach)(−cM(hz) + a)k f (M(hz)) =

∞ 

ar ( f ; n)e2π inz .

(2.4)

Further, we replace z by z/h in (2.4) and use (2.1) to obtain the desired result.



n=−∞

Definition 2.1.6 Suppose that the Fourier series expansion of f (z) at r = a/c ∈ Q ∪ {i∞} has the form (cz + d)−k f (M(z)) =

∞ 

n+κ ar ( f ; n)qc,N .

n=−∞

Then the order of vanishing of f at r is n 1 + κ, where n 1 is the greatest integer with the property that ar ( f ; n) = 0 for all n < n 1 . We denote the order of vanishing of f at r by vr ,N ( f ). Remark 2.1.1 In the remainder of the book we use the term q-series expansion of f(z) for the Fourier series expansion of f (z) at i∞. Remark 2.1.2 If f (z) satisfies the transformation property for 0 (N ) and χ then f (z) satisfies the transformation property for 0 (t N ) and χ for all positive integers t. In our definition, the order of vanishing of f (z) depends on the level f (z). Therefore we indicate that as a subscript in the notation. The explicit relation between the orders when f is considered in a different level is given by vr ,t N ( f (z)) =

t gcd(c2 , N ) vr ,N ( f (z)). gcd(c2 , t N )

18

2 Modular Forms: Definition and Some Properties

Definition 2.1.7 Let vr ,N ( f ) ≥ 0 for some r = a/c ∈ Q ∪ {i∞}. Let   ab ∈ 0 (1). We define the constant term of f at the cusp r = a/c by the limit cd

M=

Cr ( f ) := lim (cz + d)−k f (M(z)). z→i∞

We note that, if f (z) satisfies the transformation property for 0 (N ) and χ , then the value of Cr ( f ) is independent of the choice of the entries b and d in the matrix M. Remark 2.1.3 Let the Fourier series expansion of f (z) at r = a/c ∈ Q ∪ {i∞} have the form (cz + d)−k f (M(z)) =

∞ 

n+κ ar ( f ; n)qc,N .

n=0

If κ  = 0, we have Cr ( f ) = 0, and if κ = 0, we have Cr ( f ) = ar ( f ; 0). Now we are ready to give the definition of a modular form. Definition 2.1.8 Let k and N be positive integers and χ a Dirichlet character modulo N . (a) (Modular form) If f : H → C is a holomorphic function on the upper half plane of complex numbers which satisfies the transformation property for 0 (N ) and χ with weight k and vr ,N ( f ) ≥ 0 for all r ∈ Q ∪ {i∞}, then f is a modular form of weight k for 0 (N ) with Dirichlet character χ . We denote the space of all modular forms of weight k for 0 (N ) with Dirichlet character χ by Mk (0 (N ), χ ). (b) (Cusp form) If, in addition, vr ,N ( f ) > 0 (equivalently Cr ( f ) = 0) for all r ∈ Q ∪ {i∞}, then f is a cusp form of weight k for 0 (N ) with Dirichlet character χ . We denote the space of all cusp forms of weight k for 0 (N ) with Dirichlet character χ by Sk (0 (N ), χ ). (c) (Eisenstein form) The orthogonal complement of Sk (0 (N ), χ ) in Mk (0 (N ), χ ), denoted by E k (0 (N ), χ ), is the space of Eisenstein series of weight k for 0 (N ) with Dirichlet character χ. An element of E k (0 (N ), χ ) is called an Eisenstein form of weight k for 0 (N ) with Dirichlet character χ . Remark 2.1.4 We note that the orthogonality above is with respect to the Petersson inner product, see [26, Sect. 12.6] to read more on Petersson inner products. Notation 2.1.1 We have Mk (0 (N ), χ ) = E k (0 (N ), χ ) ⊕ Sk (0 (N ), χ ).

2.1 The Definition of Modular Forms

19

Therefore, each f (z) ∈ Mk (0 (N ), χ ) can be written as f (z) = E( f ; z) + S( f ; z), where E( f ; z) ∈ E k (0 (N ), χ ) and S( f ; z) ∈ Sk (0 (N ), χ ) and are uniquely determined by f (z). We refer to E( f ; z) as the Eisenstein part of f (z) and S( f ; z) as the cusp part of f (z). We denote the q-series coefficients of f (z), E( f ; z) and S( f ; z) by a( f ; n), e( f ; n) and s( f ; n), respectively, that is, we have f (z) =

∞ 

a( f ; n)q n ,

n=1

E( f ; z) =

∞ 

e( f ; n)q n ,

n=1

and S( f ; z) =

∞ 

s( f ; n)q n .

n=1

With this notation, for each n ∈ N0 , we have a( f ; n) = e( f ; n) + s( f ; n).

(2.5)

Below we give the Hecke bound for the q-series coefficients of cusp forms as it is stated in [26, Theorem 9.2.1.(a)]. Theorem 2.1.1 (Hecke bound) If f (z) ∈ Mk (0 (N ), χ ) where k > 1, then we have s( f ; n) = O(n k/2 ).

(2.6)

In Sect. 2.6 we give an explicit expression for e( f ; n). Hecke bound helps us to show that in some cases e( f ; n) dominates s( f ; n), that is, in some cases the results of Sect. 2.6 gives an asymptotic formula for the q-series coefficients of modular forms. In Theorem 2.7.3 we give a set of criteria to determine when e( f ; n) is asymptotically equivalent to a( f ; n). It is well known that the spaces E k (0 (N ), χ ) and Sk (0 (N ), χ ) are finite dimensional, see [53, Chap. 6] for formulas. These are well known and well implemented in SAGE. As a complete description of the Eisenstein part of a modular form is given in Sect. 2.6, we are only interested in the dimension of a given cusp form, which can be done by using the SAGE command given below.

20

2 Modular Forms: Definition and Some Properties

SAGE command to compute the dimensions of cusp form spaces Inputs: • chi: The element of DirichletGroup(N) which corresponds to the Dirichlet character χ . • k: The positive integer k. from sage.modular.dims import dimension_cusp_forms dimension_cusp_forms(chi,k)

Output: The value of dim(Sk (0 (N ), χ )).

Example 2.1.4 If we run G=DirichletGroup(12) chi=G[0] from sage.modular.dims import dimension_cusp_forms dimension_cusp_forms(chi,2)

in SAGE it returns the dimension of the space S2 (0 (12), χ1 ), which is 0. In the upcoming chapters sometimes we use modular form spaces whose cusp subspace dimension is 0 or 1 to give examples. Using SAGE we search for modular form spaces whose cusp subspace dimension is 0 or 1 and list some of them below for future reference. Lemma 2.1.3 If (k, N , χ ) ∈ {(2, 1, χ1 ), (2, 2, χ1 ), (2, 3, χ1 ), (2, 4, χ1 ), (2, 5, χ1 ), (2, 5, χ5 ), (2, 6, χ1 ), (2, 7, χ1 ), (2, 8, χ1 ), (2, 8, χ8 ), (2, 9, χ1 ), (2, 10, χ1 ), (2, 10, χ5 ), (2, 12, χ1 ), (2, 12, χ12 ), (2, 13, χ1 ), (2, 13, χ13 ), (2, 15, χ5 ), (2, 16, χ1 ), (2, 16, χ8 ), (2, 17, χ17 ), (2, 18, χ1 ), (2, 20, χ5 ), (2, 21, χ21 ), (2, 24, χ12 ), (2, 25, χ1 ), (2, 25, χ5 ), (2, 32, χ8 ), (3, 3, χ−3 ), (3, 4, χ−4 ), (3, 6, χ−3 ), (3, 9, χ−3 ), (4, 1, χ1 ), (4, 2, χ1 ), (4, 3, χ1 ), (4, 4, χ1 ), (4, 5, χ5 ), (5, 3, χ−3 ), (6, 1, χ1 ), (6, 2, χ1 ), (8, 1, χ1 ), (10, 1, χ1 ), (14, 1, χ1 )} then dim(Sk (0 (N ), χ ) = 0. If

2.1 The Definition of Modular Forms

21

(k, N , χ ) ∈ {(2, 11, χ1 ), (2, 14, χ1 ), (2, 15, χ1 ), (2, 17, χ1 ), (2, 19, χ1 ), (2, 20, χ1 ), (2, 21, χ1 ), (2, 24, χ1 ), (2, 27, χ1 ), (2, 32, χ1 ), (2, 36, χ1 ), (2, 49, χ1 ), (3, 7, χ−7 ), (3, 8, χ−8 ), (3, 11, χ−11 ), (3, 12, χ−3 ), (3, 16, χ−4 ), (4, 5, χ1 ), (4, 6, χ1 ), (4, 7, χ1 ), (4, 8, χ1 ), (4, 9, χ1 ), (5, 4, χ−4 ), (5, 7, χ−7 ), (6, 3, χ1 ), (6, 4, χ1 ), (6, 5, χ1 ), (7, 3, χ−3 ), (8, 2, χ1 ), (8, 3, χ1 ), (10, 2, χ1 ), (12, 1, χ1 )} then dim(Sk (0 (N ), χ ) = 1. Let  and ψ be primitive Dirichlet characters of modulus L and M, respectively. Let χ be a Dirichlet character of modulus N with L M | N and ψ = χ (mod N ) and k satisfy (−1)k = χ(−1). For k > 2 we define the Eisenstein series by the infinite sum ∞ 

Ek (, ψ; z) :=

m,n=−∞, (m,n)=(0,0)

(m)ψ(n) . (m M z + n)k

These Eisenstein series are archetypical examples of modular forms in Mk (0 (N ), χ ). Below we prove their modularity and find their constant term at the cusps. Proposition 2.1.1 Let k, , ψ and χ as above. For all t | N /L M we have Ek (, ψ; t z) ∈ Mk (0 (N ), χ ) and  Ca/c (Ek (, ψ; t z)) =

gcd(Mt, c) Mt

k   −

   c Mt 2L(ψ, k). ψ a gcd(Mt, c) gcd(Mt, c)

Proof By [43, Chap. 7] for k > 2 the infinite sum Ek (, ψ; t z) is holomorphic on the upper half plane of complex numbers. Next, we prove the transformation property,   that is, we prove that Ek (, ψ; t A(z)) = ab ∈ 0 (N ). We have χ(d)(cz + d)k Ek (, ψ; t z) for all A = cd Ek (, ψ; t A(z)) =

∞  m,n=−∞, (m,n)=(0,0)

= (cz + d)k

(m)ψ(n) k (m Mt az+b cz+d + n) ∞  m,n=−∞, (m,n)=(0,0)

(m)ψ(n) . ((m Mta + nc)z + (m Mtb + dn))k

22

2 Modular Forms: Definition and Some Properties

In the above expression, the sum is through all pairs of integers (m, n)  = (0, 0), so replacing z by z + Z for any integer Z does not change the infinite sum. Hence we replace z by z − bd in the infinite sum part of the above expression and obtain ∞  m,n=−∞, (m,n)=(0,0) ∞ 

(m)ψ(n) ((m Mta + nc)z + (m Mtb + dn))k

=

=

m,n=−∞, (m,n)=(0,0) ∞  m,n=−∞, (m,n)=(0,0)

(m)ψ(n) ((m Mta + nc)(z − bd) + (m Mtb + dn))k (m)ψ(n) . ((m Mta + nc)z + (m Mt(b − abd) + n(d − bdc)))k

Now we apply the change of variables r := m Mta + nc and s := m Mt(b − abd) + n(d − bdc), and obtain ∞  m,n=−∞, (m,n)=(0,0) ∞ 

(m)ψ(n) ((m Mta + nc)z + (m Mt(b − abd) + n(d − bdc)))k

=

r ,s=−∞, (r ,s)=(0,0)

(r d − r bdc − cs/Mt)ψ(−r Mtb(1 − ad) + as) . (r Mt z + s)k

Since L, M | N , N | c and ad − bc = 1 we have (r d − r bdc − cs/Mt) = (r d) = (d)(r ) and ψ(−r Mtb(1 − ad) + as) = ψ(as) = ψ(d)ψ(ads) = ψ(d)ψ(s). Putting these together we obtain ∞  m,n=−∞, (m,n)=(0,0)

(m)ψ(n) = (d)ψ(d) ((ma + nc)Mt z + (mb + dn))k

∞  r ,s=−∞, (r ,s)=(0,0)

(r )ψ(s) . (r Mt z + s)k

Therefore, noting that (d)ψ(d) = χ(d) (since gcd(d, N ) = 1), we obtain ∞  m,n=−∞, (m,n)=(0,0)

(m)ψ(n) = χ(d)(cz + d)k (m Mt A(z) + n)k

∞  m,n=−∞, (m,n)=(0,0)

(m)ψ(n) . (m Mt z + n)k

2.1 The Definition of Modular Forms

23

  ab ∈ 0 (N ). cd Now we prove that Ek (, ψ; t z) is holomorphic at each cusp r = a/c ∈ Q ∪ {i∞}. We start with r = i∞. Since Ek (, ψ; t z) satisfies the transformation property for 0 (N ) and χ , by Lemma 2.1.2, it has a q-series expansion of the form That is, we have Ek (, ψ; t A(z)) = χ(d)(cz + d)k Ek (, ψ; t z) for all A =

∞ 

Ek (, ψ; t z) =

a(Ek (, ψ; t z); n)q n .

n=−∞

Letting z → i∞ on the left hand side of this equation, we have lim Ek (, ψ; t z) = lim

z→i∞

z→i∞

∞  m,n=−∞, (m,n)=(0,0)

(m)ψ(n) = (0)2L(ψ, k), (m Mt z + n)k

(2.7)

that is, it is a finite complex number. Therefore Ek (, ψ; t z) is holomorphic at the cusp r = i∞, that is, vi∞,N (Ek (, ψ; t z)) ≥ 0. Now let r = a/c. Since Ek (, ψ; z) satisfies the transformation property for 0 (N ) and χ , we have lim (cz + d)−k Ek (, ψ; t A(z)) = lim (cz + d)−k

z→i∞

z→i∞

= lim

∞ 

z→i∞

m,n=−∞, (m,n)=(0,0) ∞ 

=

m,n=−∞, m Mta+nc=0 (m,n)=(0,0)

 =  =

∞  m,n=−∞, (m,n)=(0,0)

(m)ψ(n) k (m Mt az+b cz+d + n)

(m)ψ(n) ((m Mta + nc)z + (m Mtb + nd))k

(m)ψ(n) (m Mtb + nd)k

gcd(Mt, c) Mt

k   −

    ∞ (s)ψ(s) c Mt ψ a gcd(Mt, c) gcd(Mt, c) s=−∞, sk s=0

     gcd(Mt, c) k c Mt  − ψ a 2L(ψ, k), Mt gcd(Mt, c) gcd(Mt, c)

that is, it is a finite complex number. Therefore E k (, ψ; z) is holomorphic at the cusp r = a/c, that is, vr ,N (Ek (, ψ; t z)) ≥ 0. We emphasize that when r = i∞ = 1/0 this expression agrees with (2.7).  We often use the constant terms of modular forms in applications. We would therefore like to give them as simple as possible. With this goal in mind, we normalize the Eisenstein

24

2 Modular Forms: Definition and Some Properties

series Ek (, ψ; t z) by dividing it by 2L(ψ, k). That is, the normalization given by E k (, ψ; t z) :=

Ek (, ψ; t z) 2L(ψ, k)

(2.8)

is more appealing to us. We study E k (, ψ; t z) in greater detail in Sect. 2.4.

2.2

Some Properties of the Modular Forms

In this section, we give some well known properties of modular forms and prove some properties that are useful for our purposes. Theorem 2.2.1 Let f (z), g(z) ∈ Mk (0 (N ), χ ) and h(z) ∈ Mk  (0 (M), ρ). Then the following statements hold. (a) We have f ± g ∈ Mk (0 (N ), χ ). The constant terms of f (z) ± g(z) at r ∈ Q ∪ {i∞} are given by Cr ( f ± g) = Cr ( f ) ± Cr (g) and vr ,N ( f ± g) ≥ min(vr ,N ( f ), vr ,N (g)). Moreover, if both f and g are cusp forms, then f ± g ∈ Sk (0 (N ), χ ). (b) We have f h ∈ Mk+k  (0 (lcm(N , M)), χρ). The constant terms of f (z)h(z) at r ∈ Q ∪ {i∞} are given by Cr ( f h) = Cr ( f )Cr (h) and vr ,lcm(N ,M) ( f h) = vr ,lcm(N ,M) ( f ) + vr ,lcm(N ,M) (h). Moreover, if either f or h is a cusp form, then f h ∈ Sk+k  (0 (lcm(N , M)), χρ). Proof Can be deduced from the definition and is left as an exercise.



2.2

Some Properties of the Modular Forms

25

Theorem 2.2.2 Let f (z) ∈ Mk (0 (N ), χ ) and t ∈ N. Then the following statements hold. (a) We have f (t z) ∈ Mk (0 (t N ), χ ). The constant terms of f (t z) at a/c ∈ Q ∪ {i∞} are given by  Ca/c ( f (t z)) = where a  =

gcd(c, t) t

k Ca  /c ( f (z))

(2.9)

at c and c = ; and we have gcd(c, t) gcd(c, t) vr ,t N ( f (t z)) =

t 2 gcd(c2 , N ) vr ,N ( f (z)). gcd(c2 , t N )

Moreover, if f (z) is a cusp form so is f (t z). (b) In addition assume that the conductor of χ divides N /t. If the q-series expansion of  f is of the form n≥0 an q tn , then there exists a g(z) ∈ Mk (0 (N /t), χ ) such that f (z) = g(t z). If f (z) is a cusp form, so is g(z). Proof All these assertions can be deduced easily by using the definition except perhaps (2.9), whose proof is as follows. By the definition of constant terms, we have   at z + bt −k . (2.10) Ca/c ( f (t z)) = lim (cz + d) f z→i∞ cz + d

at c at c Since gcd gcd(c,t) = 1, there exist β and δ ∈ Z such that gcd(c,t) , gcd(c,t) δ − gcd(c,t) β = 1,   at/ gcd(c, t) β that is, ∈ 0 (1). We have c/ gcd(c, t) δ      at bt at/ gcd(c, t) β gcd(c, t) u = , c d c/ gcd(c, t) δ 0 t/ gcd(c, t) where u = δbt − βd. Now let z 1 :=

(2.11)

gcd(c, t)z + u . We have z → i∞ if and only if z 1 → t/ gcd(c, t)

i∞. Also, we have (cz + d) =

t gcd(c, t)



 c z1 + δ , gcd(c, t)

(2.12)

26

2 Modular Forms: Definition and Some Properties

and by (2.11) we have     at bt at/ gcd(c, t) β (z) = (z 1 ). c d c/ gcd(c, t) δ

(2.13)

Using (2.12) and (2.13) in (2.10), we obtain  Ca/c ( f (t z)) =

gcd(c, t) t

k

 lim

z 1 →i∞

c z1 + δ gcd(c, t)

−k f

   at/ gcd(c, t) β (z 1 ) , c/ gcd(c, t) δ 

from which (2.9) follows by the definition of constant terms.

We now prove a useful statement concerning the constant terms of modular forms at equivalent cusps. Lemma 2.2.1 Let f ∈ Mk (0 (N ), χ ). If r , r  ∈ Q ∪ {i∞} are equivalent cups under 0 (N ), then we have Cr ( f ) = ωCr  ( f ) for some root of unity ω. In particular, we have Cr ( f ) = 0 if and only if Cr  ( f ) = 0. Proof If r = a/c and r  = a  /c are equivalent cups under 0 (N ), then there is a matrix   a b αβ be a matrix in 0 (1) with A= ∈ 0 (N ) such that r  = A(r ). Let M = cd γ δ M(i∞) = a/c. Let M  = AM. This implies that M  (i∞) = a  /c . Hence, by definition, we have Cr  ( f ) = lim (c z + d  )−k f (M  (z)).

(2.14)

z→i∞

Since f ∈ Mk (0 (N ), χ ) and A ∈ 0 (N ), we deduce that 

c z + d  f (M (z)) = f (AM(z)) = χ (δ)(γ M(z) + δ) f (M(z)) = χ (δ) cz + d 

k

k f (M(z)).

Putting this in (2.14) we obtain Cr  ( f ) = χ(δ) lim (cz + d)−k f (M(z)) = χ (δ)Cr ( f ). z→i∞

Since gcd(δ, N ) = 1, we have χ(δ)  = 0. The first assertion follows from χ (δ) being a root of unity. The second assertion is clear as χ(δ)  = 0. 

2.2

Some Properties of the Modular Forms

27

Next, we give an explicit relationship between the constant terms of a modular form at certain cusps. This is sometimes useful in optimizing the SAGE programs in applications of Theorem 2.6.1. Lemma 2.2.2 ([13, Lemma 6]) Let f (z) ∈ Mk (0 (N ), χ ) and c be a divisor of N . Let the positive integers a and a  both be coprime to c with a ≡ a  (mod gcd(c, N /c)). If (, ψ) ∈ E(N , χ ) with M | c, then we have ψ(a)[0]a/c f = ψ(a  )[0]a  /c f . The following result is proved in [15]. Theorem 2.2.3 ([15, Proposition 1.3]) Let t and N be positive integers and let M = lcm(t 2 , N ). Let χ be a Dirichlet character with a conductor dividing M/t. Let f (z) ∈ Mk (0 (N ), χ ) and suppose that ai = a j for all integers δ that are coprime to M, where j ≡ iδ 2 (mod t) and j ∈ {1, . . . , t}. Then we have t 

 ai f

i=1

z+

i t

 ∈ Mk (0 (M), χ ).

As an immediate consequence of this, we obtain the following result. Corollary 2.2.1 Let t and N be positive integers and let M = lcm(t 2 , N ). Let χ be a Dirichlet character with a conductor dividing M/t. If f (z) ∈ Mk (0 (N ), χ ) and has a  q-series expansion n≥0 a( f ; n)q n , then we have g(z) =



a( f ; tn)q n ∈ Mk (0 (M/t), χ ).

n≥0

Proof We deduce this by letting ai = 1/t for all i = 1, . . . , t in Theorem 2.2.3 and by applying Theorem 2.2.2 (b) afterwards.  The next result, which is referred to as Sturm bound, gives a very strong tool to check if given modular forms are equal. This was first proven by Sturm [54], we use the version given by [26, Corollary 5.6.14]. Theorem 2.2.4 (Sturm bound) Let f and g both be in Mk (0 (N ), χ ). If a( f ; n) = a(g; n) for each

28

2 Modular Forms: Definition and Some Properties

n ≤1+

kN  p + 1 , 12 p p|N

then f (z) = g(z). It is sometimes useful to write a modular form as a linear combination of other modular forms. Below we write a SAGE function that compares q-series expansions of given modular forms up to a given precision. This SAGE function is useful in obtaining modular identities. SAGE function to find modular identities Inputs: • forms: The q-series expansions of the modular forms f 1 , . . ., fl ∈ Mk (0 (N ), χ ) given as a list. • f: The q-series expansion of the modular form f ∈ Mk (0 (N ), χ ). • prec: An integer, for the number of terms to compare in the q-series expansions. • unique: An optional input with default value 1.

def ModIdFinder(forms,f,prec,unique=1): M=matrix(QQbar,prec+1,len(forms)) for i in range(len(forms)): for j in range(prec+1): M[j,i]=forms[i].coefficient(q,j) b=matrix(prec+1,1) for j in range(prec+1): b[j,0]=f.coefficient(q,j) VV=M.right_kernel(basis='pivot').basis() try: out=(M\b) if len(VV)==0: return [QQbar(at[0]).radical_expression()\ for at in out] elif len(VV)!=0 and unique==1: print('Solution is not unique.') return False elif len(VV)!=0 and unique!=1: return([(M\b).transpose(), [vv for vv in VV]]) except(ValueError): return False

2.3

Newforms

29

Output: • If the optional input unique is 1,  – and if the q-series coefficients of f (z) and lj=1 a j f j (z) agree up to the coefficient of q pr ec for a unique set of a j ∈ C, then it returns the values of a j as an ordered list; – and if the above fails, then it returns False. • If the optional input unique is different from 1,  – and if the q-series coefficients of f (z) and lj=1 a j f j (z) agree up to the coefficient of q pr ec for a unique set of a j ∈ C, then it returns the values of a j as an ordered list; – and if there is no a j ∈ C with the above property then it returns False; – and if a j ∈ C is not unique, then it returns a particular solution as a list together with the vectors that are helpful in determining the general solution. We note that if this is the case, then either the user has chosen a small precision and the results are not conclusive or the set of modular forms { f j (z) : 1 ≤ j ≤ l} is linearly dependent. We note that one should use this option with caution. Additionally, if the precision of this SAGE function is set to be at least the Sturm bound, then it will be useful in giving proofs for modular identities, see Example 4.6.2 for an example.

2.3

Newforms

In this section, we introduce the concept of newforms. The ideas, theorems and definitions of this section follow the ideas from Sects. 13.2 and 13.3 of [26]. By Theorem 2.2.2 we have that if f (z) ∈ Mk (0 (N ), χ ) then f (t z) ∈ Mk (0 (t N ), χ ) for all positive integers t. On the other hand it is clear that if f (z) ∈ Mk (0 (N ), χ ) then f (z) ∈ Mk (0 (t N ), χ ) for all positive integers t. But these forms are not ‘new’ in Mk (0 (t N ), χ ), therefore we would like to call them and their linear combinations as ‘oldforms’. Definition 2.3.1 The space of oldforms Mkold (0 (N ), χ ) is defined to be the vector space generated by all forms in  ({ f (z) : f (z) ∈ Mk (0 (N /t), χ )} ∪ { f (t z) : f (z) ∈ Mk (0 (N /t), χ )}) . 1 2 the q-series expansion of the Eisenstein series defined by (2.8) can be given by combining [26, Corollary 8.5.5] with Lemma 1.5.1. For k = 1 and 2 it turns out the Eisenstein series in Mk (0 (N ), χ ) have similar q-series expansions. Therefore to unify the notation for all k ≥ 1 we give the following definition. Definition 2.4.1 We define the weight k Eisenstein series associated to  and ψ by E k (, ψ; t z) := (0) + Nk (, ψ)

∞ 

σk−1 (, ψ; n)q tn ,

n=1

where σk−1 (, ψ; n) :=

⎧ ⎪ (n/d)ψ(d)d k−1 if n ∈ N0 , ⎨

and  Nk (, ψ) :=

Mω M

d|n

⎪ ⎩0

k

if n ∈ / N0 ,

⎞   k  p G(ψ) ⎠ −2k ⎝ G(ω) p k − ω( p) Bk (ω) ⎛

p|L M

with ω := (ψ)o , and Mω is the conductor of ω. For k = 1, 2 the modularity conditions are a little different and given in Theorem 2.4.1. Before stating the Theorem 2.4.1 we fix the notation

32

2 Modular Forms: Definition and Some Properties

 R,ψ (A, B) :=

gcd(A, B) B

k   −

   A B . ψ gcd(A, B) gcd(A, B)

In the statement below we combine the results concerning the Eisenstein series that appear in [26, Chap. 8] and [43, Chap. 7]. Theorem 2.4.1 For all t | N /L M the following statements hold. (a) If k ≥ 2 and (k, , ψ)  = (2, χ1 , χ1 ), then E k (, ψ; t z) ∈ Mk (0 (N ), χ ) and we have Ca/c (E k (, ψ; t z)) = ψ(a)R,ψ (c, t M). (b) If (k, , ψ) = (2, χ1 , χ1 ), then E 2 (χ1 , χ1 ; z) − t E 2 (χ1 , χ1 ; t z) ∈ M2 (0 (N ), χ1 ) and we have Ca/c (E 2 (χ1 , χ1 ; z) − t E 2 (χ1 , χ1 ; t z)) = Rχ1 ,χ1 (c, 1) − tRχ1 ,χ1 (c, t). (c) If k = 1 and  is an even Dirichlet character (that is, (−1) = 1), then E 1 (, ψ; t z) ∈ M1 (0 (N ), χ ) and we have Ca/c (E 1 (, ψ; t z)) = ψ(a)R,ψ (c, t M) + (a)

G()G(ψ)L(ψ, 1) Rψ, (c, t L) · . M L(ψ, 1)

Remark 2.4.1 We note that when k = 1, for the character pairs (, ψ) where ψ is even E 1 (, ψ; t z) is not a modular form in M1 (0 (N ), χ ). For this reason, almost all the results we have in this book have a separate statement for the case k = 1. The results given in Theorem 2.4.1 are very important to our approach. The collection of all Eisenstein series described in Theorem 2.4.1 that are in Mk (0 (N ), χ ) constitutes a basis for E k (0 (N ), χ ). This was first proven by J. R. Weisinger in [56], see [26, Theorems 8.5.17, 8.5.22 and 8.5.23] for refined versions. Theorem 2.4.2 A basis for E k (0 (N ), χ ) is given by (a) the set {E k (, ψ; t z) : (, ψ) ∈ E(N , χ ) and t | N /L M}, except when k = 1 or (k, χ )  = (2, χ1 ); (b) the set {E 2 (χ1 , χ1 ; z) − t E 2 (χ1 , χ1 ; t z) : t | N } ∪ {E 2 (, ψ; t z) : (, ψ) ∈ E(N , χ1 )\{(χ1 , χ1 )} and t | N /L M}, if (k, χ ) = (2, χ1 ); (c) the set {E k (, ψ; t z) : (, ψ) ∈ E(N , χ ), (−1) = 1 and t | N /L M}, if k = 1. Next we give a SAGE function that determines the character pairs that can be used to give a basis for E k (0 (N ), χ ).

2.4

Eisenstein Series

33

SAGE function that determines the character pairs for bases of E k (0 (N ), χ ) Inputs: • k, N: The positive integers k and N , respectively. • chi: The element of DirichletGroup(N) which corresponds to the Dirichlet character χ.

def EisBaseFind(k,N,chi): if chi(-1)!=(-1)ˆk: return [] chars=CharPairsFind(N,chi) out=[] if k==1: for char in chars: if char[0](-1)==1: out.append(char) else: out=chars return out

Output: A list of pairs of primitive Dirichlet characters corresponding to the Eisenstein series which gives a basis for E k (0 (N ), χ ). Each Dirichlet character in the pair is an element of DirichletGroup, where each pair is given as a list as well.

As a result of Theorem 2.4.2 for any f ∈ Mk (0 (N ), χ ) we have   E( f ; z) = a f (, ψ, t)E k (, ψ; t z) (,ψ)∈E(N ,χ) t|N /L M

for some a f (, ψ, t) ∈ C, where if k = 1 the value of a f (, ψ, t) is implied to be 0 whenever (−1) = −1. In Sect. 2.6 we determine a f (, ψ, t) explicitly in terms of the constant terms of f (z). Next, we determine explicit expressions for the Eisenstein series associated with some Dirichlet characters  and ψ. In all these examples we assume that k is a positive integer that satisfies ψ(−1) = (−1)k and for k = 1, we assume  is an even Dirichlet character. Example 2.4.1 If (, ψ) = (χ1 , χ1 ) then we have the classical Eisenstein series ∞

E k (χ1 , χ1 ; t z) = 1 +

−2k  σk−1 (χ1 , χ1 ; n)q tn . Bk (χ1 ) n=1

34

2 Modular Forms: Definition and Some Properties

Example 2.4.2 If gcd(L, M) = 1 then Mω = L M. Additionally, by Lemma 1.4.1, we have 1 G(ψ) G(ψ) = = , G(ω) (M)ψ(L)G() (M)ψ(L)G()G(ψ) therefore we have  E k (, ψ; t z) = (0) +

Lk (M)ψ(L)G()



−2k Bk (ω)

 ∞

σk−1 (, ψ; n)q tn .

n=1

Example 2.4.3 If ψ =  and  is a nonreal Dirichlet character, then we have ω =  2 and Mω = L. Therefore we obtain  ∞    −2k G() σk−1 (, ; n)q tn . E k (, ; t z) = (0) + 2 G( ) Bk ( 2 ) n=1 Example 2.4.4 If both  and ψ are real Dirichlet characters then we have  = χd1 and ψ = χd2 where d1 and d2 are fundamental discriminants which satisfy |d1 | = L and |d2 | = M. d1 d2 If we denote j = gcd(L,M) 2 and let

d3 =

⎧ ⎪ ⎪4 j if 4 | d1 , d2 and d1 d2 /16 ≡ 2 (mod 4), ⎨ ⎪ ⎪ ⎩j

or, if 8 | d1 , d2 and d1 d2 /64 ≡ 3 (mod 4),

otherwise,

then we have ω = χd3 . By using Lemma 1.4.2 we have ⎞ ⎛  k−1/2 k   d3  p ⎠ ⎝ e(d2 ) E k (χd1 , χd2 ; t z) = χd1 (0) +   d2 e(d3 ) p k − χd3 ( p)  ×

−2k Bk (χd3 )

 ∞

p|L M

σk−1 (χd1 , χd2 ; n)q tn .

(2.16)

n=1

Next, we investigate some special cases of this. (a) If gcd(L, M) = 1, then we have d3 = d1 d2 . Therefore (2.16) simplifies to E k (χd1 , χd2 ; t z) = χd1 (0) +

e(d2 ) k−1/2 L e(d3 )



 ∞ −2k σk−1 (χd1 , χd2 ; n)q tn . Bk (χd1 d2 ) n=1

(b) If L = M and 8  L or, L = M, 8 | L and d1 d2 > 0, then j = 1. Therefore in this case (2.16) simplifies to

2.4

Eisenstein Series

35

E k (χd1 , χd1 ; t z) = χd1 (0)

⎞ ⎛   ∞ e(d1 ) ⎝  p k ⎠ −2k + k−1/2 σk−1 (χd1 , χd1 ; n)q tn . L pk − 1 Bk (χ1 ) n=1

p|d1

(c) If we have L = M, 8 | L and d1 d2 < 0, then j = −4. Therefore in this case (2.16) simplifies to ⎞ ⎛   −2k pk −ie(d1 ) ⎝ ⎠ E k (χd1 , χd1 ; t z) = (L/4)k−1/2 p k − χ−4 ( p) Bk (χ−4 ) p|L

×

∞ 

σk−1 (χd1 , χd1 ; n)q tn .

n=1

2.4.2

Constant Terms of Eisenstein Series

In this subsection, we study the constant terms of the Eisenstein series at each cusp. These results are mostly from [13]. To motivate this study we start with an example. This example additionally provides a blueprint to the method used to prove Theorem 2.6.1. Example 2.4.5 Let f (z) ∈ M4 (0 (6), χ1 ). Then by Theorem 2.4.2 we have  f (z) = E( f ; z) + S( f ; z) = at E 4 (χ1 , χ1 ; t z) + S( f ; z). t|6

Below we find the Eisenstein part of f explicitly in terms of the Eisenstein series defined in Sect. 2.4. Because S( f ; z) is a cusp form, Ca/c (S( f ; z)) = 0 for all cusps a/c. Therefore by using Theorems 2.4.1 and 2.2.1 we obtain Ca/c ( f ) =



at Ca/c (E 4 (χ1 , χ1 ; t z)) =

t|6



at Rχ1 ,χ1 (c, t) =

t|6

 t|6

 at

gcd(c, t) t

for all cusps a/c. Letting a/c ∈ {1/1, 1/2, 1/3, 1/6}, we obtain the linear equations a2 a3 a6 + 4 + 4, 4 2 3 6 a3 a6 C1/2 ( f ) = a1 + a2 + 4 + 4 , 3 3 a2 a6 C1/3 ( f ) = a1 + 4 + a3 + 4 , 2 2 C1/6 ( f ) = a1 + a2 + a3 + a6 . C1/1 ( f ) = a1 +

Solving this system of linear equations for a1 , a2 , a3 and a6 we find

4

36

2 Modular Forms: Definition and Some Properties

a1 = a2 = a3 = a6 =

  C1/2 ( f ) C1/3 ( f ) C1/6 ( f ) 27 C1/1 ( f ) − , − + 25 24 34 64   C1/3 ( f ) C1/6 ( f ) 27 −C1/1 ( f ) + C1/2 ( f ) + , − 25 34 34   C1/2 ( f ) C1/6 ( f ) 27 −C1/1 ( f ) + , + C1/3 ( f ) − 25 24 24  27  C1/1 ( f ) − C1/2 ( f ) − C1/3 ( f ) + C1/6 ( f ) . 25

This means for any f (z) ∈ M4 (0 (6), χ1 ) we have  E( f ; z) = at E 4 (χ1 , χ1 ; t z), t|6

where at are as above. This gives Eisenstein part of any modular form in M4 (0 (6), χ1 ). The previous example illustrates how to compute the Eisenstein part of an f (z) ∈ Mk (0 (N ), χ ). The general case, of course, is more challenging. The structure of the constant terms Ca/c (E k (, ψ; t z)) is complicated. Upon numerical experiments, we discover that working with the weighted average of the constant terms, defined by Cc,ρ ( f ) :=

1 φ(c)

c 

ρ(a)Ca/c ( f ),

a=1, gcd(a,c)=1

where ρ is a Dirichlet character, is more convenient. If ρ is chosen carefully, this gives the constant terms a very nice shape. In a way, this is analogous to applying elementary row operations to a set of linear equations to obtain a simpler system. Now we compute weighted averages of constant terms of the Eisenstein series. This result, except for the case k = 1, is from [13]. Theorem 2.4.3 Let ρ be a primitive Dirichlet character with conductor Mρ which satisfies Mρ | c, where c is a positive integer. Then the following statements hold. (a) If k ≥ 2 and (k, , ψ)  = (2, χ1 , χ1 ), then we have R,ψ (c, t M) if ρ = ψ and M | c, Cc,ρ (E k (, ψ; t z)) = 0 otherwise. (b) If (k, , ψ) = (2, χ1 , χ1 ), then we have Cc,ρ (E 2 (χ1 , χ1 ; z) − t E 2 (χ1 , χ1 ; t z)) =

Rχ1 ,χ1 (c, 1) − tRχ1 ,χ1 (c, t) if ρ = χ1 , 0

otherwise.

2.4

Eisenstein Series

37

(c) If k = 1 and  is the even Dirichlet character, that is, (−1) = 1, then we have R,ψ (c, t M) if ρ = ψ and M | c, Cc,ρ (E 1 (, ψ; t z)) = 0 otherwise. Proof If k ≥ 2 and (k, , ψ)  = (2, χ1 , χ1 ), then by Theorem 2.4.1 the constant terms are given by Ca/c (E k (, ψ; t z)) = ψ(a)R,ψ (c, Mt). Using this we obtain Cc,ρ (E k (, ψ; t z)) =

1 φ(c)

c 

ρ(a)Ca/c (E k (, ψ; t z))

a=1, gcd(a,c)=1

⎛

⎜ 1 = R,ψ (c, Mt) ⎜ ⎝ φ(c)

⎞ c  a=1, gcd(a,c)=1

⎟ ρ(a)ψ(a)⎟ ⎠.

If M  c, then R,ψ (c, Mt) = 0. Therefore we have Cc,ψ (E k (, ψ; t z)) = 0. If M | c, Mρ | c and gcd(a, c) = 1 then gcd(a, M) = 1, gcd(a, Mρ ) = 1 and lcm(M, Mρ ) | c. Therefore, if ψ  = ρ, we have 1 φ(c)

c 

ρ(a)ψ(a) = 0.

a=1, gcd(a,c)=1

If ψ = ρ, then we have ψ(a)ψ(a) = 1 for each a coprime to c. That is, we have Cc,ψ (E k (, ψ; t z)) = R,ψ (c, Mt). If (k, , ψ) = (2, χ1 , χ1 ), then, by Theorem 2.4.1, the constant terms are Ca/c (E 2 (χ1 , χ1 ; z) − t E 2 (χ1 , χ1 ; t z)) = Rχ1 ,χ1 (c, 1) − tRχ1 ,χ1 (c, t). The proof now follows similarly. If k = 1 and  is the even Dirichlet character, that is, (−1) = 1, then by Theorem 2.4.1 the constant terms are given by

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2 Modular Forms: Definition and Some Properties

Ca/c (E 1 (, ψ; t z)) = ψ(a)R,ψ (c, Mt) + (a)

G()G(ψ)L(ψ, 1) Rψ, (c, Lt) · . M L(ψ, 1) (2.17)

Since k = 1 and  is the even Dirichlet character, ψ has to be an odd Dirichlet character. Therefore,   = ψ. If L  c then we have Rψ, (c, Lt) = 0, and if L | c then c 

ψ(a)(a) = 0.

a=1, gcd(a,c)=1

In either case, if we take the weighted average of the constant terms in (2.17), the second part sums up to 0. Now the proof follows by arguments similar to the proof of (a) when applied to the first summand on the right hand side of (2.17).  In the next section we use the multiplicative properties of the sum of divisors functions to obtain some results that are useful in the upcoming discussions.

2.5

The Sum of Divisors Function

In this section, we work on the sum of divisors function and derive results that help in obtaining simplified expressions for the Eisenstein part of a modular form. The results of this section are also helpful in computations and determining the asymptotic behavior of the Eisenstein series. We denote by ν p (n) the biggest integer that satisfies p ν p (n) | n. If n is clear from the context we just use ν p to denote ν p (n). We denote the M-free part of n by r M (n), that  is, r M (n) = p ν p (n) ; and r LM (n) denotes the M-full L-free part of n, that is, r LM (n) = 

p

p|n, pM ν p (n)

. Finally, ζ (k) denotes the Riemann zeta function.

p|n, p|M pL

Proposition 2.5.1 Let  and ψ be primitive Dirichlet characters of modulus L and M, respectively. Then for positive integers k the following statements hold. (a) We have ⎧ (ψ( p) p k )ν p +1 − ( p)ν p +1 ⎪ ⎨ if gcd(n, L, M) = 1, ψ( p) p k − ( p) σk (, ψ; n) = p|n ⎪ ⎩ 0 if gcd(n, L, M) > 1.

2.5 The Sum of Divisors Function

39

(b) We have σk (, ψ; n) ψ(r M (n))r M (n)

k

= ψ(gcd(n, L, M))(r LM (n))

 ψ(d) . dk

d|r M (n)

(c) If gcd(n, M) = 1, then we have σk (, ψ; n)  ψ(d) = . ψ(n)n k dk d|n

(d) If gcd(n, L, M) = 1 and k is greater than 1, then we have     σk (, ψ; n) M  ≤ ζ (k) − 1.  − (r (n)) L   ψ(r (n))r (n)k M M (e) If gcd(n, L) and gcd(n, M) = 1 and k is greater than 1, then we have    σk (, ψ; n)   − ψ(n) ≤ ζ (k) − 1.  k n Proof We start by proving (a). The sum of the divisors function is a multiplicative function of n, that is, we have σk (, ψ; n) =



(n/d)ψ(d)d = k

d|n

νp 

( p ν p − j )ψ( p j ) p jk .

(2.18)

p|n j=0

If gcd(n, L, M) > 1, then there exists at least one prime p that divides n, L and M. For that ν p ( p ν p − j )ψ( p j ) p jk = 0, that is, σk (, ψ; n) = 0. If gcd(n, L, M) = prime p we have j=0 1, then for each p | n we have νp  i=0

( p ν p − j )ψ( p j ) p jk =

(ψ( p) p k )ν p +1 − ( p)ν p +1 . ψ( p) p k − ( p)

The proof of (a) follows by using this in (2.18). Next we use the vanishing of Dirichlet characters  and ψ at prime divisors of L and M and obtain

40

2 Modular Forms: Definition and Some Properties

⎛ ⎜ σk (, ψ; n) = ⎜ ⎝

⎞⎛  p|n, p|gcd(L,M)

⎞

νp ⎟⎜  ⎟ νp ⎟ ⎟ ⎜ 0⎠ ⎝ ( p )⎠ ( p ν p −i )ψ( pi ) pik p|n, p|M pL

⎛

⎜ = ψ(r M (n))r M (n)k ⎜ ⎝

 p|n, p|gcd(L,M)

p|n, i=0 pM

⎞⎛

⎞

⎟⎜  ⎟  ψ(d) ⎜ 0⎟ ( p ν p )⎟ , ⎠⎝ ⎠ dk p|n, p|M pL

d|r M (n)

where in the final step we change the variable i to variable ν p − i and use the multiplicative properties. Observing that ψ(r M (n)) is never equal to 0 and using the identity  0 = ψ(gcd(n, L, M)) p|n, p|gcd(L,M)

above concludes the proof of (b). The proof of (c) is by part (b) because if gcd(n, M) = 1 then r M (n) = n. To prove (d), assuming k > 1, we use the the triangle inequality to obtain               ψ(d) ψ(d)  1 =  ≤ − 1 ≤ ζ (k) − 1.     k k k d d d     d|r M (n)

d|r M (n),d>1

d|r M (n),d>1

To conclude the proof we use the identities ψ(gcd(n, L, M)) = ψ(1) = 1 and |(r LM (n))| = 1 in the inequality above. Part (e) is an immediate consequence of part (d), since, in this case, r M (n) = n and  r LM (n) = 1. From part (a) of Proposition 2.5.1 we obtain the following formulas. Corollary 2.5.1 We have σk (, ; n) =

(n)σ (χ1 , χ1 ; n) if gcd(n, L) = 1, 0

if gcd(n, L) > 1,

and σk (, ψ; n) = ψ(n)σk (ψ, ; n). Next, we use Proposition 2.5.1 to write a SAGE function that gives the value of σk (, ψ; n).

2.5 The Sum of Divisors Function

41

SAGE function to compute σk (, ψ; n) Inputs: • k: The positive integer k. • eps, psi: The elements of DirichletGroup(N) which corresponds to the primitive Dirichlet characters  and ψ, respectively. • n: The positive rational number n.

def sigma(k,eps,psi,n): L,M=eps.conductor(),psi.conductor() eps=eps.primitive_character() psi=psi.primitive_character() if k==0 and n not in ZZ: return 0 elif k==0 and n in ZZ: return sum(eps(n/d)*psi(d) for d in divisors(n)) if n not in ZZ: return 0 elif gcd(gcd(L,M),n)!=1: return 0 else: return prod(((psi(ps[0])*ps[0]ˆk)ˆ(ps[1]+1)-eps\ (ps[0])ˆ(ps[1]+1))/(psi(ps[0])*ps[0]\ ˆk-eps(ps[0])) for ps in factor(n))

Output: The value of σk (, ψ; n).

In the remainder of this section, we use these results to write a SAGE function to compute the q-series expansion of the Eisenstein series up to a given precision. First, we give a SAGE function which computes Nk (, ψ) of Definition 2.4.1. SAGE function to compute Nk (, ψ) Inputs: • k: The positive integer k. • eps, psi: The elements of DirichletGroup(N) which corresponds to the primitive Dirichlet characters  and ψ, respectively.

42

2 Modular Forms: Definition and Some Properties

def EisNormalCoeff(k,eps,psi): L1,M1=eps.modulus(),psi.modulus() if L1!=M1: eps=eps.extend(lcm(L1,M1)) psi=psi.extend(lcm(L1,M1)) omega=(eps*psi.bar()).primitive_character() eps=eps.primitive_character() psi=psi.primitive_character() L,M=eps.conductor(),psi.conductor() Mo=omega.conductor() C1=(Mo/M)ˆk LMf=factor(L*M) C2=prod(p[0]ˆk/(p[0]ˆk-omega(p[0])) for p in LMf) C3=-2*k/(omega.bar()).bernoulli(k) if omega.multiplicative_order() 1. Then we have (1, p b , p ν ) =

ν 

S,ψ (1, pi )R,ψ (1, pi )R,ψ ( pi , p b )

i=0

= S,ψ (1, 1)R,ψ (1, 1)R,ψ (1, p b ) + S,ψ (1, p)R,ψ (1, p)R,ψ ( p, p b ).

46

2 Modular Forms: Definition and Some Properties

Putting in the values of R,ψ and S,ψ from (2.20) and (2.21) in the above equation, and assuming b  = 0, we obtain (1, p b , p ν ) =



1 pb

k

 1 k

ψ pb − ψ p b = 0; pb

if b = 0 (recall that a = 0 as well), then we have (1, p b , p ν ) =  (−1)  (−1) −

 k 1 p k − ψ ( p)  (−1) ψ ( p)  (− p) = . p pk

This proves the statement when a = 0. The final case is when a = ν > 0. By (2.21) we have S,ψ ( p ν , pi ) = 0 if i < n − 1. Then we have ν

ν

( p , p , p ) = b

ν 

S,ψ ( p ν , pi )R,ψ ( p ν , pi )R,ψ ( pi , p b )

i=0

= S,ψ ( p ν , p ν−1 )R,ψ ( p ν , p ν−1 )R,ψ ( p ν−1 , p b ) + S,ψ ( p ν , p ν )R,ψ ( p ν , p ν )R,ψ ( p ν , p b ) = − (− p) R,ψ ( p ν−1 , p b ) +  (−1) R,ψ ( p ν , p b ). If b < ν, then, by further manipulations, we obtain ( p ν , p b , p ν ) = − (− p) R,ψ ( p ν−1 , p b ) +  (−1) R,ψ ( p ν , p b )  ν  ν p p + = − pb pb = 0; and if b = ν (recall that a = ν as well), then we have ( p ν , p b , p ν ) = − (− p) R,ψ ( p ν−1 , p b ) +  (−1) R,ψ ( p ν , p b )  k 1  (−1) ψ ( p) +  (−1)  (−1) = − (− p) p =

pk − ψ ( p) . pk

This finishes the proof. The proof of Theorem 2.6.1 hinges on the following orthogonal relation.



2.6

Projections of Modular Forms on the Eisenstein Series

47

Lemma 2.6.1 ([13, Theorem 3]) Let N be a positive integer and A and B be divisors of N . Then we have ⎧ ⎪ if A  = B, ⎪ ⎨0  S,ψ (N , A, C)R,ψ (A, C)R,ψ (C, B) =  p k − ψ( p) if A = B. ⎪ ⎪ C|N ⎩ pk p|N

Proof Using the multiplicativity of Dirichlet characters we put the terms together in the expression and obtain    gcd(A, C) gcd(B, C) k  AC  BC gcd(A, C) gcd(B, C) C|N     AC BC μ ×ψ gcd(A, C) gcd(B, C) gcd(A, C)2    p k + ψ( p) × . pk

(A, B, N ) =

p|gcd(A,B), 0 2, A( f )  = 0 and if for some m coprime to N we have |B f (m)| > ζ (k − 1) − 1, A( f ) then there exist two positive constants B1 and B2 such that B1 n k−1 ≤ |e( f ; n)| ≤ B2 n k−1 hold for all n ≡ m (mod N ).

60

2 Modular Forms: Definition and Some Properties

If needed a pair of suitable B1 and B2 can be determined from the proof below. Proof Using Theorem 2.7.1 and the triangle inequality we have     e( f ; n)     −  B f (n) ≤ (ζ (k − 1) − 1)A f .   n k−1   This yields the inequality      e( f ; n)       (1 − ζ (k − 1))A f + B f (n) ≤  k−1  ≤ (ζ (k − 1) − 1)A f +  B f (n) . n

(2.24)

For each n satisfying |B f (n)| > ζ (k − 1) − 1 A( f )

(2.25)

both sides of the inequality (2.24) is positive. On the other hand, each ψ is a Dirichlet character whose conductor divides N , therefore they are N periodic. Hence B f (n) is N periodic, that is, if (2.25) is satisfied for some m coprime to N then it holds for each n that is congruent to m modulo N . Therefore the theorem follows.  Theorem 2.7.3 If k > 2, A( f )  = 0 and for some m coprime to N |B f (m)| > ζ (k − 1) − 1 A( f ) is satisfied, then we have    a( f ; n)    lim  e( f ; n)  = 1, n→∞ n≡m (mod N ) that is, |e( f ; n)| is asymptotically equal to |a( f ; n)| when n ≡ m (mod N ). If, additionally, a( f ; n) ∈ R for each n ≡ m (mod N ), then we have lim

n→∞ n≡m (mod N )

a( f ; n) = 1, e( f ; n)

that is, e( f ; n) is asymptotically equal to a( f ; n) when n ≡ m (mod N ). Proof By (2.5) we have a( f ; n) = e( f ; n) + s( f ; n). Therefore we have         a( f ; n)   e( f ; n) + s( f ; n)   =  = 1 + s( f ; n)  .     e( f ; n)   e( f ; n) e( f ; n) 

(2.26)

2.7 The Size of the q-Series Coefficients of the Eisenstein Part of a Modular Form

61

By Hecke bound (Theorem 2.1.1) we have s( f ; n) = O(n k/2 ), and, if k > 2 then k − 1 > k/2. Therefore by using Theorem 2.7.2 we obtain    s( f ; n)    lim  e( f ; n)  = 0, n→∞ n≡m (mod N ) that is, we have     

 s( f ; n)  lim  = 0. n→∞ e( f ; n)  n≡m (mod N )

Using this in (2.26) finishes the proof of the first part. As k > 2, if a( f ; n) is real for each n ≡ m (mod N ) then so is e( f ; n) and s( f ; n). Therefore, by the first part of the theorem we have a( f ; n) = ±1. n→∞ e( f ; n) gcd(n,N )=1 lim

Negative sign is impossible since s( f ; n) = O(n k/2 ) and k > 2. Thus, the second part of the statement follows.  Remark 2.7.1 The bounds we use to prove Theorems 2.7.2 and 2.7.3 are generous. Despite that, our tests suggest that it is still quite useful in determining the asymptotical equivalence of a( f ; n) and e( f ; n) in many cases. Remark 2.7.2 In special cases we may be able to derive the size of e( f ; n) even if n is not coprime to N . Below we would like to draw attention to two cases where this may be the case: (a) Theorems 2.7.1, 2.7.2 and 2.7.3 hold if we replace N by N  = lcm({lcm(L, M) : (, ψ) ∈ E(N , χ ) and b f (, ψ, 1)  = 0}). In some applications using this may provide results for more equivalence classes in the aforementioned theorems. (b) In the special cases, navigating through b f (, ψ, t)σk−1 (, ψ; n/t) carefully, we may be able to say something about the size of e( f ; n), see Example 3.4.3 or [1] for executions of this idea. Next, we give a SAGE function that checks if the conditions of Theorem 2.7.3 are satisfied, that is, checks if e( f ; n) is asymptotically equivalent to a( f ; n) on certain arithmetic progressions modulo N of n.

62

2 Modular Forms: Definition and Some Properties

SAGE function that determines if a( f ; n) is asymptotically equal to e( f ; n) We note that the inputs below assumes that q-series coefficients of the Eisenstein part of f is given by  e( f ; n) = b f (, ψ, t)σk−1 (, ψ, t). (,ψ)∈E(N ,χ)

Inputs: • k, N: The positive integers k and N , respectively. • chi: The element of DirichletGroup(N) which corresponds to the Dirichlet character χ. • bfts: A list of lists with the entries [, ψ, t, b f (, ψ, t)] for each (, ψ) ∈ E(N , χ ) and t | N /L M, where  and ψ are elements of DirichletGroup(N), t is a positive integer and b f (, ψ, t) is a complex number.

def EisPartAsymptotic(k,N,chi,bfts): if k(zeta(k-1)-1): print('The formula is asymptotical if n is\ congruent to ',m[i],' modulo ',N) else: print('This program cannot determine the\ asymptotic behavior if n is congruent to',\ m[i],' modulo ',N)

2.7 The Size of the q-Series Coefficients of the Eisenstein Part of a Modular Form

63

Output: None, only prints the integers m between 0 and N that are coprime to N for which    a( f ; N n + m)    = 1, lim n→∞  e( f ; N n + m)  holds, and the integers m between 0 and N that are coprime to N for which the above limit cannot be determined by using Theorem 2.7.3.

This finishes a rather lengthy introduction to modular forms. The rest of the book is devoted to applications of the results of this chapter.

3

Application: Quadratic Forms

3.1

The Modularity and the Constant Terms

Let j be a positive integer and Q be an j × j positive definite matrix, that is, all eigenvalues of Q are positive. Additionally, assume that Q is symmetric with integer entries and that the diagonal entries of Q are even. Then f Q (X ) :=

1 T X QX 2

defines a positive definite quadratic form with integer coefficients, where ⎛ ⎞ x1 ⎜ .. ⎟ X := ⎝ . ⎠ = (x1 , . . . , x j ) ∈ Z j xj and X T denotes the transpose of X . We denote by r (Q; n) the number of representations of an integer n by f Q (X ), that is,  1 r (Q; n) := # X ∈ Z j : X T Q X = n . 2 The generating function of the number of representations by the quadratic form f Q (X ) is given by

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-32629-5_3.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 Z. S. Aygin, Introduction to Applications of Modular Forms, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-32629-5_3

65

66

3 Application: Quadratic Forms

θ (Q; z) :=

∞

r (Q; n)q n =



q f Q (X ) .

X ∈Z j

n=0

The function θ (Q; z) is referred to as the theta function corresponding to the quadratic form f Q (X ). If j is even, then θ (Q; z) is an integer weight modular form and if j is odd, then θ (Q; z) is a half integer weight modular form, see [55, Theorem 10.1]. The latter is out of the scope of this book. Therefore from now on we direct our attention to the case when j is even. We first give the statement on the modularity of θ (Q; z) and on its constant coefficients at cusps. This statement is a combination of [55, Theorem 10.1] and [55, Eq. (10.2)]. Theorem 3.1.1 Let k be a positive integer and Q a 2k × 2k positive definite symmetric matrix with integer entries where the diagonal entries of Q are even. Then we have θ (Q; z) ∈ Mk (0 (N ), χ D ), where N is the smallest positive integer such that the matrix N × Q −1 has even diagonal entries and  (−1)k S if S is odd and (−1)k S ≡ 1 (mod 4), D := (−1)k 4S otherwise, where S denotes the squarefree part of det(Q). The constant terms of θ (Q; z) are given by Ca/c (θ (Q)) =

(−i)k Ga/c (Q), √ ck det(Q)

where Ga/c (Q) :=

c−1

x1 =0

···

c−1

e2π i f Q (X )a/c .

(3.1)

x2k =0

The following SAGE function uses Theorem 3.1.1 to check the modularity of a given quadratic form. SAGE function to determine modularity of θ (Q; z) Input: • Q: The matrix Q with integer entries. def isQFModular(Q): if Q.determinant()==0: print('Warning: Matrix is singular!')

3.1 The Modularity and the Constant Terms

67

return False if Q!=Q.transpose(): print('WARNING: Matrix is not symmetric.') return False if all(eig>0 for eig in Q.eigenvalues()): print( 'Quadratic form is positive definite.') else: print('WARNING: Quadratic form is NOT positive\ definite.') return False k=len(Q[0])/2 if (k in ZZ) and all((Q[i][i])%2==0 for i in\ range(len(Q[0]))): Qinv=Q.inverse() N=1 for i in range(len(Qinv[0])): N=lcm(Qinv[i][i].denominator(),N) if all((N*Qinv[i][i])%2==0 for i in\ range(len(Qinv[0]))): N=N else: N=2*N print('The theta function of Q is modular form of') print( 'Level ', N) print('Weight ', k) S=squarefree_part(Q.determinant()) if ((-1)ˆk*S)%4==1: char=(-1)ˆk*S print( 'Character ', char) else: char=(-1)ˆk*4*S print( 'Character ', char) return [k,N,char] else: print ('WARNING: The theta function of Q is not an\ integer weight modular form.') return False

Output: If θ (Q; z) is an integer weigh modular form in Mk (0 (N ), χ D ) then it returns [k, N , D] in a list as well as printing those values out; otherwise it returns False.

In the next section, we discuss Ga/c (Q), the Gauss sums associated with quadratic forms, and E(θ (Q); z), the Eisenstein part of θ (Q; z).

68

3.2

3 Application: Quadratic Forms

Gauss Sums Associated with Quadratic Forms and the Eisenstein Part of θ( Q; z)

In this section, we study Gauss sums associated with quadratic forms and the Eisenstein part of θ (Q; z). We start with the Gauss sums. Definition 3.2.1 The finite sum Ga/c (Q) defined in (3.1) is called the Gauss sum associated with the quadratic form f Q at a/c . The challenging part in computing Ca/c (θ (Q)) is computing Ga/c (Q). Our primary objective for the rest of this chapter is to compute Ca/c (θ (Q)) for various quadratic forms, so that we can apply Theorem 2.6.1 to obtain formulas concerning r (Q; n). The computation of Ga/c (Q) in various settings is a lively research topic in number theory, fur further reading see [22]. The following SAGE function computes Ga/c (Q). SAGE function to compute Ca/c (θ (Q; z)) Inputs: • Q: The matrix Q with integer entries. • a, c: The coprime integers a and c. def QFConstantTerm(Q,a,c): k=Q.nrows() if c==1: return Iˆ(k/2)/cˆ(k/2)/sqrt(Q.det()) all_cases=cˆk X=matrix([[0 for i in range(c)]]) result=0 increment=0 while increment < all_cases: X=increment.digits(c) l=len(X) if len(X)0: if gcd(a-i1*gcd(c,N/c),c)==1: i2=i2+1 i1=i1+1 if i2==1 and gcd(a,c)==1: Cstc.update({a: QFConstantTerm(Q,a,c)}) Cst.update({c: Cstc}) Cst.update({N: {1:1}}) ConstantsAll={} for char in chars: Constants={} eps,psi=char[0],char[1] L=eps.conductor() M=psi.conductor() for c in divisors(N/L/M): val=0 for a in list(Cst[c*M].keys()): val+=QQbar(psi(a)).radical_expression()\ *Cst[c*M][a] Constants.update({c*M: val/euler_phi(gcd(c*M\ ,N/(c*M)))}) ConstantsAll.update({psi: Constants}) return EisPart(k,N,chi,ConstantsAll,bftout=bftout\ ,prec=prec,chars=chars)

Output: • If the input bftout is not 0, then it returns a list where each element in the list is a list of elements [, ψ, t, bθ(Q) (, ψ, t)] where  and ψ are elements of DirichletGroup. • If the input bftout is 0, and the input prec is a positive integer, then it returns the q-series expansion of E(θ (Q); z) up to and including the term q pr ec .

Example 3.2.2 Let ⎛

4 ⎜0 Q=⎜ ⎝0 0

0 4 0 0

0 0 6 2

⎞ 0 0⎟ ⎟. 2⎠ 6

The quadratic form corresponding to Q is f Q = 2x12 + 2x22 + 3x32 + 3x42 + 2x3 x4 , which was studied in [8, Theorem 1.1]. We run the SAGE commands

3.2

Gauss Sums Associated with Quadratic Forms and the Eisenstein Part of θ(Q; z)

71

Q=matrix([[4,0,0,0], [0,4,0,0], [0,0,6,2], [0,0,2,6]]) EisPartofQF(Q)

and from the output, we determine that θ (Q; z) ∈ M2 (0 (32), χ8 ) and we obtain the Eisenstein part of the theta function as E(θ (Q; z)) = 1 +

∞

(−σ1 (χ1 , χ8 ; n) + σ1 (χ1 , χ8 ; n/2) − 2σ1 (χ1 , χ8 ; n/4)

n=1

−σ1 (χ−4 , χ−8 ; n) + 2σ1 (χ8 , χ1 ; n)) q n . By Lemma 2.1.3 we have dim(S2 (0 (32), χ8 )) = 0, therefore for all n > 0 the above modular identity yields the formula r (Q; n) = − σ1 (χ1 , χ8 ; n) + σ1 (χ1 , χ8 ; n/2) − 2σ1 (χ1 , χ8 ; n/4) − σ1 (χ−4 , χ−8 ; n) + 2σ1 (χ8 , χ1 ; n). This agrees with the result of [8, Theorem 1.1]. Please beware of the limitations of these two SAGE functions. Depending on the computing power there is a very narrow window for k and N for this function to execute in a reasonable amount of time. We tested this in an M2 MacBook Air with 8GB RAM, and we find that if N and k satisfy N k ≤ 644 then the runtime was acceptable (less than 5 mins). There are certain quadratic forms whose associated Gauss sums are known. Using these known values of Gauss sums associated with certain families of quadratic forms we devise a remedy to solve the runtime problem for larger k and N in certain cases. Next, we discuss this.

3.2.1

Separable Quadratic Forms

In this subsection, we introduce the concept of separable quadratic forms. Definition 3.2.2 Let k be a positive integer and Q be a k × k positive definite symmetric matrix with integer entries where the diagonal entries of Q are even. If Q is a block diagonal matrix, then we call the quadratic form f Q associated with Q a separable (positive definite) quadratic form.

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3 Application: Quadratic Forms

We note that this separation is in terms of variables of corresponding quadratic forms as demonstrated by the next example. This helps with breaking the Gauss sum into manageable sub-Gauss sums, see Lemma 3.2.1. Example 3.2.3 (A separable quadratic form) Let ⎛

2 ⎜1 Q=⎜ ⎝0 0

1 2 0 0

0 0 2 1

⎞ 0

0⎟ ⎟ = Q1 0 , 0 Q2 1⎠ 4.



21 21 where Q 1 = and Q 2 = , and 0’s denotes the 2 × 2 matrix with all entries 12 14 equal to 0. Then we write f Q = f Q1 + f Q2 , where f Q = x12 + x22 + x1 x2 + x32 + 2x42 + x3 x4 , f Q 1 = x12 + x22 + x1 x2 , and f Q 2 = x32 + 2x42 + x3 x4 . Next, we prove a result that helps in the computation of Gauss sums associated with separable quadratic forms. Lemma 3.2.1 If f Q is a separable positive definite quadratic form with ⎛

Q1 ⎜0 ⎜ Q=⎜ . ⎝ .. 0

0 Q2 .. . 0

0 0 .. . ···

··· ··· .. . 0

⎞ 0 0 ⎟ ⎟ .. ⎟ , . ⎠ Qw

where Q j are k j × k j matrices and the matrix 0 in jth row denotes the k j × k j 0 matrix, then we have Ga/c (Q) =

w  j=1

Ga/c (Q j ).

3.3

A Family of Binary Quadratic Forms

73

Proof Since f Q is separable, we write fQ =

w

fQ j .

j=1

Therefore we have Ga/c (Q) =

c−1

···

x1 =0

=

c−1

= =

x1 =0 w 

e2π i f Q (X )a/c

x2k =0

···

x1 =0 c−1

c−1

c−1

e

2π i



w i=1

 f Q j a/c

x2k =0

···

c−1  w

e

2π i f Q j a/c

x2k =0 j=1

Ga/c (Q j ),

j=1

where the last line is due to the fact that Q j and Q j  do not share any variable unless j = j  .  The rest of this chapter is devoted to working out Ga/c (Q) for certain families of quadratic forms f Q , then giving the Eisenstein part of θ (Q; z) and finally applying these results to various problems in number theory.

3.3

A Family of Binary Quadratic Forms

Let R be a positive integer and α be a divisor of R. Let

2α 1 B(R, α) := . 1 2R/α Then the quadratic form corresponding to B(R, α) is f B(R,α) = αx12 + x1 x2 + (R/α)x22 . Theorem 3.3.1 If R is a positive integer and if α is a divisor of R, then we have θ (B(R, α); z) ∈ M1 (0 (4R − 1), χ−S ), where S is the squarefree part of 4R − 1. We additionally have

74

3 Application: Quadratic Forms c−1

c−1

−i 2 2 Ca/c (θ (B)) = √ e2π i(αx1 +x1 x2 +(R/α)x2 )a/c . c 4R − 1 x =0 x =0 1

2

Proof We have det(B(R, α)) = 4R − 1 and B(R, α)−1 =



1 2R/α −1 . 4R − 1 −1 2α

Therefore, the diagonal entries of (4R − 1)B(R, α)−1 are even. Since det(B(R, α)) = 4R − 1 ≡ 3 (mod 4), and square of any odd number is congruent to 1 modulo 4, we have the squarefree part of 4R − 1, say S, is also congruent to 3 modulo 4. That is, −S ≡ 1 (mod 4). Therefore by Theorem 3.1.1 we have the desired result.  Now finding the Eisenstein part of θ (B(R, α); z) for each R and α | R is a straightforward application of Theorem 2.6.1. Next, we illustrate this by discussing the cases when R = ( p + 1)/4 where p is a prime congruent to 3 modulo 4. We note that the following result goes back to Dirichlet (when n and p are coprime), see [28, p. 78], and see [33, Theorem 9.1] for an elementary proof (including the cases when n and p are not coprime). Theorem 3.3.2 Let R = ( p + 1)/4 where p is a prime congruent to 3 modulo 4. Then for all divisors α of R we have ⎛ ⎞ ∞



2 ⎝ χ− p (d)⎠ q n . (3.2) E(θ (B(R, α)); z) = 1 − B1 (χ− p ) n=1

d|n

Proof Let R = ( p + 1)/4 where p is a prime congruent to 3 modulo 4. Then for all positive divisors α of R we have θ (B(R, α); z) ∈ M1 (0 ( p), χ− p ). On the other hand we have C1/ p (θ (B(R, α); z)) = 1. Therefore, we obtain the desired result by applying Theorem 2.6.1.  Special cases of Theorem 3.3.2 have interesting connections. We discuss them below. Example 3.3.1 If p = 3, 7, 11, 19, 43, 67 or 163, then dim(S1 (0 ( p), χ− p )) = 0. That is, the formula (3.2) is an exact formula for θ (B(R, α); z). On the other hand we compute that B1 (χ− p ) = −1/3 if p = 3, and B1 (χ− p ) = −1 if p = 7, 11, 19, 43, 67 or 163. Putting these together for n > 0 we have

3.3

A Family of Binary Quadratic Forms

75

r (B(1, 1); n) = 6



χ−3 (d),

d|n

and r (B(( p + 1)/4, α); n) = 2



χ− p (d),

d|n

where p = 7, 11, 19, 43, 67 or 163, and α is any divisor of ( p + 1)/4. Example 3.3.2 If p = 23, 31, 59, 83, 107, 139, 211, 307, 379 or 499, then dim(S1 (0 ( p), χ− p )) = 1. In each of these cases, except p = 211, the number R = ( p + 1)/4 has 4 divisors. Let α1 = 1, and β be a divisor of R different than 1 or R. Then we have r (B(R, 1); 1) = 2,

(3.3)

r (B(R, β); 1) = 0.

(3.4)

and

That is, θ (B(R, 1))  = θ (B(R, β)). As the Eisenstein parts of θ (B(R, 1)) and θ (B(R, β)) are equal, we have that the nonzero form θ (B(R, 1)) − θ (B(R, β)) is in S1 (0 ( p), χ− p ). Since the dimension of the space is 1, by Remark 2.3.1 the difference θ (B(R, 1)) − θ (B(R, β)) is a newform in S1new (0 ( p), χ− p ). From (3.3) and (3.4) we obtain ∞

1 a( f ( p; z); n)q n , (θ (B(R, 1)) − θ (B(R, β))) = q + 2 n=2

that is, N(1, p, χ− p ; z) =

1 (θ (B(R, 1)) − θ (B(R, β))). 2

If p = 211, then we have R = (211 + 1)/4 = 53 a prime, that is, it has only two divisors that yield the same quadratic form. We consider the modular form ⎞ ⎛ ∞ 2 ⎝

χ− p (d)⎠ q n ∈ S1 (0 (211), χ−211 ). θ (B(53, 1); z) − 1 − 3 n=1

d|n

We have r (B(53, 1); 0) = 1, and

76

3 Application: Quadratic Forms

r (B(53, 1); 1) = 2. Therefore

⎞ ⎞ ⎛ ∞



3 2 ⎝ N(1, 211, χ−211 ; z) = ⎝θ (B(53, 1); z) − 1 − χ− p (d)⎠ q n ⎠ . 4 3 ⎛

n=1

d|n

Below we briefly discuss congruence relations modulo p between the representation numbers of these binary quadratic forms and q-series coefficients of some cusp forms of S( p+1)/2 (0 (1), χ1 ). Recall that (z) defined by (2.15) is the normalized newform in M12 (0 (1), χ1 ). The q-series coefficients of (z) are denoted by τ (n). The function τ (n) is called Ramanujan’s tau function. From a result of Serre in [51] when p = 23 we have a(N(1, 23, χ−23 ; z); n) =

1 (r (B(6, 1); n) − r (B(6, 2); n)) ≡ τ (n) (mod 23). 2

Next, let us denote the q-series coefficients of (z)E 4 (χ1 , χ1 ; z) by τ16 (n), which is the weight 16 analog of the Ramanujan-τ function. When p = 31, in [19] it is proven that a(N(1, 31, χ−31 ; z); n) =

1 (r (B(8, 1); n) − r (B(8, 2); n)) ≡ τ16 (n) (mod 31). 2

The next prime p for which dim(S1 (0 ( p), χ− p )) = 1 is 59. Our calculations suggest that the following equivalence relation modulo 59 holds: a(N(1, 59, χ−59 ; z); n) =

1 (r (B(15, 1); n) − r (B(15, 3); n)) ≡ τ30 (n) (mod 59), 2 (3.5)

where τ30 (n) are q-series coefficients of the cusp form (z)(E 8 (χ1 , χ1 ; z)E 10 (χ1 , χ1 ; z) + 44(z)E 6 (χ1 , χ1 ; z)). To the best of our knowledge, the identity (3.5) has neither appeared before nor been proven yet. We leave the proof of this to the interested reader as an open problem. On the other hand, we may have similar equivalence relations between representation numbers and qseries coefficients of weight ( p + 1)/2 cusp forms modulo p, for p = 83, 107, 139, 211, 307, 379, 499, . . . It should be an interesting challenge to try and write these weight ( p + 1)/2 cusp forms in terms of Eisenstein series and eta quotients. (See Chap. 4 for the definition of eta quotients.)

3.4

Diagonal Quadratic Forms

3.4

77

Diagonal Quadratic Forms

In this section, we discuss diagonal quadratic forms. These forms are positive definite quadratic forms whose corresponding matrix is diagonal. In Sects. 3.4.1 and 3.4.2, we set the stage for the main result, which is presented in Sect. 3.4.3. Our results in Theorem 3.4.3 are explicit enough to devise a very fast SAGE function that determines the Eisenstein part of any given diagonal form. In the following subsection, we introduce the Kronecker symbol in a more general way, as these symbols appear in the formulas for Gauss sums associated with diagonal quadratic forms.

3.4.1

Generalized Kronecker Symbol

Before we delve into diagonal quadratic forms, we extend the definition of the Kronecker symbol to include all integers. Let a and n be integers. If the prime decomposition of n is given by  n = (−1)ν p ν p (n) , p

then we have

a  n where the symbol

a  n K

K

:=

a −1

ν   a ν2 (n) K 2 K

 p≡1 (mod 2)

ν p (n) a , p L

for n = −1, 2, and 0 is given by

⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨0 a  = 1 n K ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎩ 0

if n = −1 and a > 0, if n = −1 and a < 0, if n = 2 and a ≡ 0 (mod 2), if n = 2 and a ≡ ±1 (mod 8), if n = 2 and a ≡ ±3 (mod 8), if n = 0 and a = ±1, if n = 0 and a  = ±1,

  and for odd primes p the symbol

a p L

denotes the Legendre symbol (see Chap. 1 for the   definition). Note that if a is a quadratic discriminant, then an K = χa (n), therefore it is   a primitive Dirichlet character. But the Kronecker symbol an K is not always a Dirichlet 3 character, as it is not always periodic. For example n K is not periodic. In the context of   modular forms and Eisenstein series we always use χd notation. The an K notation appears

78

3 Application: Quadratic Forms

in constant terms of certain modular forms and in this context they do not have to be Dirichlet characters. So to make this distinction we opted to use two different notations. However, when they appear together in the applications we convert as many terms as possible to χd notation by using the properties given in the following lemma. Lemma 3.4.1 Let d be a quadratic discriminant. Then the following statements hold.     n (a) If d is odd, then we have |d| = χd (n) and dn K = χd (n). K (b) Let a be an integer. If ad > 0 and the squarefree parts of a and d agree, then we have a  n K = χd (n) (mod lcm(a, d)). Proof Proofs can be obtained by using the definition and are left as an exercise.

3.4.2



Constant Terms of Theta Functions Corresponding to the Diagonal Quadratic Forms

We now turn our attention to diagonal quadratic forms. Let ⎛

2α1 ⎜ 0 ⎜ D(α1 , . . . , α2k ) := ⎜ . ⎝ .. 0

0 2α2 .. . 0

0 0 .. . 0

⎞ ··· 0 ··· 0 ⎟ ⎟ .. ⎟ , .. . ⎠ . · · · 2α2k

(3.6)

where α j are positive integers. Then we have

1 T α j x 2j . X DX = 2 2k

fD =

j=1

Thus, f D is a separable quadratic form with the diagonal blocks Q j being the 1 × 1 matrix   2α j . Therefore by Lemma 3.2.1 the value of Ga/c (D) can be given by studying the Gauss sum defined by G(α, β) :=

β−1

e2π ix

2 α/β

.

x=0

By [22, Theorems 1.5.2 and 1.5.4] for gcd(α, β) = 1 we have

(3.7)

3.4

Diagonal Quadratic Forms

79

⎧ ⎪ 0 ⎪ ⎪   √ ⎪ ⎪ α ⎪ ⎪ β ⎪ ⎪ ⎨ β K √ α β G(α, β) = i β K  ⎪ ⎪ √ β ⎪ ⎪ β (1 + i) α ⎪ ⎪  K √ ⎪ ⎪ ⎪ ⎩(1 − i) β β α K

if β ≡ 2 (mod 4), if β ≡ 1 (mod 4), if β ≡ 3 (mod 4),

(3.8)

if β ≡ 0 (mod 4) and α ≡ 1 (mod 4), if β ≡ 0 (mod 4) and α ≡ 3 (mod 4).

and if gcd(α, β)  = 1, then

α β . G(α, β) = gcd(α, β)G , gcd(α, β) gcd(α, β)

Using this result we write the SAGE function that computes G(α, β) as follows. SAGE function to compute G(α, β) Inputs: • alp, bet: The integers α and β, respectively. def G(alp,bet): alp1,bet1=alp/gcd(alp,bet),bet/gcd(alp,bet) if bet1%4==2: return 0 if bet1%4==1: return gcd(alp,bet)*kronecker(alp1,bet1)*sqrt(bet1) if bet1%4==3: return I*gcd(alp,bet)*kronecker(alp1,bet1)\ *sqrt(bet1) if bet1%4==0 and alp1%4==1: return (1+I)*gcd(alp,bet)*kronecker(bet1,alp1)\ *sqrt(bet1) if bet1%4==0 and alp1%4==3: return (1-I)*gcd(alp,bet)*kronecker(bet1,alp1)\ *sqrt(bet1)

Output: The value of G(α, β).

Next we use Lemma 3.2.1 and (3.7) to give a formula for Ga/c (D).

(3.9)

80

3 Application: Quadratic Forms

Proposition 3.4.1 If D is as in (3.6), then for all cusps a/c we have   2k    αja c  ,   , gcd α j , c G Ga/c (D) = gcd α j , c gcd α j , c j=1 where the values of G(α, β) are given by (3.8) Proof As explained above f D is a separable positive definite quadratic form with blocks   Q j = 2α j . Therefore by Lemma 3.2.1 and (3.7), we have Ga/c (D) =

2k 

G(α j a, c).

(3.10)

j=1

Then we use (3.9) in (3.10) to obtain the desired result.



Our next result concerns the modularity of θ (D; z) and the constant terms of θ (D; z) at each cusp a/c. Theorem 3.4.1 If D is as in (3.6), then θ (D; z) ∈ Mk (0 (4N ), χ D ) where N = lcm(α1 , . . . , α2k ),

(3.11)

and D :=

 (−1)k S

if S is odd and (−1)k S ≡ 1 (mod 4),

(−1)k 4S otherwise,

with S denoting the squarefree part of Ca/c (θ (D; z)) =

−i 2c

2k

j=1 α j . The constant terms of θ (D; z) are given by

   

k  2k gcd α j , c αja c  ,   . G √ αj gcd α j , c gcd α j , c j=1

Proof First the inverse of D is given by

D−1

⎛ ⎞ 1/2α1 0 0 · · · 0 ⎜ 0 1/2α2 0 · · · 0 ⎟ ⎜ ⎟ =⎜ . .. ⎟ . .. .. .. ⎝ .. . ⎠ . . . 0 0 0 · · · 1/2α2k

We observe that the diagonal entries of 4 lcm(α1 , . . . , α2k )D−1 are even. Therefore by Theorem 3.1.1 the level of the space is 4 lcm(α1 , . . . , α2k ).

3.4

Diagonal Quadratic Forms

81

To compute the character, we find that det(D) = 22k

2k 

αj,

(3.12)

j=1

and that the squarefree part of the right side of the equation in (3.12) is equal to the squarefree  part of 2k j=1 α j . Therefore by Theorem 3.1.1 the character of the space is as described in the statement. Again by Theorem 3.1.1, the constant terms of θ (D; z) are given by Ca/c (θ (D)) =

(−i)k Ga/c (D). √ ck det(D)

Therefore using (3.12) and Proposition 3.4.1, we obtain the second part of the statement.  The following SAGE function computes Ca/c (θ (D)) using the results of Theorem 3.4.1. SAGE function to compute Ca/c (θ (D)) Inputs: • R: A list of the positive integers α1 , …, α2k . • a, c: The coprime integers a and c, respectively. def DiagonalFirstTerm(R,a,c): k,det=len(R)/2,prod(2*r for r in R) return (-I/c)ˆk/sqrt(det)*prod(G(a*r,c) for r in R)

Output: The value of Ca/c (θ (D)).

Our final result in this subsection is very helpful in speeding up the calculations as it paves the way to prove that the Eisenstein series associated with nonreal Dirichlet characters do not appear in the Eisenstein part of θ (D). Theorem 3.4.2 Let D be as in (3.6), and N be as in (3.11). Let ψ be a nonreal Dirichlet character of modulus M and c be a divisor of N . If M | c, then we have Cc,ψ (θ (D)) = 0.

82

3 Application: Quadratic Forms

Proof We start by recalling that Cc,ψ (θ (D)) =

Below we show that

(−i)k √ ck det(D)

c

c

ψ(a)Ga/c (D).

a=1 gcd(a,c)=1

ψ(a)Ga/c (D) = 0,

a=1 gcd(a,c)=1

whenever ψ is a nonreal Dirichlet character of modulus M, where M | c. Recall that we have   2k    αja c  ,   . gcd α j , c G Ga/c (D) = gcd α j , c gcd α j , c j=1 If gcd(a, c) = 1, then by (3.8), we have ⎧ ⎪ if c ≡ 2 (mod 4), ⎪0 ⎨ a  G (aα, c) = G c) if c ≡ 1 (mod 2), (α, c K ⎪ ⎪ ⎩e(a)−α  c  G (α, c) if c ≡ 0 (mod 4). a K

(3.13)

  Therefore, if there is an α j such that c/gcd α j , c ≡ 2 (mod 4), then Ga/c (D) = 0 for all a coprime to c. Hence, the result is trivial in these cases. Below we disregard these cases. We also assume that 2  gcd(α1 , . . . , α2k ) for the first part of the proof. We discuss the case 2 | gcd(α1 , . . . , α2k ) at the end of the proof. Before moving on we clarify that if c ≡ 2 (mod 4), then there exists at least one α j such   that c/gcd α j , c ≡ 2 (mod 4), since 2  gcd(α1 , . . . , α2k ). Therefore below we study the   cases when c  ≡ 2 (mod 4), where there are no α j such that c/gcd α j , c ≡ 2 (mod 4). Using (3.13) we obtain Ga/c (D) = e(a) A ρD (c; a)G1/c (D),

(3.14)

where 2k 

ρD (c; a) :=

j=1 c/gcd(α j ,c)≡1 (mod 2)

×



a 

c/gcd α j , c

2k  j=1 c/gcd(α j ,c)≡0 (mod 4)



  K

  c/gcd α j , c , a K

3.4

Diagonal Quadratic Forms

83

and 2k

A := −

j=1 c/gcd(α j ,c)≡0 (mod 4)

αj  . gcd α j , c

If c is odd then, A = 0 and ρD (c; a) is a real Dirichlet character with modulus dividing c. That is, we have c

ψ(a)Ga/c (D) = G1/c (D)

a=1 gcd(a,c=1)

c

ψ(a)ρD (c; a).

a=1 gcd(a,c=1)

As ψ is a nonreal Dirichlet character, we have ψ  = ρD , and since, M | c, the desired result follows from Lemma 1.3.1. If c ≡ 0 (mod 4), by (3.14) we have c

a=1 gcd(a,c=1)

ψ(a)Ga/c (D) = G1/c (D)

ψ(a)e(a) A ρD (c; a)

a=1 gcd(a,c)=1

⎛ ⎜ ⎜ = G1/c (D) ⎜ ⎜ ⎝

c

c

a=1 gcd(a,c)=1 a≡1 (mod 4)

ψ(a)ρD (c; a) + I

⎞ A

c

a=1 gcd(a,c)=1 a≡3 (mod 4)

⎟ ⎟ ψ(a)ρD (c; a)⎟ ⎟. ⎠

Now ρD is a real Dirichlet character with modulus dividing c except maybe the cases 2 || c and 4 || c. If 2 || c, then c ≡ 2 (mod 4), that is, by the arguments at the beginning of the proof   the desired result is trivial. If 4 || c and if there is no α j with c/gcd α j , c ≡ 2 (mod 4) then ρD (c; ∗) = χ1 (4; ∗)χ (∗), where χ(∗) is a Dirichlet character of modulus dividing c/4. This means ρD (c; ∗) is a real Dirichlet character of modulus dividing c. Now since 4 | c, ρD is a real Dirichlet character with modulus dividing c, and ψ is a complex valued Dirichlet character with modulus dividing c, the character ρD ψ is not equal to χ1 or χ−4 . Therefore the desired result follows from Corollary 1.3.1. If 2ν || gcd(α1 , . . . , α2k ), then fix β j := α j /2ν . Therefore we have θ (D(α1 , . . . , α2k ); z) = θ (D(β1 , . . . , β2k ); 2ν z). Now since 2  gcd(β1 , . . . , β2k ), the proof follows from the first part and (2.9).



84

3 Application: Quadratic Forms

3.4.3

An Explicit Formula for Representation Numbers of Diagonal Forms

By Theorem 3.4.2, in the Eisenstein part of θ (D), only Eisenstein series associated with real Dirichlet characters are involved. Using this we give the Eisenstein part of θ (D) in Theorem 3.4.3 below. This refined theorem has a tremendous computational advantage. Because determining the real Dirichlet character pairs in E(N , χ ) is much faster than determining all the pairs in E(N , χ ). For example for the diagonal quadratic form D(1, 1, 1, 1, 1, 1, 29, 100) a SAGE function written without using Theorem 3.4.3 takes about 197 s to compute the Eisenstein part of the corresponding theta function, whereas the SAGE function we give on page 89 takes about 16 s to do the same (both on an M2 MacBook Air with 8GB of RAM). Let us denote the real Dirichlet character pairs in E(N , χ ) by ER (N , χ ), then we have  d1 d2 = d , ER (N , χd ) = (χd1 , χd2 ) : d1 d2 | N , gcd(d1 , d2 )2 where d1 and d2 are quadratic discriminants. Theorem 3.4.3 Let k be a positive integer and D be as in (3.6). If we let N = 4lcm(α1 , . . . , α2k ), then the following statements hold.  k (a) If the squarefree part of 2k j=1 α j is odd, say D, and 1 < k satisfies χ Dq (−1) = (−1) , meaning either D ≡ 1 (mod 4) and k ≡ 0 (mod 2) or D ≡ 3 (mod 4) and k ≡ 1 (mod 2), then we have



E(θ (D); z) = aθ (, ψ, t)E k (, ψ; t z) (,ψ)∈ER (N ,χ Dq ) t|N /L M

where aθ (, ψ, t) are as in (2.19).  (b) If the squarefree part of 2k j=1 α j is odd, say D, and 1 < k satisfies χ−4Dq (−1) = k (−1) , meaning either D ≡ 1 (mod 4) and k ≡ 1 (mod 2) or D ≡ 3 (mod 4) and k ≡ 0 (mod 2), then we have



E(θ (D); z) = aθ (, ψ, t)E k (, ψ; t z) (,ψ)∈ER (N ,χ−4Dq ) t|N /L M

where aθ (, ψ, t) are as in (2.19).  (c) If the squarefree part of 2k j=1 α j is even, say 2D and 1 < k satisfies χ±8Dq (−1) = k (−1) , then we have



E(θ (D); z) = aθ (, ψ, t)E k (, ψ; t z) (,ψ)∈ER (N ,χ±8Dq ) t|N /L M

where aθ (, ψ, t) are as in (2.19).

3.4

Diagonal Quadratic Forms

85

(d) In either case if k = 1, then the statements above hold if we replace the coefficient aθ (, ψ; t) by 0 whenever (−1) = −1. Proof By Theorem 3.4.1, we have θ (D) ∈ Mk (0 (N ), χ ), where if the squarefree part of χ= and if the squarefree part of

2k

j=1 α j

 χ Dq

if (−1)k = χ Dq (−1),

χ−4Dq if (−1)k = χ−4Dq (−1),

2k

χ=

is odd, say D, then

j=1 α j

is even, say 2D, then

 χ8Dq

if (−1)k = χ8Dq (−1),

χ−8Dq if (−1)k = χ−8Dq (−1).

Therefore, in each case, χ is a real Dirichlet character. By Theorem 2.6.1, we have



a(, ψ, t)E k (, ψ; t z), E(θ (D); z) = (,ψ)∈E(N ,χ) t|N /L M

and by Theorem 3.4.2, for all nonreal Dirichlet characters ψ we have CcM,ψ (θ (D)) = 0. Therefore, a f (, ψ; t) = 0 whenever ψ is a nonreal Dirichlet character, that is, we have E(θ (D); z) =





a(, ψ, t)E k (, ψ; t z).

(3.15)

(,ψ)∈E(N ,χ) t|N /L M ψ is real

Since ψ = χ is a real Dirichlet character,  must be real as well. Now the theorem follows from observing that the set of pairs of primitive real Dirichlet characters (, ψ) in E(N , χ Dq ), E(N , χ−4Dq ) and E(N , χ±8Dq ) are given by ER (N , χ Dq ), ER (N , χ−4Dq ) and  ER (N , χ±8Dq ), respectively. Next, we use Theorem 3.4.3 to give a SAGE function that computes the Eisenstein part of any given diagonal quadratic form. In SAGE the Dirichlet characters as elements of DirichletGroup, are different from the Kronecker symbols. There is no time saving way to go between them. Therefore most of our previous functions need to be rewritten for Kronecker symbols. Below we start with doing that and then conclude with the desired SAGE function.

86

3 Application: Quadratic Forms

SAGE function to compute Bk (χd ) Inputs: • k: The positive integer k. • d: The quadratic discriminant d. def BernoulliKS(k,d): out,N=0,abs(d) for j in range(k+1): S=sum(kronecker(d,r)*rˆ(k-j) for r in range(N)) out+=factorial(k)/factorial(j)/factorial(k-j)\ *bernoulli(j)*Nˆ(j-1)*S return out

Output: The value of Bk (χd ).

SAGE function to compute σk (χd1 , χd2 ; n) Inputs: • k: The positive integer k. • d1, d2: The quadratic discriminants d1 and d2 , respectively. • n: The nonnegative rational number n. def sigmaR(k,d1,d2,n): L,M=abs(d1),abs(d2) if k==0 and n not in ZZ: return 0 elif k==0 and n in ZZ: return(sum(kronecker(d1,n/d)*kronecker(d2,d)\ for d in divisors(n))) if n not in ZZ: return 0 elif gcd(gcd(L,M),n)!=1: return 0 else: return prod(((kronecker(d2,ps[0])*ps[0]ˆk)ˆ(ps[1]+1)\ -kronecker(d1,ps[0])ˆ(ps[1]+1))\ /(kronecker(d2,ps[0])*ps[0]ˆk\ -kronecker(d1,ps[0])) for ps in\ factor(n))

Output: The value of σk (χd1 , χd2 ; n).

3.4

Diagonal Quadratic Forms

SAGE function to determine if an integer is a quadratic discriminant Input: • d: An integer d. def isQD(d): if d==squarefree_part(d) and d%4==1: return True elif d%4==0 and (d/4)==squarefree_part(d/4) and\ (d/4)%4==2: return True elif d%4==0 and (d/4)==squarefree_part(d/4) and\ (d/4)%4==3: return True else: return False

Output: Returns True, if d is a quadratic discriminant, False, if not.

SAGE function to compute Rχd1 ,χd2 (A, B) Inputs: • k, A, B: The positive integers k, A and B, respectively. • d1, d2: The quadratic discriminants d1 and d2 , respectively. def CalRR(k,d1,d2,A,B): return (gcd(A,B)/B)ˆk*kronecker(d1,-A/gcd(A,B))\ *kronecker(d2,B/gcd(A,B))

Output: The value of Rχd1 ,χd2 (A, B).

SAGE function to compute Sχd1 ,χd2 (N , A, B) Inputs: • k, N, A, B: The positive integers k, N , A and B, respectively. • d1, d2: The quadratic discriminants d1 and d2 , respectively. def CalSR(k,N,d1,d2,A,B): prod,D=1,A*B/gcd(A,B)ˆ2

87

88

3 Application: Quadratic Forms if moebius(D)==0: return 0 else: for ps in factor(gcd(A,B)): p=ps[0] nu_p=vp(p,N) if 00 and d31: for d2 in divisors(N/d1): if isQD(d2) and ((-d1*d2/gcd(d1,d2)ˆ2)==\ char or (-4*d1*d2/gcd(d1,d2)ˆ2)==char): pairs.append([-d1,d2]) if isQD(-d2) and ((d1*d2/gcd(d1,d2)ˆ2)==\ char or (4*d1*d2/gcd(d1,d2)ˆ2)==char): pairs.append([-d1,-d2]) out_sgm_c=[] for char in pairs: d1,d2=char[0],char[1]

90

3 Application: Quadratic Forms L,M=abs(d1),abs(d2) Constants={} for c in divisors(N/L/M): val=0 for a in range(1,c*M+1): if gcd(a,c*M)==1: val+=QQbar(kronecker(d2,a))\ .radical_expression()\ *DiagonalFirstTerm(R,a,c*M) Constants.update({c*M: val/euler_phi(c*M)}) for t in divisors(N/L/M): bftt=bftR(k,N,d1,d2,t,Constants) out_sgm_c.append([d1,d2,t,QQbar(bftt)\ .radical_expression()]) if bftout!=0: return out_sgm_c elif prec!=0: out=1 q=var('q') for n in range(1,prec+1): efn=0 for VV in out_sgm_c: efn=efn+VV[3]*sigmaR(k-1,VV[0],VV[1],n/VV[2]) out=out+QQbar(efn).radical_expression()*qˆn return out.series(q,prec+1)

Output: • If the input bftout is not 0, then it returns a list where each element in the list is a list of elements [, ψ, t, bθ(D) (, ψ, t)] where  and ψ are elements of DirichletGroup. • If the input bftout is 0, and the input prec is a positive integer, then it returns the q-series expansion of E(θ (D); z) up to and including the term q pr ec .

In the next section, we give explicit formulas for certain diagonal quadratic forms using this SAGE program.

3.4.4

Diagonal Quadratic Forms Examples

In this subsection, we derive explicit formulas for r (D; n) for certain diagonal quadratic forms D. In some cases, we point out some interesting connections with other arithmetic functions. For these explicit examples, we use θ (α1 , . . . , α2k ; z) to denote θ (D) and r (α1 , . . . , α2k ; n) to denote r (D; n).

3.4

Diagonal Quadratic Forms

91

3.4.4.1 The Cases Where dim(Sk (0 (N), χ)) = 0 Referring to Lemma 2.1.3 if (k, N , χ ) =(2, 4, χ1 ), (2, 8, χ8 ), (2, 12, χ1 ), (2, 12, χ12 ), (2, 16, χ1 ), (2, 32, χ8 ), (2, 24, χ12 ), (2, 32, χ8 ), (3, 4, χ−4 ), or (4, 4, χ1 ), then the dimensions of Sk (0 (N ), χ ) and its subsets are 0. On the other hand by Theorem 3.4.1 we have θ (1, 1, 1, 1; z) ∈ M2 (0 (4), χ1 ), θ (1, 1, 1, 2; z) ∈ M2 (0 (8), χ8 ), θ (1, 1, 1, 3; z) ∈ M2 (0 (12), χ12 ), θ (1, 1, 1, 4; z) ∈ M2 (0 (16), χ1 ), θ (1, 1, 1, 8; z) ∈ M2 (0 (32), χ8 ), θ (1, 1, 3, 3; z) ∈ M2 (0 (12), χ1 ), θ (1, 1, 1, 1, 1, 1; z) ∈ M3 (0 (4), χ−4 ), θ (1, 1, 1, 1, 2, 2; z) ∈ M3 (0 (8), χ−8 ), θ (1, 1, 1, 1, 1, 1, 1, 1; z) ∈ M4 (0 (4), χ1 ). Therefore, for these generating functions, by using the SAGE function written by using Theorem 3.4.3, we obtain an exact formula for r (α1 , . . . , α2k ; n) in terms of divisor functions. Example 3.4.1 If we run the code EisPartofDiagForm([1,1,1,1])

the output is [[1, 1, 1, 8], [1, 1, 2, 0], [1, 1, 4, -32]]

Using this we write the formula r (1, 1, 1, 1; n) = 8(σ1 (χ1 , χ1 ; n) − 4σ1 (χ1 , χ1 ; n/4)), which is due to Jacobi [35]. See [52] for an elementary arithmetic proof. Similarly, we run the function EisPartofDiagForm()

92

3 Application: Quadratic Forms

with proper parameters and from the output we obtain the formulas r (1, 1, 1, 2; n) = −2(σ1 (χ1 , χ8 ; n) − 4σ1 (χ8 , χ1 ; n)),

(3.16)

r (1, 1, 1, 3; n) = − (σ1 (χ1 , χ12 ; n) + 2σ1 (χ−4 , χ−3 ; n)) + 3(σ1 (χ−3 , χ−4 ; n) + 2σ1 (χ12 , χ1 ; n)),

(3.17)

r (1, 1, 1, 4; n) =4σ1 (χ1 , χ1 ; n) − 20σ1 (χ1 , χ1 ; n/4) + 24σ1 (χ1 , χ1 ; n/8) − 32σ1 (χ1 , χ1 ; n/16) + 2σ1 (χ−4 , χ−4 ; n),

(3.18)

r (1, 1, 1, 8; n) = − σ1 (χ1 , χ8 ; n) + 4σ1 (χ8 , χ1 ; n) + σ1 (χ1 , χ8 ; n/2) + 4σ1 (χ8 , χ1 ; n/2) − 2σ1 (χ1 , χ8 ; n/4) − 16σ1 (χ8 , χ1 ; n/4) + σ1 (χ−4 , χ−8 ; n) + 2σ1 (χ−8 , χ−4 ; n),

(3.19)

r (1, 1, 3, 3; n) =4(σ1 (χ1 , χ1 ; n) − 3σ1 (χ1 , χ1 ; n/3)) − 8(σ1 (χ1 , χ1 ; n/2) − 3σ1 (χ1 , χ1 ; n/6)) + 16(σ1 (χ1 , χ1 ; n/4) − 3σ1 (χ1 , χ1 ; n/12)),

(3.20)

r (1, 1, 1, 1, 1, 1; n) = −4(σ2 (χ1 , χ−4 ; n) − 4σ2 (χ−4 , χ1 ; n)),

(3.21)

r (1, 1, 1, 1, 2, 2; n) = −4(σ2 (χ1 , χ−4 ; n/2) − 2σ2 (χ−4 , χ1 ; n)),

(3.22)

r (1, 1, 1, 1, 1, 1, 1, 1; n) =16(σ3 (χ1 , χ1 ; n) − 2σ3 (χ1 , χ1 ; n/2) + 16σ3 (χ1 , χ1 ; n/4)).

(3.23)

None of the formulas (3.16)–(3.23) are new, for Liouville’s results of this type see [40], also see [3, 5, 7, 8, 11, 57, 58] and their references for similar results.

3.4.4.2 The Cases Where dim(Sk (0 (N), χ)) = 1 Referring to Lemma 2.1.3 if (k, N , χ ) =(2, 20, χ1 ), (2, 24, χ1 ), (2, 32, χ1 ), (2, 36, χ1 ), (3, 8, χ−8 ), (3, 12, χ−3 ), (3, 16, χ−4 ), (4, 8, χ1 ), (5, 4, χ−4 ), or (6, 4, χ1 ). then the dimension of Sk (0 (N ), χ ) is 1. On the other hand by Theorem 3.4.1 we have

3.4

Diagonal Quadratic Forms

93

θ (1, 1, 5, 5; z) ∈ M2 (0 (20), χ1 ) θ (1, 1, 6, 6; z) ∈ M2 (0 (24), χ1 ), θ (1, 1, 8, 8; z) ∈ M2 (0 (32), χ1 ), θ (1, 1, 9, 9; z) ∈ M2 (0 (36), χ1 ), θ (1, 1, 1, 1, 1, 2; z) ∈ M3 (0 (8), χ−8 ), θ (1, 1, 1, 1, 1, 3; z) ∈ M3 (0 (12), χ−3 ), θ (1, 1, 1, 1, 1, 4; z) ∈ M3 (0 (16), χ−4 ), θ (1, 1, 1, 1, 1, 1, 2, 2; z) ∈ M4 (0 (8), χ1 ), θ (1, 1, 1, 1, 1, 1, 1, 1, 1, 1; z) ∈ M5 (0 (4), χ−4 ), θ (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; z) ∈ M6 (0 (4), χ1 ). Therefore by Remark 2.3.1 for each of these generating functions, we have that θ − E(θ ; z) is a newform. Using this below we list the normalized newforms of Sknew (0 (N ), χ ) where (k, N , χ ) =(2, 20, χ1 ), (2, 24, χ1 ), (2, 32, χ1 ), (2, 36, χ1 ), (3, 8, χ−8 ), (3, 12, χ−3 ), (3, 16, χ−4 ), (4, 8, χ1 ), (5, 4, χ−4 ), or (6, 4, χ1 ). We also use the multiplicative properties of normalized newforms and present an interesting formula concerning r (1, 1, 5, 5; n). Example 3.4.2 First, we run the code EisPartofDiagForm([1,1,5,5])

the output is [[1, [1, [1, [1, [1, [1,

1, 1, 1, 1, 1, 1,

1, 4/3], 2, 0], 4, -16/3], 5, 20/3], 10, 0], 20, -80/3]]

Using this and the arguments above we conclude that the function ∞

A1 (z) :=θ (1, 1, 5, 5; z) − 1 −

4

(σ1 (χ1 , χ1 ; n) − 4σ1 (χ1 , χ1 ; n/4) 3 n=1

+ 5σ1 (χ1 , χ1 ; n/5) − 20σ1 (χ1 , χ1 ; n/20))q n 8 = q + ··· 3

(3.24)

94

3 Application: Quadratic Forms

3 is in S2new (0 (20), χ1 ), from which we deduce that N(2, 20, χ1 ; z) = A1 (z) is a normal8 ized newform. Therefore by Theorem 2.3.1 we have ⎧ ⎪ ⎪a(N(2, 20, χ1 ; z); p)a(N(2, 20, χ1 ; z); n) ⎨ a(N(2, 20, χ1 ; z); pn) = if p | n, − p k−1 a(N(2, 20, χ1 ; z); n/ p) ⎪ ⎪ ⎩a(N(2, 20, χ ; z); p)a(N(2, 20, χ ; z); n) if p  n. 1

1

(3.25) If m and n are coprime integers each coprime to 20, using (3.25) in (3.24) we obtain the formula 3 r (1, 1, 5, 5; m · n) = r (1, 1, 5, 5; m)r (1, 1, 5, 5; n) + 2σ1 (χ1 , χ1 ; m · n) 8 1 − (σ1 (χ1 , χ1 ; n)r (1, 1, 5, 5; m) + σ1 (χ1 , χ1 ; m)r (1, 1, 5, 5; n)). 2 Arguing similarly we obtain the normalized newforms ∞

1 1

(σ1 (χ1 , χ1 ; n) − σ1 (χ1 , χ1 ; n/2) θ (1, 1, 6, 6; z) − − 2 2

N(2, 24, χ1 ; z) =

n=1

− 3σ1 (χ1 , χ1 ; n/3) − 2σ1 (χ1 , χ1 ; n/4) + 3σ1 (χ1 , χ1 ; n/6) + 8σ1 (χ1 , χ1 ; n/8) + 6σ1 (χ1 , χ1 ; n/12) − 24σ1 (χ1 , χ1 ; n/24))q n , ∞

N(2, 32, χ1 ; z) =

1 1 1

(σ1 (χ1 , χ1 ; n) − σ1 (χ1 , χ1 ; n/2) θ (1, 1, 8, 8; z) − − 2 2 2 n=1

+ 8σ1 (χ1 , χ1 ; n/16) − 32σ1 (χ1 , χ1 ; n/32) + σ1 (χ−4 , χ−4 ; n) + 2σ1 (χ−4 , χ−4 ; n/2))q n , ∞

3 3 1

N(2, 36, χ1 ; z) = θ (1, 1, 9, 9; z) − − (σ1 (χ1 , χ1 ; n) − 4σ1 (χ1 , χ1 ; n/3) 8 8 2 n=1

− 4σ1 (χ1 , χ1 ; n/4) + 9σ1 (χ1 , χ1 ; n/9) + 16σ1 (χ1 , χ1 ; n/12) − 36σ1 (χ1 , χ1 ; n/36))q n ,

N(3, 16, χ−4 ; z) =

∞ 1 1

1 θ(1, 1, 1, 1, 1, 4; z) − − (−σ2 (χ1 , χ−4 ; n) + σ2 (χ1 , χ−4 ; n/2) 4 4 2 n=1

− 2σ2 (χ1 , χ−4 ; n/4) + 4σ2 (χ−4 , χ1 ; n) + 4σ2 (χ−4 , χ1 ; n/2) − 32σ2 (χ−4 , χ1 ; n/4))q n ,

3.4

Diagonal Quadratic Forms

95 ∞

1 1 N(4, 8, χ1 ; z) = θ (1, 1, 1, 1, 1, 1, 2, 2; z) − − 2 (σ3 (χ1 , χ1 ; n) − σ3 (χ1 , χ1 ; n/2) 4 4 n=1

− 2σ3 (χ1 , χ1 ; n/4) + 32σ3 (χ1 , χ1 ; n/8))q n , ∞

N(5, 4, χ−4 ; z) =

1 1 1

(σ4 (χ1 , χ−4 ; n) θ (1, 1, 1, 1, 1, 1, 1, 1, 1, 1; z) − − 12 12 15 n=1

+ 16σ4 (χ−4 , χ1 ; n))q n , ∞

N(6, 4, χ1 ; z) =

1 1 1

(σ5 (χ1 , χ1 ; n) θ (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1; z) − − 16 16 2 n=1

− 64σ5 (χ1 , χ1 ; n/4))q n . Example 3.4.3 (Ramanujan–Mordell formula) In our next example, we let α j = 1 for 1 ≤ j ≤ 2k and reproduce the Ramanujan–Mordell formula, [45]. By Theorem 3.4.1 we have  Mk (0 (4), χ1 ) if k ≡ 0 (mod 2), θ (1, . . . , 1; z) ∈ Mk (0 (4), χ−4 ) if k ≡ 1 (mod 2), and C1/c (θ (1, . . . , 1; z)) =

−i 2c

k G (1, c)2k

⎧ ⎪ 1 if c ≡ 0 (mod ⎪ ⎪ ⎪ ⎨ −i k if c ≡ 1 (mod 2 = ⎪ 0 if c ≡ 2 (mod ⎪ ⎪ ⎪ ⎩ i k if c ≡ 3 (mod 2

4), 4), 4), 4).

Therefore by Theorem 3.4.3, if k is even, then we have −2k(2k − 1)  (−1)k/2 σk−1 (χ1 , χ1 ; n) Bk (χ1 ) ∞

E(θ (1, . . . , 1); z) =1 +

n=1

− (1 − (−1)

k/2

 )σk−1 (χ1 , χ1 ; n/2) + 2k σk−1 (χ1 , χ1 ; n/4) q n ;

if k = 1, then we have ∞

E(θ (1, 1); z) = 1 +

−2k

σ0 (χ1 , χ−4 ; n)q n ; B1 (χ−4 ) n=1

96

3 Application: Quadratic Forms

and if 1 < k is odd, then we have  −2k  (2i)k−1 σk−1 (χ−4 , χ1 ; n) + σk−1 (χ1 , χ−4 ; n) q n . Bk (χ−4 ) ∞

E(θ (1, . . . , 1); z) = 1 +

n=1

Therefore for each n > 0 we have r (1, . . . , 1; n) = e(θ ; n) + O(n k/2 ), where e(θ ; n) ⎧  ⎪ −2k(2k − 1)  k ⎪ ⎪ (χ , χ ; n) + 2 σ (χ , χ ; n/4) σ k−1 1 1 k−1 1 1 ⎪ ⎪ Bk (χ1 ) ⎪ ⎪ ⎪ ⎪ k − 1)  ⎪ 2k(2 ⎪ ⎪ σk−1 (χ1 , χ1 ; n) + 2σk−1 (χ1 , χ1 ; n/2) ⎪ ⎪ ⎪ Bk (χ1 ) ⎪ ⎪  ⎪ ⎪ ⎨ −2k σk−1 (χ1 , χ1 ; n/4) =  ⎪ −2k  k−1 ⎪ ⎪ ⎪ σ (χ , χ ; n) + σ (χ , χ ; n) 2 k−1 −4 1 k−1 1 −4 ⎪ ⎪ Bk (χ−4 ) ⎪ ⎪ ⎪   ⎪ −2k ⎪ ⎪ ⎪ − 2k−1 σk−1 (χ−4 , χ1 ; n) + σk−1 (χ1 , χ−4 ; n) ⎪ ⎪ Bk (χ−4 ) ⎪ ⎪ ⎪ ⎪ ⎩4σ (χ , χ ; n) 0

1

−4

if k ≡ 0 (mod 4),

if k ≡ 2 (mod 4), if k ≡ 1 (mod 4), if k ≡ 3 (mod 4), if k = 1.

If k = 1, 2, 3 or 4, then the dimension of the corresponding cusp form space is 0. Therefore, in these cases, the formulas above give an exact formula, that is, if k = 1, 2, 3 or 4, then r (1, . . . , 1; n) = e(θ ; n). Below we discuss the asymptotic behavior if k > 4. In this case, for m ∈ {1, 3} we have ⎧ ⎪ 1 if k ≡ 0 (mod 2), ⎪ ⎪ ⎪ k−1 + χ (m) ⎪ 2 ⎨ −4 |Bθ (m)| if k ≡ 1 (mod 4), k−1 + 1 = 2 ⎪ A(θ ) ⎪ ⎪ ⎪ 2k−1 − χ−4 (m) ⎪ ⎩ if k ≡ 3 (mod 4). 2k−1 + 1 For k = 5, we have |Bθ (m)| > ζ (4) − 1, A(θ )

3.4

Diagonal Quadratic Forms

97

θ (m)| for m ∈ {1, 3}. For fixed m, the value of |BA(θ) is nondecreasing as k increases, and ζ (k − 1) − 1 is decreasing. Therefore, for each k > 4, we have

|Bθ (m)| > ζ (k − 1) − 1. A(θ ) Hence, by Theorem 2.7.3, we have lim

n→∞

r (1, . . . , 1; 4n + m) =1 e(θ ; 4n + m)

for m ∈ {1, 3}. For the cases when m = 0 or 2 we study the divisor functions in greater detail as promised in Remark 2.7.2. If k ≡ 0 (mod 4) and ν2 (n) ≥ 2 then, by using Proposition Proposition 2.5.1(a), we find that σk−1 (χ1 , χ1 ; n) + 2k σk−1 (χ1 , χ1 ; n/4)   3 · 2(k−1)ν2 − (2k + 1) (k−1)ν2 . = σk−1 (χ1 , χ1 ; r2 (n)) 2 + 2k−1 − 1 Therefore, if k ≡ 0 (mod 4) and ν2 (n) ≥ 2, we have  

3 · 2(k−1)ν2 − (2k + 1) σk−1 (χ1 , χ1 ; ) −Bk (χ1 ) e(θ ; n) 1+ = . 2k(2k − 1) n k−1 2(k−1)ν2 (2k−1 − 1) r2 (n)k−1 By similar manipulations, if k is even and 2 | n, we obtain 

 −(−1)k/2 Bk (χ1 ) e(θ ; n) 2k(2k − 1) n k−1 ⎧ σk−1 (χ1 , χ1 ; n) ⎪ ⎪ ⎪ ⎪ n k−1  ⎪ ⎪ ⎪ ⎪ σk−1 (χ1 , χ1 ; r2 (n)) ⎪ ⎪ 1+ ⎪ ⎪ ⎨ r2 (n)k−1  = σ k−1 (χ1 , χ1 ; r2 (n)) ⎪ ⎪ 1+ ⎪ ⎪ ⎪ r2 (n)k−1 ⎪  ⎪ ⎪ ⎪ ⎪ σk−1 (χ1 , χ1 ; r2 (n)) ⎪ ⎪ 1+ ⎩ r2 (n)k−1

3 · 2(k−1)ν2 − (2k + 1) 2(k−1)ν2 (2k−1 − 1) 

3 · 2(k−1)ν2 − 3

2(k−1)ν2 (2k−1 − 1)  2(k−1)ν2 + 2k − 3 2(k−1)ν2 (2k−1 − 1)



if k ≡ 0(mod 4) and ν2 (n) = 1, if k ≡ 0(mod 4) and ν2 (n) ≥ 2, if k ≡ 2(mod 4) and ν2 (n) = 1, if k ≡ 2(mod 4) and ν2 (n) ≥ 2,

and if k is odd and 2 | n we obtain

−Bk (χ−4 ) 2k

⎧ σ (χ , χ ; r (n)) ⎪ ⎪ k−1 −4 1 2 e(θ ; n) ⎨ r2 (n)k−1 = σ ⎪ k−1 (χ−4 , χ1 ; r2 (n)) n k−1 ⎪ ⎩ r2 (n)k−1



χ−4 (r2 (n)) 1 + (ν +1)(k−1) ifk ≡ 1(mod 4), 2 2

χ−4 (r2 (n)) −1 + (ν +1)(k−1) if k ≡ 3(mod 4). 2 2

98

3 Application: Quadratic Forms

Since k > 4, by using Proposition Proposition 2.5.1(d), for n ≡ 0 (mod 2) we deduce n k/2 = 0. n→∞ e(θ ; n) lim

Therefore we have lim

n→∞

r (1, . . . , 1; 4n + m) =1 e(θ ; 4n + m)

for m ∈ {0, 2} as well. We note that these are well known results that were previously proven by using the circle method, see [20, p. 72]. For q-series coefficients a( f ; n) with gcd(n, N ) = 1, our results provide an alternative way to conclude the asymptotic behavior in a much simpler way. Now we give some results concerning a family of diagonal quadratic forms. These results, to the best of our knowledge, are new to the literature. Theorem 3.4.4 Let P := D(α1 , . . . , α2k ), where each α j is either 1 or a prime and α j1  = α j2 unless j1 = j2 . Let S2 (s) := #{α j : α j | s, α j ≡ 3 (mod 4)} S4 (s) := #{α j : α j  s, α j ≡ 3 (mod 4)}.  2k k (a) Let 2k j=1 α j be odd and let N := j=1 α j . If k > 1 and (−1) = χ Nq (−1) is satisfied, then, for n > 0, we have −2k/Bk (χ Nq ) N k−1  r (P; n) = k Aσk−1 (χ(N /s)q , χsq ; n) 2 − χ Nq (2) s s|N   − χsq (2) χ Nq (2)A + B σk−1 (χ(N /s)q , χsq ; n/2)  + 2k Bσk−1 (χ(N /s)q , χsq ; n/4) + O(n k/2 ), where A = χ(N /s)q (−2s)(−1)(6k+N −s+2S2 (s))/4 (= ±1), B = χ(N /s)q (−2s)(−1)(N −s+6S4 (s))/4 (= ±1).  2k k (b) Let 2k j=1 α j be odd and let N := j=1 α j . If k > 1 and (−1) = −χ Nq (−1) is satisfied, then, for n > 0, we have

3.4

Diagonal Quadratic Forms

99





N k−1  −2k 2k−1 Cσk−1 (χ−4(N /s)q , χsq ; n) r (P; n) = Bk (χ−4Nq ) s s|N  + Dσk−1 (χ(N /s)q , χ−4sq ; n) + O(n k/2 ), where C = χ(N /s)q (−2s)(−1)(6k+3N −s+6S2 (s))/4 (= ±1), D = χ(N /s)q (−2s)(−1)(3N −3s+6S4 (s))/4 (= ±1).  1 2k k (c) Let 2k j=1 α j be even and let N := 2 j=1 α j . If k > 1 and (−1) = χ±8Nq (−1) is satisfied, then, for n > 0, we have  

N k−1  −2k 4k−1 Eσk−1 (χ±8(N /s)q , χsq ; n) r (P; n) = Bk (χ±8Nq ) s s|N  + Fσk−1 (χ(N /s)q , χ±8sq ; n) + O(n k/2 ), where E = χ(N /s)q (−s)(−1)(2s+N s−N +2S2 (s)−1+(−1) F = χ(N /s)q (−s)(−1)

((−1)k −(−1)k N s+6S4 (s))/4

k )/4

(= ±1),

(= ±1).

(d) If k = 1, then the statements hold in each case if we replace the coefficient of σ0 (, ψ; n/t) by 0 whenever (−1) = −1. Proof As the proofs of (a), (b), (c) and (d) are all similar, we prove only (a) and give the nontrivial identities we use to obtain the rest of the results. 2k  k Letting 2k j=1 α j be odd and N := j=1 α j , if k > 1 and (−1) = χ Nq (−1) is satisfied, then we have θ (P) ∈ Mk (0 (4N ), χ Nq ) ⊂ Mk (0 (8N ), χ Nq ). Next, we compute the constant terms of θ (P). By Proposition Proposition 3.4.1 we have 2k 







αj c  ,   gcd α j , c G G1/c (P) = gcd α j , c gcd α j , c j=1 If c ≡ 2 (mod 4), then by (3.8) we have G 0.



αj c , gcd(α j ,c) gcd(α j ,c)





= 0. Therefore G1/c (P) =

100

3 Application: Quadratic Forms

If c ≡ 1 (mod 4), then we write G1/c (P) =

2k  j=1 α j |c α j ≡1(4)



2k 2k  c c   α j G 1, α j G 1, G αj, c . αj αj j=1 α j |c α j ≡3(4)

j=1 α j c

By (3.9), if α j | c, we have

√ cα j if α j ≡ 1(4), c = √ α j G 1, αj i cα j if α j ≡ 3(4), and if α j  c we have  αj  √  c. G αj, c = c K Therefore we obtain G1/c (P) = ck

2k  √ αj j=1 α j |c α j ≡1(4)

2k 

2k √  i αj χ(α j )q (c).

j=1 α j |c α j ≡3(4)

j=1 α j c

Noting that we have 2k  √ √ α j = c, j=1 α j |c

and 2k 

α j = N /c,

j=1 α j c

we obtain G1/c (P) = χ(N /c)q (c)i S2 (c) ck+1/2 . When c ≡ 3 (mod 4), similar calculations yield G1/c (P) = χ(N /c)q (c)ck+1/2 i S2 (c) (−1)k+S2 (c)+S4 (c) . Now observe that if N ≡ 1 (mod 4), then k + S2 (c) + S4 (c) ≡ 0 (mod 2). That is, for all c | N we have

3.4

Diagonal Quadratic Forms

101

G1/c (P) = χ(N /c)q (c)i S2 (c) ck+1/2 . Therefore for all c | N we have (−i)k G1/c (P) = χ(N /c)q (c)i3k+S2 (c) C1/c (θ (P)) = √ ck 22k N

√ c/N . 2k

Similarly for each c | N we show that  C1/4c (θ (P)) = χ(N /c)q (c)(−i) S4 (c) c/N ,  C1/8c (θ (P)) = χ(N /c)q (2c)(−i) S4 (c) c/N . Altogether we have C1/c (θ (P)) = χ(N /c)q (c)i3k+S2 (c)

√ c/N , 2k

C1/2c (θ (P)) = 0,

 C1/4c (θ (P)) = χ(N /c)q (c)i3S4 (c) c/N ,  C1/8c (θ (P)) = χ(N /c)q (2c)i3S4 (c) c/N .

We then use these values in Corollary 2.6.2 and obtain E(θ (P); z) =



2k a f (χ(N /s)q , χsq , t)E k (χ(N /s)q , χsq ; t z), k 2 − χ Nq (2)

(3.26)

s|N t|4

where for each s | N we have √ s/N a f (χ(N /s)q , χsq , 1) = i χ(N /s)q (−s) , (−2)k  √s/N  3k+S2 (s) 3S4 (s) +i , a f (χ(N /s)q , χsq , 2) = −χ(N /s)q (−s)χsq (2) χ Nq (2)i 2k  a f (χ(N /s)q , χsq , 4) = χ(N /s)q (−s)i3S4 (s) s/N . k+S2 (s)

By Example 2.4.4 we have e(sq ) E k (χ(N /s)q , χsq ; t z) = χ(N /s)q (0) + e(Nq ) ×

∞

n=1

with the previously defined notation



−2k(N /s)k−1/2 Bk (χ Nq )

σk−1 (χ(N /s)q , χsq ; n)q tn ,



(3.27)

102

3 Application: Quadratic Forms

 e(d) =

1 if d > 0, if d < 0.

i

Next, we prove some helpful identities for further manipulation. Observing that  0 (mod 2) if s ≡ 1 (mod 4), S2 (s) ≡ 1 (mod 2) if s ≡ 3 (mod 4), and e(sq ) = χ(s)q (2)eπ i(1−s)/4 , we show that e(sq )i S2 (s) = χsq (2)(−1)(2S2 (s)+1−s)/4 .

(3.28)

Additionally, observing that if N ≡ 1 (mod 4), then k ≡ 0 (mod 2), and if N ≡ 3 (mod 4), then k ≡ 1 (mod 2), we prove ik = χ Nq (2)(−1)(2k+N −1)/4 , e(Nq )

(3.29)

e(sq ) = χ(N /s)q (2)(−1)(N −s)/4 . e(Nq )

(3.30)

and

Next we put (3.27) in (3.26), use the identities (3.28)–(3.30) and use Theorem 2.1.1 to obtain the desired result. With the assumptions of (b), we list the nontrivial identities we use to obtain the formula in (b) below: e(−4Nq ) = −χ8 (N )eπi(1−3N )/4 , e(sq ) = −χ8 (N /s)eπi(3N −s)/4 , e(−4Nq ) e(−4sq ) = χ8 (N /s)eπi(3N −3s)/4 . e(−4Nq ) and, with the assumptions of (c), we list the nontrivial identities we use to obtain the formula in (c) below: e(±8Nq ) = eπ i(1−(−1) )/4 , e(±8sq ) k k = χ(N /s)q (2)eπ i((−1) −(−1) N s)/4 . e(±8Nq ) k

3.4

Diagonal Quadratic Forms

103

We note that a case analysis show that A, B, C, D, E and F are always ±1, as expected.  This finalizes our discussion on applications of modular forms to the representation numbers of quadratic forms. In the next chapter we discuss applications of modular forms to eta quotients.

4

Application: Eta Quotients

4.1

The Definition, the Modularity and the Constant Terms

We start with the definition of Dedekind’s eta function. Definition 4.1.1 We define the Dedekind’s eta function by the infinite product η(z) := eπ iz/12

∞ 

(1 − e2π inz ) = q 1/24

n=1

∞ 

(1 − q n ).

n=1

The q-series expansion of Dedekind’s eta function is well known and given in the next statement. Theorem 4.1.1 We have η(z) = q 1/24

∞ 

(−1)n q n(3n−1)/2 .

n=−∞



Proof See [21, Corollary 1.3.4 and Eq. (1.2.4)]. We write the following SAGE function using Theorem 4.1.1.

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-32629-5_4.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 Z. S. Aygin, Introduction to Applications of Modular Forms, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-32629-5_4

105

106

4 Application: Eta Quotients

SAGE function for finding q-series expansion of Dedekind’s eta function Inputs: • t: The positive integer t. • prec: A positive integer, for the number of terms to compute in the q-series expansion.

def EtaExpand(t,prec): q=var('q') out=0 bd=floor(sqrt(2*(prec+1)/3))+1 for n in range(-bd,bd): out=out+(-1)ˆn*qˆ(t*n*(3*n-1)/2) return out.series(q,prec+1)

Output: The q-series expansion of η(t z)/q t/24 up to and including the term q pr ec .

Next, we give the transformation formula for Dedekind’s eta function, which is originally due to Ligozat [38]. Over the years this formula has been refined, and extended to work for all matrices in 0 (1), see [36, Theorem 1.7]. Those versions are notation heavy and require defining a new type of character. To keep the notation light we give Ligozat’s version as it is stated in [26, Proposition 5.8.3] and work with it.   ab Lemma 4.1.1 For all M = ∈ 0 (1), with c ≥ 0 and gcd(a, 6) = 1 we have cd 

az + b η cz + d

 = w(m)

c a

K

((cz + d)/i)1/2 η(z).

where 

 πi w(M) := exp (a(b − c + 3)) . 12 We now manipulate this to obtain a transformation formula for η(t z), where t is a positive integer. The following lemma is a version of Proposition 2.1 of [36] refined to fit our purposes. Lemma 4.1.2 Let M =

  ab ∈ 0 (1) with c > 0. Then we have cd

4.1 The Definition, the Modularity and the Constant Terms ⎛

107

⎛

   ⎜ πi ⎜ at + n t c ⎜ 3 − c − ht − n t + δt η(t M(z)) = exp ⎜ ⎝ 12 ⎝ ht c ht  ×

c/h t at/h t + n t c/h t

  K

ht t



1/2 ((cz + d)/i)1/2 η

where h t := gcd(c, t), n t is an integer satisfying gcd so that

atδt −h t c





at+cn t ht

2

 −1

c/h t

⎞⎞ +d

⎟ h 2t ⎟ ⎟⎟ ct ⎠⎠

 h 2t z , t

at+cn t ht , 6



= 1, and δt is an integer

is also an integer.

  ab Proof Let M = ∈ 0 (1). Let h t = gcd(c, t). And assume δt is an integer such that cd βt :=

atδt − h t c

(4.1)

is an integer. We note that, since gcd(at/h t , c/h t ) = 1, such δt always exists and we have at c δt − βt = 1. ht ht Using this we write      at bt at/h t βt ht ut t M(z) = (z) = (z), c/h t δt 0 t/h t c d where

  at/h t βt ∈ 0 (1) and c/h t δt u t := δt bt − βt d.

(4.2)

Therefore we have η(t M(z)) = η

   at/h t βt (z 1 ) , c/h t δt

where z 1 :=

ht z + ut . t/h t

Assume c > 0 and let n t be an integer that satisfies   c at + n t , 6 = 1. gcd ht ht  πi  n η(z) we have Then since η(z + n) = exp 12

(4.3)

108

4 Application: Eta Quotients

     πi at/h t βt (z 1 ) + n t η(t M(z)) = exp − n t η c/h t δt 12      πi at/h t + n t c/h t βt + n t δt = exp − n t η (z 1 ) . c/h t δt 12 Therefore by Lemma 4.1.1 we have     c/h t πi ((c/h t z 1 + δt )/i)1/2 η (z 1 ) , η(t M(z)) = exp − n t w(Bt ) 12 at/h t + n t c/h t K where   at/h t + n t c/h t βt + n t δt Bt := . c/h t δt By using (4.2) and (4.3) we obtain c/h t z 1 + δt =

ht (cz + d). t

Hence using (4.3) again we have   πi η(t M(z)) = exp − (n t − u t h t /t) w(Bt ) 12   1/2  2   ht ht c/h t ((cz + d)/i)1/2 η × z . at/h t + n t c/h t K t t

(4.4)

Using (4.1) and (4.2) we get   πi exp − (n t − u t h t /t) w(Bt ) 12   ⎞⎞ ⎛ ⎛ 2 at+cn t    − 1 2 h t ⎜ πi ⎜ c at + n t c ht h ⎟⎟ ⎜ ⎟. − n t + δt +d t ⎟ = exp ⎜ ⎝ 12 ⎝ 3 − h t − c ht c/h t ct ⎠⎠ Putting this in (4.4) we obtain the desired result.



The quotients of Dedekind’s eta functions η(t z) correspond to interesting functions. Furthermore, they are sometimes useful in giving a basis for modular form spaces. In the remainder of this chapter, we study eta quotients, which are defined as follows. Definition 4.1.2 Let N be a positive integer and t a divisor of N . An eta quotient of level N is quotients of η(t z) of the form

4.1 The Definition, the Modularity and the Constant Terms



109

η(t z)rt ,

t|N

where rt are integers (not all zero). For a shorthand notation to represent eta quotients, we use  η N [r1 , . . . , r N ](z) := η(t z)rt , t|N

where [r1 , . . . , r N ] is an ordered list of the exponents rt placed in increasing order of the t’s. When we are talking about an arbitrary eta quotient we usually tend to shorten this notation to η N . Remark 4.1.1 In the literature usually the level of an eta quotient is taken to be the lowest common multiple of t’s for which rt is nonzero. As this sometimes does not agree with the level of the modular form space the eta quotient is in, we do not use that convention. We emphasize that in our definition the level of an eta quotient is not unique. The q-series expansion of an eta quotient up to a given precision can be calculated by the following SAGE function. SAGE function to find the q-series expansion of an eta quotient Inputs: • N: The positive integer N . • etaq: A list of integers rt (t | N ) placed in the list in an increasing order of t’s. Note 1  that 24 t|N trt must be a nonnegative integer. • prec: A positive integer, for the number of terms to compute in the q-series expansion.

def EtaQExpand(N,etaq,prec): Divs=divisors(N) q=var('q') out=qˆ(sum(Divs[i]*etaq[i] for i in range(len(Divs)))/24) for i in range(len(Divs)): out=out*EtaExpand(Divs[i],prec)ˆ(etaq[i]) return out.series(q,prec+1)

Output: The q-series expansion of η N [r1 , . . . , r N ](z) up to and including the term q pr ec .

Next, we discuss the modularity of eta quotients. We start by giving the order of vanishing at each cusp, which is also due to Ligozat [38, Proposition 3.2.8, p. 34].

110

4 Application: Eta Quotients

Lemma 4.1.3 The order of the vanishing of η N at the cusp a/c is given by va/c,N (η N ) =

 gcd(c, t)2 N rt . 24 gcd(c2 , N ) t t|N

Proof Let M =

  ab ∈ 0 (1). Using the transformation formula derived in Lemma 4.1.2, cd

we have 

η (t M(z)) = C(cz + d) rt

t|N

k



 η

rt

t|N

 h 2t z , t

(4.5)

where we combine all constants under C as, for the purposes of this proof, the value of that constant is not important. Next using Theorem 4.1.1 in (4.5) we obtain  ∞ rt    h2   2 2 t ηr t q rt h t /24t (−1)n q (h t /t)(n(3n−1)/2) . z = t n=−∞ t|N

t|N

Recalling that qc,N = q gcd(c

2 ,N )/N

,

we have (cz + d)−k



(N /24 gcd(c2 ,N ))

ηrt (t M(z)) = Cqc,N



2 t|N rt h t /t



 1 + O(qc,N ) .

t|N



The result now follows from Definition 2.1.6.

Remark 4.1.2 By Bhattacharya [23, p. 884] the coefficient matrix of the set of linear equations v1/c,N (η N ) =

 gcd(c, t)2 N rt , 2 24 gcd(c , N ) t t|N

for c | N , is invertible. This means the order uniquely determines the eta quotient. Using Lemma 4.1.3 we write the following SAGE code.

4.1 The Definition, the Modularity and the Constant Terms

111

SAGE function to compute the order of vanishing of an eta quotient Inputs: • N: The positive integer N . • etaq: A list of integers rt (t | N ) placed in the list in an increasing order of t’s.

def OrderEtaQ(N,etaq): Divs=divisors(N) return [N/24/gcd(cˆ2,N)*sum(gcd(c,Divs[i])ˆ2\ *etaq[i]/Divs[i] for i in range(len(Divs)))\ for c in Divs]

Output: A list of the values of v1/c,N (η N [r1 , . . . , r N ](z)) placed in the list in an increasing order of the c’s (c | N ).

We are now ready to state the result on the modularity of eta quotients. The first part of the following theorem is a well known result, see [36, Sect. 2.4], [26, Proposition 5.9.2] for some versions of it. The second part is proved using Lemma 4.1.2.  Theorem 4.1.2 If k = 21 t|N rt is a positive integer, v1/1,N (η N ) and v1/N ,N (η N ) are nonnegative integers and v1/c,N (η N ) ≥ 0 for each c | N , then η N [r1 , . . . , r N ](z) ∈ Mk (0 (N ), χ D ), where D=

 (−1)k S

if S is odd and (−1)k S ≡ 1 (mod 4),

(−1)k 4S otherwise,

 with S denoting the squarefree part of t|N t |rt | . If η N ∈ Mk (0 (N ), χ D ) and c > 0, then for all positive a coprime to c we have ⎧  rt  rt /2 ⎪ c/h t ht ⎪(−i)k ω(a, c)  ⎨ if va/c ( f ) = 0, at/h t + n t c/h t K t Ca/c (η N ) = t|N ⎪ ⎪ ⎩0 if v ( f ) > 0, a/c

where   ⎞⎞ ⎛ 2 at+cn t     − 1 h t ⎟⎟ ⎜ πi  ⎜ at + n t c ht c ⎟⎟ , ω(a, c) = exp ⎜ rt ⎜ − n t + δt ⎝ 3 − ht − c ⎠⎠ ⎝ 12 ht c/h t ⎛

t|N

112

4 Application: Eta Quotients

where h t = gcd(c, t), n t is an integer such that gcd that

atδt −h t c



at+cn t ht , 6



= 1, and δt is an integer so

is an integer.

Proof The modularity criteria given in the first part of the statement is well known, see [26, Proposition 5.9.2].  Below we prove the formula for the constant terms. ab Let M = ∈ 0 (1). Using Lemma 4.1.2 we obtain cd η N (M(z)) = ⎛



η(t M(z))rt

t|N

⎛

   ⎜ πi  ⎜ at + n t c ⎜ 3 − c − ht − n t + δt = exp ⎜ r t ⎝ ⎝ 12 ht c ht



at+cn t ht

2

c/h t

t|N

 −1

⎞⎞ +d

⎟ h 2t ⎟ ⎟⎟ ct ⎠⎠

rt  rt /2  2 rt ht z c/h t ht η −i (cz + d) at/h t + n t c/h t K t t t|N   ⎛ ⎞⎞ ⎛ 2 at+cn t     − 1 2 h t ⎜ πi  ⎜ c at + n t c ht h ⎟⎟ ⎟ − n t + δt = exp ⎜ rt ⎜ +d t ⎟ ⎝ 3 − ht − c ⎝ 12 ht c/h t ct ⎠⎠

×



t|N

× q(



h2 t t|N rt t

)/24

 t|N

rt  rt /2  ∞ h2 ht t (1 − q t n )rt . −i (cz + d) t K

c/h t at/h t + n t c/h t

n=1

If 

h 2t > 0, t

rt

t|N

then lim q (



2 t|N rt h t /t)/24

z→i∞

Therefore we have Ca/c (η N ) = 0. But if  h2 t

t|N

t

rt = 0,

then d  h 2t rt = 0. c t t|N

= 0.

4.1 The Definition, the Modularity and the Constant Terms

113

Using this we obtain ⎛

⎛

   ⎜ πi  ⎜ at + n t c ⎜ 3 − c − ht − n t + δt η N (M(z)) = exp ⎜ r t ⎝ ⎝ 12 ht c ht



t|N

× (−i(cz + d))k

 t|N

c/h t at/h t + n t c/h t

rt  K

ht t

 ⎞⎞ −1 ⎟⎟ ⎟⎟ ⎠⎠ c/h t

at+cn t ht

2

rt /2  2 rt ht z η . t

The result now follows from Definition 2.1.7.



Next, we use Theorem 4.1.2 to give a SAGE function that checks the modularity of a given eta quotient. SAGE function to check modularity of η N Inputs: • N: The positive integer N . • etaq: A list of integers rt (t | N ) placed in the list in an increasing order of t’s.

def isEtaQModular(N,etaq): Ord=OrderEtaQ(N,etaq) k=sum(etaq)/2 if all(o>=0 for o in Ord) and (Ord[0] in ZZ) and\ (Ord[len(Ord)-1] in ZZ) and (k in ZZ): char=squarefree_part(prod(divisors(N)[i]\ ˆabs(etaq[i]) for i in range(len(etaq)))) if char%2==1 and (-1)ˆk*char%4==1: char=(-1)ˆk*char else: char=(-1)ˆk*4*char print('Weight:',k,'Level:',N,'Character:',char) return True else: print('Not an integer weight modular form!') return False

Output: Returns False, if the eta quotient η N [r1 , . . . , r N ](z) is not a modular form. Returns True and prints k, N and D if the eta quotient η N [r1 , . . . , r N ](z) is a modular form in Mk (0 (N ), χ D ).

114

4 Application: Eta Quotients

Finally we use Theorem 4.1.2 again to write a SAGE function that computes Ca/c (η N ). SAGE function to compute Ca/c (η N ) Inputs: • N: The positive integer N . • etaq: A list of integers rt (t | N ) placed in the list in an increasing order of t’s. Note 1  that 24 t|N trt must be a nonnegative integer. • a, c: The coprime integers a and c, respectively.

def EtaQConstantTerm(N,etaq,a,c): Divs=divisors(N) if sum(gcd(c,Divs[i])ˆ2*etaq[i]/Divs[i]\ for i in range(len(Divs)))!=0: return 0 else: k=sum(etaq)/2 w,C1,C2=0,1,1 for i in range(len(Divs)): t,rt=Divs[i],etaq[i] ht,nt=gcd(c,t),0 while gcd(a*t/ht+nt*c/ht,6)!=1: nt=nt+1 deltat=0 while (a*t*deltat-ht)/c not in ZZ or\ gcd(deltat,(a*t*deltat-ht)/c)!=1: deltat=deltat+1 C1=C1*(ht/t)ˆ(rt/2) C2=C2*kronecker_symbol(c/ht,(a*t+nt*c)/ht)ˆrt w=w+rt*((3-c/ht-ht/c)*((a*t+nt*c)/ht)-nt\ +deltat*(((a*t+nt*c)/ht)ˆ2-1)/(c/ht)) return exp(pi*I/12*w)*C1*C2*(-I)ˆk

Output: The value of Ca/c (η N [r1 , . . . , r N ](z)).

4.2

Finding the Complete Sets of Eta Quotients in Mk (0 (N), χ)

In this section, we first state the necessary results to establish an algorithm to find all eta quotients in a modular form space with the given weight, level and character. The algorithm we use is a generalized version of the algorithm given in [18]. As noted in Remark 4.1.2, the order of an eta quotient uniquely determines the eta quotient. The following result determines

4.2

Finding the Complete Sets of Eta Quotients in Mk (0 (N ), χ )

115

the bounds for the limits of the search for appropriate orders which may yield an eta quotient. This result is the backbone of our algorithm. Lemma 4.2.1 If η N ∈ Mk (0 (N ), χ ), then for each divisor c of N and a coprime to c we have 

φ(gcd(N /c, c))v1/c,N (η N ) =

c|N

kN  p + 1 , 12 p

(4.6)

p|N

0 ≤ va/c,N (η N ) ≤

 p+1 kN 12φ(gcd(N /c, c)) p

(4.7)

p|N

and va/c,N (η N ) ∈

 N0 N0 +

if χ(1 + a N / gcd(c, N /c)) = 1, 1 2

if χ(1 + a N / gcd(c, N /c)) = −1,

(4.8)

where   1 1 N0 + := n + : n ∈ N0 . 2 2 Proof The equation (4.6) is from [9, Eq. (4.2.9)], or can alternatively be proven by using the valence formula. Next we note that va/c,N (η N ) = v1/c,N (η N ), and as each va/c,N (η N ) ≥ 0 (4.7) follows from (4.6). The result given by (4.8) follows from (2.1).  Recall that an eta quotient is uniquely determined by its orders of vanishing at cusps (Remark 4.1.2). In the SAGE function below, we search for eta quotients in a given modular form space Mk (0 (N ), χ ) by running through all possible values of v1/c,N (η N ) within the constraints given by Lemma 4.2.1. SAGE function to find all eta quotients in Mk (0 (N ), χ ) Inputs: • k, N: The positive integers k and N , respectively. • chi: The element of DirichletGroup(N) which corresponds to the Dirichlet character χ.

def FindAllEtaQ(k,N,chi): Divs=divisors(N) l_Divs=len(Divs)

116

4 Application: Eta Quotients CS=2*k*N/24; for i in list(factor(N)): CS=CS*(i[0]+1)/i[0] if CS not in ZZ: print('No eta quotients!') return [] if chi(-1)!=(-1)ˆk: print('Character is not compatible with the weight,\ no eta quotients!') return [] if all( [V in ZZ for V in chi.values_on_gens()])==0: print('Character is not real, no eta quotients!') return [] U1=CS for i in range(l_Divs): c=Divs[i] if chi(1+N/gcd(N/c,c))==-1: U1=U1-1/2*euler_phi(gcd(N/c,c)) EQ=zero_matrix(QQ,l_Divs,l_Divs) for i in range(l_Divs): for j in range(l_Divs): EQ[i,j]=(((N)/(24*gcd(Divs[i]ˆ2,N)))\ *(((gcd(Divs[i],Divs[j])ˆ2)/(Divs[j])))) EQinv=EQ.inverse() out=[] if N==1: exps=EQinv*matrix([U1]) if all(exps[i][0] in ZZ for i in range(l_Divs)): out.append(list(exps.transpose()[0])) return out lmt=floor(U1/euler_phi(gcd(Divs[1],N/Divs[1]))) X2=zero_matrix(QQ,l_Divs,1) for i in range(1,l_Divs): c=Divs[i] X2[i,0]=(1-chi(1+N/gcd(c,N/c)))/4 while X2[1,0] < lmt: X2[0,0]=CS-sum(X2[i,0]*euler_phi(gcd(Divs[i]\ ,N/Divs[i])) for i in range(1,l_Divs)) exps=EQinv*X2 if all(exps[i][0] in ZZ for i in range(l_Divs))\ and squarefree_part(prod(Divs[i]ˆexps[i][0]\ for i in range(l_Divs)))\ ==squarefree_part(chi.conductor()): out.append(list(exps.transpose()[0])) X2[l_Divs-1,0]=X2[l_Divs-1,0]+1 i2=l_Divs-2 while sum(X2[i,0]*euler_phi(gcd(Divs[i]\ ,N/Divs[i])) for i in range(1,l_Divs))>CS: X2[i2,0]=X2[i2,0]+1

4.3 The Eisenstein Part of Eta Quotients

117

for i3 in range(i2+1,l_Divs): c=Divs[i3] X2[i3,0]=(1-chi(1+N/gcd(c,N/c)))/4 i2=i2-1 X2[0,0]=CS-sum(X2[i,0]*euler_phi(gcd(Divs[i]\ ,N/Divs[i])) for i in range(1,l_Divs)) exps=EQinv*X2 if all(exps[i][0] in ZZ for i in range(l_Divs))\ and squarefree_part(prod(Divs[i]ˆexps[i][0]\ for i in range(l_Divs)))\ ==squarefree_part(chi.conductor()): out.append(list(exps.transpose()[0])) return out

Output: The list of all eta quotients in Mk (0 (N ), χ ). Each entry in the list is a list of exponents rt placed in the list in an increasing order of the divisors t of N .

4.3

The Eisenstein Part of Eta Quotients

Below we combine Theorems 2.6.1 and 4.1.2 to write a SAGE function that gives the Eisenstein part of any eta quotient which is a modular form. SAGE function to compute Eisenstein part of an eta quotient in Mk (0 (N ), χ ) Inputs: • • • •

N: The positive integer N . etaq: A list of integers rt (t | N ) placed in the list in an increasing order of t’s. bftout: An optional input with a default value of 0. prec: An optional input of an integer value for the number of terms to compute in the q-series expansion. The default value is 0. • chars: A value of 0 or the character pairs ( , ψ) ∈ E(N , χ ) given as a list. This is an optional input with a default value of 0. This is useful when working with multiple modular forms in the same space, as EisBaseFind(k,N,chi) can be time consuming to run each time.

def EisPartofEtaQ(N,etaq,bftout=1,prec=0,chars=0): k=sum(etaq)/2 G=DirichletGroup(N) Divs=divisors(N) D=prod(Divs[i]ˆetaq[i] for i in range(len(Divs)))

118

4 Application: Eta Quotients charinit=(-1)ˆ(k)*squarefree_part(D) j=0 while all(G[j].primitive_character()(i)==\ kronecker(charinit,i) for i in\ G.unit_gens())==False: j=j+1 chi=G[j] if chars==0: chars=EisBaseFind(k,N,chi) ConstantsAll={} for char in chars: Constants={} eps,psi=char[0],char[1] L=eps.conductor() M=psi.conductor() for c in divisors(N/L/M): val=0 for a in range(1,c*M+1): if gcd(a,c*M)==1: val+=QQbar(psi(a))\ .radical_expression()\ *EtaQConstantTerm(N,etaq,a,c*M) Constants.update({c*M: val/euler_phi(c*M)}) ConstantsAll.update({psi: Constants}) return EisPart(k,N,chi,ConstantsAll,bftout=bftout\ ,prec=prec,chars=chars)

Output: • If the input bftout is not 0, then it returns a list where each element in the list is a list of elements [ , ψ, t, bη N ( , ψ, t)] where and ψ are elements of DirichletGroup. • If the input bftout is 0, and the input prec is a positive integer, then it returns the q-series expansion of E(η N ; z) up to and including the term q pr ec .

Next, we showcase some of the SAGE functions we have devised in the previous sections with an example. Example 4.3.1 Let f (z) = η12 [9, −6, −7, 3, 12, −5](z). We run the SAGE command isEtaQModular(12,[9, -6, -7, 3, 12, -5])

and from the output we determine that f (z) ∈ M3 (0 (12), χ−4 ). Next, we run the SAGE command

Eta Quotients in E k (0 (N ), χ )

4.4

119

EisPartofEtaQ(12,[9, -6, -7, 3, 12, -5])

and from the output, we write the modular identity ∞

E( f ; z) =1 −

1 (σ2 (χ1 , χ−4; n) + 27σ2 (χ1 , χ−4; n/3) − 16σ2 (χ−4 , χ1; n) 7 n=0

+432σ2 (χ−4 , χ1; n/3)) q n . Lastly, we run the SAGE command bfts=EisPartofEtaQ(12,[9, -6, -7, 3, 12, -5]) EisPartAsymptotic(3,12,DirichletGroup(12)[1],bfts)

and the output is The The The The

formula formula formula formula

is is is is

asymptotical asymptotical asymptotical asymptotical

if if if if

n n n n

is is is is

congruent congruent congruent congruent

to to to to

1 modulo 12 5 modulo 12 7 modulo 12 11 modulo 12

From this, we determine that lim

n→∞

a( f ; N n + m) =1 e( f ; N n + m)

for m = 1, 5, 7 and 11. In Sects. 4.4 and 4.5 below we give more results using the SAGE functions.

4.4

Eta Quotients in E k (0 (N), χ)

In this section, we find q-series coefficients of some eta quotients. Infinite product to infinite sum formulas has always been an active research area going back to Jacobi’s Triple Product Identity. The majority of the known results on the eta quotients are given in terms of the sum of divisor functions, see [3, 30, 59] and their references for some examples. This means these eta quotients can be given in terms of the Eisenstein series. On the other hand, if an eta quotient is in E k (0 (N ), χ ), then it can be written in terms of the Eisenstein series. In this section we give examples of eta quotients that are in E k (0 (N ), χ ), hence their q-series coefficients can be given in terms of the sum of divisor functions. Consulting to Theorem 2.4.2 and the SAGE function given on page 28 we write the following SAGE code. When the precision is set to be the Sturm bound, this code determines if there are any eta quotients in E k (0 (N ), χ ).

120

4 Application: Eta Quotients

EQS=FindAllEtaQ(k,N,chi) PureEis=[] chars=EisBaseFind(k,N,chi) for i in range(len(EQS)): etaq=EQS[i] BB=((EtaQExpand(N,etaq,prec)-EisPartofEtaQ(N,etaq\ ,bftout=0,prec=prec,chars=chars))).series(q,prec+1) if sum(BB.coefficient(q,i) for i in range(prec+1))==0: print(etaq) print(EisPartofEtaQ(N,etaq,bftout=1,chars=chars))

We tried this code for various spaces (most of the ones listed in Lemma 2.1.3). This has returned many examples. It is not possible to present them all in this book. In the subsections below, we present the ones we find interesting. Before moving on to the examples we would like to draw attention to an interesting coincidence we have discovered while working on these examples. When k > 1 there are no new eta quotients in E k (0 (N ), χ ) if the sum of vanishing orders at inequivalent cusps is greater than the number of inequivalent cusps, that is, if kN  p + 1  φ(gcd(c, N /c)). > 12 p p|N

c|N

When k = 1 there are no new eta quotients in E k (0 (N ), χ ) if the sum of vanishing orders at inequivalent cusps is greater than twice the number of inequivalent cusps, that is, if  kN  p + 1 φ(gcd(c, N /c)). >2 12 p p|N

c|N

The findings of [25] D. Choi, B. Kim, and S. Lim supports this claim for E k (0 (N ), χ1 ), where N is a squarefree number. The findings of [2] supports this claim for E k (0 (N ), χ1 ), where N is a prime power. A more general version of this problem seems to be very challenging to solve.

4.4.1

The Level 12 Examples

In this section, we give six eta quotients in E 3 (0 (12), χ−3 ). Then we use them to give a basis for M3 (0 (3), χ−3 ) in terms of eta quotients.

4.4

Eta Quotients in E k (0 (N ), χ )

121

Example 4.4.1 We have η12 [3, −9, −9, 3, 27, −9](z) =

∞ 

(σ2 (χ−3 , χ1 ; n) − 6σ2 (χ−3 , χ1 ; n/2)

n=1

−8σ2 (χ−3 , χ1 ; n/4)) q n ,

η12 [4, −7, −4, 4, 13, −4](z) =

∞ 

(σ2 (χ−3 , χ1 ; n) − 7σ2 (χ−3 , χ1 ; n/2)

n=1

−8σ2 (χ−3 , χ1 ; n/4)) q n ,

η12 [−1, 7, −5, −1, 11, −5](z) =1 +

∞ 

(σ2 (χ1 , χ−3 ; n) − 2σ2 (χ1 , χ−3 ; n/2)

n=1

−8σ2 (χ1 , χ−3 ; n/4)) q n ,

η12 [−4, 13, 4, −4, −7, 4](z) =

∞ 

(σ2 (χ1 , χ−3 ; n) + 7σ2 (χ1 , χ−3 ; n/2)

n=1

−8σ2 (χ1 , χ−3 ; n/4)) q n ,

η12 [−9, 27, 3, −9, −9, 3](z) =1 + 9

∞ 

(σ2 (χ1 , χ−3 ; n) + 6σ2 (χ1 , χ−3 ; n/2)

n=1

−8σ2 (χ1 , χ−3 ; n/4)) q n , and η12 [−5, 11, −1, −5, 7, −1](z) =

∞ 

(σ2 (χ−3 , χ1 ; n) + 2σ2 (χ−3 , χ1 ; n/2)

n=1

−8σ2 (χ−3 , χ1 ; n/4)) q n . ∞ n and Next we combine these and isolate the terms n=1 σ2 (χ−3 , χ1 ; n/2)q ∞ n n=1 σ2 (χ1 , χ−3 ; n/2)q to obtain the following statement.

122

4 Application: Eta Quotients

Corollary 4.4.1 We have ∞ 

σ2 (χ−3 , χ1 ; n/2)q n = η12 [3, −9, −9, 3, 27, −9](z) − η12 [4, −7, −4, 4, 13, −4](z),

n=1

(4.9) and ∞

1  σ2 (χ1 , χ−3 ; n/2)q n = η12 [−4, 13, 4, −4, −7, 4](z) − + 9 n=1

1 − η12 [−9, 27, 3, −9, −9, 3](z). 9

(4.10)

We use (4.9), (4.10) and the fact that the dimension of the space M3 (0 (3), χ−3 ) is 2 to conclude that it is generated by the modular forms B(2)|(η12 [3, −9, −9, 3, 27, −9](z) − η12 [4, −7, −4, 4, 13, −4](z)) and 1 B(2)|(η12 [−4, 13, 4, −4, −7, 4](z) − η12 [−9, 27, 3, −9, −9, 3](z)), 9 ∞ where B(t) is the operator on f (z) = n=0 an q n that acts on the q-series expansion of f (z) as follows B(t)|

∞  n=0

an q n =

∞ 

ant q n .

n=0

This result is interesting, because there are no eta quotients in M3 (0 (3), χ−3 ), yet it is generated by using eta quotients coming from a superspace of M3 (0 (3), χ−3 ). We believe there are more spaces like this. We encourage the interested reader to search for modular form spaces that cannot be generated by using eta quotients that are in them, but can be generated by eta quotients coming from a superspace.

4.4.2

The Level 16 Examples

The following eta quotients in E 3 (0 (16), χ−4 ) combine well and that combination yields a nontrivial interesting relationship between their q-series coefficients.

Eta Quotients in E k (0 (N ), χ )

4.4

123

Example 4.4.2 We have ∞

η16 [2, −1, −4, 7, 2](z) = −

1  (σ2 (χ1 , χ−4 ; n) − σ2 (χ1 , χ−4 ; n/2) 16 n=1

−σ2 (χ−4 , χ1 ; n) + 4σ2 (χ−4 , χ1 ; n/2) +32σ2 (χ−4 , χ1 ; n/4)) q n , and ∞

η16 [2, −1, −6, 13, −2](z) =

1 (σ2 (χ1 , χ−4 ; n) − σ2 (χ1 , χ−4 ; n/2) 8 n=1

−σ2 (χ−4 , χ1 ; n) + 12σ2 (χ−4 , χ1 ; n/2) −32σ2 (χ−4 , χ1 ; n/4)) q n . Combining these eta quotients we obtain the following result. Corollary 4.4.2 We have 2η16 [2, −1, −4, 7, 2](z) + η16 [2, −1, −6, 13, −2](z) =

∞ 

(σ2 (χ−4 , χ1 ; n/2) − 8σ2 (χ−4 , χ1 ; n/4)) q n .

n=1

That is, for the q-series coefficients of these eta quotients we have 2a(η16 [2, −1, −4, 7, 2]; 2n + 1) + a(η16 [2, −1, −6, 13, −2]; 2n + 1) = 0 for all nonnegative integers n. The following eta quotient is in E 3 (0 (16), χ−4 ) and has a beautiful symmetry. Example 4.4.3 We have η16 [2, −9, 20, −9, 2](z) =

∞ 

(2σ2 (χ1 , χ−4 ; n/2) − 2σ2 (χ1 , χ−4 ; n/4)

n=1

+σ2 (χ−4 , χ1 ; n) − 8σ2 (χ−4 , χ1 ; n/2)) q n .

124

4 Application: Eta Quotients

4.4.3

The Level 25 Examples

Recall that by Theorem 3.4.3, in the Eisenstein part of theta functions corresponding to diagonal forms there are no Eisenstein series associated with nonreal Dirichlet characters. As demonstrated by the next example this is not the case for eta quotients. In other words, in the following example, we give an eta quotient that can be written in terms of the Eisenstein series including some associated with nonreal Dirichlet characters. This is quite interesting considering the fact that q-series coefficients of all eta quotients are integers. Example 4.4.4 Let ρ denote the primitive Dirichlet character of conductor 5 with ⎧ ⎪ 0 if n ≡ 0 (mod 5), ⎪ ⎪ ⎪ ⎪ ⎪ 1 if n ≡ 1 (mod 5), ⎪ ⎨ ρ(n) = i if n ≡ 2 (mod 5), ⎪ ⎪ ⎪ ⎪ −i if n ≡ 3 (mod 5), ⎪ ⎪ ⎪ ⎩−1 if n ≡ 4 (mod 5). We have ∞

η25 [3, −1, 2](z) =

1 (σ1 (χ1 , χ5 ; n) − σ1 (χ1 , χ5 ; n/5) − σ1 (χ5 , χ1 ; n) 4 n=1

+25σ1 (χ5 , χ1 ; n/5) − iσ1 (ρ, ρ; n) + iσ1 (ρ, ρ; n)) q n . Next, we show that the sums of the divisor functions above combine well and yield rational numbers in the q-series coefficients. Recall that r M (n) denotes the M-free part of n and ν5 (n) denotes the maximal integer so that 5ν5 (n) | n. Corollary 4.4.3 We have η25 [3, −1, 2](z) =

1 2

∞ 

(σ1 (χ1 , χ5 ; n) + σ1 (χ1 , χ1 ; n)) q n

n=1, n≡2 (mod 5) ∞ 

+

+

1 2

(σ1 (χ1 , χ5 ; n) − σ1 (χ1 , χ1 ; n)) q n

n=1, n≡3 (mod 5) ∞  ν5

5 σ1 (χ5 , χ1 ; r5 (n))q n .

n=1, n≡0 (mod 5)

Proof We use formulas given by Proposition 2.5.1 and Corollary 2.5.1 to prove this. Details are left as an exercise. 

4.5

4.5

Eight Families of Eta Quotients Whose Eisenstein Part is a Single Eisenstein Series

125

Eight Families of Eta Quotients Whose Eisenstein Part is a Single Eisenstein Series

In [42] Martin has determined all eta quotients with multiplicative q-series coefficients. In Table I of [42] the eta quotients which are not cusp forms are Eisenstein forms and their Eisenstein part consists of a single Eisenstein series. Next natural object of study is the eta quotients whose Eisenstein part consists of a single Eisenstein series. In this section, we find infinite families of eta quotients whose Eisenstein part is a single Eisenstein series, and hence, although their q-series coefficients are not multiplicative, their Eisenstein part’s q-series coefficients are multiplicative. In the example below we give eight infinite families of eta quotients whose Eisenstein part is a single Eisenstein series. The experimental process of finding these examples is given after the statement. Theorem 4.5.1 If 3 < p ≡ 3 (mod 4) is a prime, then we have   η(5 p−11)/2 (2z)η p−1 ( pz)η2 p−5 (4 pz) ;z E η p−1 (z)η p−4 (4z)η3( p−3)/2 (2 pz) ∞

=

2(4 p)( p−3)/4 ( p − 1)  σ p−2 (χ4 p , χ1 ; n)q n , B p−1 (χ4 p ) n=1

 E

η2 p−5 (z)η p−1 (4z)η(5 p−11)/2 (2 pz) ;z η3( p−3)/2 (2z)η p−4 ( pz)η p−1 (4 pz)



∞

=1−

2( p − 1)  σ p−2 (χ1 , χ4 p ; n)q n , B p−1 (χ4 p ) n=1

 E

η p−1 (z)η2 p−5 (4z)η(5 p−11)/2 (2 pz) ;z η3( p−3)/2 (2z)η p−1 ( pz)η p−4 (4 pz)



∞

=

2( p−1)/2 ( p − 1)  σ p−2 (χ−4 , χ− p ; n)q n , B p−1 (χ4 p ) n=1

and  E

 η(5 p−11)/2 (2z)η2 p−5 ( pz)η p−1 (4 pz) ;z η p−4 (z)η p−1 (4z)η3( p−3)/2 (2 pz)  ∞  2 p ( p−3)/4 ( p − 1)  σk−1 (χ− p , χ−4 ; n)q n . = B p−1 (χ4 p ) n=1

126

4 Application: Eta Quotients

If p ≡ 1 (mod 4) is a prime, then we have   η p+11 (z)η( p−23)/2 (2z)η p+3 (4z)η(5 p−19)/2 (2 pz) E ;z η p−9 ( pz)η p−1 (4 pz) ∞

=1−

 3p + 3 σ(3 p+1)/2 (χ1 , χ−4 p ; n)q n , B(3 p+3)/2 (χ−4 p ) n=1

 E

 η(5 p−19)/2 (2z)η p+3 ( pz)η( p−23)/2 (2 pz)η p+11 (4 pz) ;z η p−1 (z)η p−9 (4z)   ∞ 3p + 3 σ(3 p+1)/2 (χ−4 p , χ1 ; n)q n , = 2(3 p−15)/2 (− p)( p−1)/4 B(3 p+3)/2 (χ−4 p ) n=1

 E

 η(5 p−19)/2 (2z)η p+11 ( pz)η( p−23)/2 (2 pz)η p+3 (4 pz) ;z η p−9 (z)η p−1 (4z)   ∞ 3p + 3 ( p−1)/4 σ(3 p+1)/2 (χ p , χ−4 ; n)q n , = −p B(3 p+3)/2 (χ−4 p ) n=1

and  E

 η p+3 (z)η( p−23)/2 (2z)η p+11 (4z)η(5 p−19)/2 (2 pz) ;z η p−1 ( pz)η p−9 (4 pz)   ∞ 3p + 3 (3 p−15)/2 σ(3 p+1)/2 (χ−4 , χ p ; n)q n . = χ p (2)2 B(3 p+3)/2 (χ−4 p ) n=1

Remark 4.5.1 The proofs of all these identities are similar and straightforward. Less straightforward is finding the identities, or rather knowing where to look. Therefore, before giving the proofs, we would like to explain the process of finding these examples by demonstrating it for finding the eta quotient whose Eisenstein part is a constant multiple of E k (χ4 p , χ1 ; z). This process is experimental and requires a trial and error analysis at each step. Slight changes in this process could help determine eta quotients with another prescribed Eisenstein part. We first state our purpose: For each prime p greater than 3, we would like to find eta quotients of level 4 p whose Eisenstein parts can be expressed as a single Eisenstein series. Therefore we examine the Eisenstein series in Mk (0 (4 p), χ ). As it is a representative case we only give the details for E k (χ4 p , χ1 ; z) where p ≡ 3 (mod 4) and k is even. From (2.1) we have

4.5

Eight Families of Eta Quotients Whose Eisenstein Part is a Single Eisenstein Series

v1/c,4 p (E k (χ4 p , χ1 ; z)) ∈

127

⎧ ⎨N0

if c = 1, 4, p or 4 p, 1 ⎩N0 + if c = 2 or 2 p. 2

By Theorem 2.4.1 we have C1/c (E k (χ4 p , χ1 ; z)) =

 0 if c = 2, 4, p, 2 p or 4 p, 1 if c = 1.

Therefore for any even weight eta quotient η4 p satisfying ⎧ ⎨N0 if c = 1, 4, p or 4 p, v1/c,4 p (η4 p ) ∈ 1 ⎩N0 + if c = 2 or 2 p, 2

(4.11)

and C1/c (η4 p )

 = 0 if c = 2, 4, p, 2 p or 4 p,  = 0 if c = 1,

(4.12)

by Corollary 2.6.2 (b) we have η4 p = AE k (χ4 p , χ1 ; z) + C(z), where A=

C1/1 (η4 p ) . C1/1 (E k (χ4 p , χ1 ; z))

(4.13)

Hence we turn our attention to finding even weight eta quotients that satisfy (4.11) and (4.12). Upon some experiments, we notice that if k is fixed there are only finitely many p for which this is possible, and we additionally notice that increasing k allows us to find examples for bigger p. Therefore to obtain an infinite family of examples we conclude that k should increase as p increases. By Lemma 4.1.3 we have ⎞⎛ ⎞ ⎛ ⎛ ⎞ r1 v1/1 4p 2p p 4 2 1 ⎜ p 2p p 1 2 1 ⎟ ⎜ r ⎟ ⎜ v ⎟ ⎟ ⎜ 2 ⎟ ⎜ 1/2 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ 1 ⎜ ⎜ p 2 p 4 p 1 2 4 ⎟ ⎜ r4 ⎟ ⎜ v1/4 ⎟ ⎟⎜ ⎟ = ⎜ ⎜ ⎟ 24 ⎜ 4 2 1 4 p 2 p p ⎟ ⎜ r p ⎟ ⎜ v1/ p ⎟ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎝ 1 2 1 p 2 p p ⎠ ⎝r2 p ⎠ ⎝v1/2 p ⎠ 1 2 4 p 2p 4p r4 p v1/4 p where for notational convenience we write v1/c for v1/c,4 p (η4 p ). Solving this system for rt (t | 4 p) we obtain

128

4 Application: Eta Quotients

⎛

2p ⎜− p ⎜ 4 ⎜ ⎜ 0 ⎜ p 2 − 1 ⎜ −2 ⎜ ⎝ 1 0

−2 p 5p −2 p 2 −5 2

0 −p 2p 0 1 −2

−2 1 0 2p −p 0

2 −5 2 −2 p 5p −2 p

⎞⎛ ⎞ ⎛ ⎞ v1/1 r1 0 ⎜ ⎟ ⎜ ⎟ 1 ⎟ ⎟ ⎜ v1/2 ⎟ ⎜ r2 ⎟ ⎟⎜ ⎟ ⎜ ⎟ −2 ⎟ ⎜ v1/4 ⎟ ⎜ r4 ⎟ ⎟⎜ ⎟ = ⎜ ⎟. 0 ⎟ ⎜ v1/ p ⎟ ⎜ r p ⎟ ⎟⎜ ⎟ ⎜ ⎟ − p ⎠ ⎝v1/2 p ⎠ ⎝r2 p ⎠ 2p v1/4 p r4 p

(4.14)

Now our goal is to determine v1/c,4 p (η4 p ) that would make each rc an integer and which satisfy (4.11) with v1/1,4 p (η4 p ) = 0. Additionally to have the right character squarefree part of c|rc | must be p. By (4.6) we must also have 

v1/c,4 p (η4 p ) = k

c|4 p

p+1 . 2

Therefore by examining (4.14), we find that putting multiples of p − 1 for each v1/c,4 p (η4 p ) could be helpful. If all v1/c,4 p (η4 p ) are multiples of p − 1, then k have to be a multiple of p − 1. As we like to have the weight be small, even and increase as p increases, we first try k = p − 1. Hence we are now looking for v1/c,4 p (η4 p ) that would make each rc an integer and which satisfy (4.11) with v1/1,4 p (η4 p ) = 0 and 

v1/c,4 p (η4 p ) =

c|4 p

p2 − 1 . 2

Using these observations we set v1/1 = 0, v1/2 = ( p − 1)(a2 p + b2 ), v1/4 = ( p − 1)(a4 p + b4 ), v1/ p = ( p − 1)(a p p + b p ), v1/2 p = ( p − 1)(a2 p p + b2 p ), v1/4 p = ( p − 1)(a4 p p + b4 p ), where 1 , 2 1 = . 2

a2 + a4 + a p + a2 p + a4 p = b2 + b 4 + b p + b 2 p + b 4 p

Note that all ac have to be nonnegative, as otherwise ac p + bc > 0 can be correct only for finitely many p, and hence v1/c,4 p (η4 p ) can be positive only for finitely many p. At this 1 point, we list sets of nonnegative rational numbers that add up to and try the rational 2 numbers in

4.5

Eight Families of Eta Quotients Whose Eisenstein Part is a Single Eisenstein Series



129

 1 1 1 1 . , , , 8 8 4 4

We use this choice and use that p ≡ 3 (mod 4) to list a few possibilities for v1/c,4 p (η4 p ): 1 ∈ N, 2 1 1 ( p − 1) ∈ N + , 4 2 p+1 ( p − 1) ∈ N, 4 p+1 ( p − 1) ∈ N, 8 p−1 ( p − 1) ∈ N, 2 p−1 ( p − 1) ∈ N, 4 p−1 1 ( p − 1) ∈N+ . 8 2 ( p − 1)

After trying a few suitable combinations among these choices, we find that the choice v1/1 = 0, ( p − 1)2 , 8 p−1 v1/4 = , 2 p2 − 1 v1/ p = , 8 p−1 v1/2 p = , 4 ( p − 1)2 v1/4 p = 4 v1/2 =

satisfies all the requirements. In the steps above we made some random choices, different choices in those steps may yield different examples. We encourage the interested reader to explore other choices. Specifically what happens if one chooses a weight other than p − 1? Proof After the explanations in Remark 4.5.1 for the proof it remains to compute C1/1 (η N ) E k (χ4 p , χ1 ; z). C1/1 (E k (χ4 p , χ1 ; z))

(4.15)

130

4 Application: Eta Quotients

We start by computing the ratio of the constant terms in (4.15). By Theorem 2.4.1 we have C1/1 (E k (χ4 p , χ1 ; z)) = χ1 (1)Rχ4 p ,χ1 (1, 1) = 1.

(4.16)

Now we compute C1/1 (η N ) using Theorem 4.1.2. If a = 1 and c = 1, then in the notation of Theorem 4.1.2 we have h t = gcd(1, t) = 1. Therefore since p is greater than 3 and congruent to 3 modulo 4 we can choose n 1 = 0, n 2 = 3, n 4 = 1, n p = 0, n 2 p = 3, n4 p = 3 and δ1 = 0, δ2 = 0, δ4 = 0, δ p = 0, δ2 p = 0, δ4 p = 0. Therefore as p ≡ 3 (mod 4) we have   ⎛ ⎞⎞ 2 at+cn t     − 1 h t ⎜ πi  ⎜ ⎟⎟ c at + n t c ⎟ + δt ω(1, 1) = exp ⎜ rt ⎜ − 2n t ⎟ ⎝ 12 ⎝ 3 − ht ⎠⎠ ht c/h t ⎛

t|N



πi = exp ( p − 1)2 2



= 1. Next, we compute that  t|N

c/h t at/h t + n t c/h t

rt  K

ht t

rt /2

=

 t|N

1 t + nt

rt  rt /2  3( p−1)/4 1 1 = . t 4 p K (4.17)

4.6

Eta Quotients and Newforms

131

Putting ω(1, 1) and (4.17) in Theorem 4.1.2 we obtain C1/1 (η N ) = (−i)k ω(1, 1)

 t|N

1 t + nt

rt  rt /2  3( p−1)/4 1 1 = (−1)( p−1)/2 . t 4 p K

Since p ≡ 3 (mod 4), ( p − 1)/2 is odd, and hence we have 

1 C1/1 (η N ) = − 4p

3( p−1)/4 .

(4.18)

Putting (4.16) and (4.18) together we obtain  3( p−1)/4 C1/1 (η N ) 1 . =− C1/1 (E k (χ4 p , χ1 ; z)) 4p

(4.19)

Now using Example 2.4.4 we have  E p−1 (χ4 p , χ1 ; z) = (4 p)

p−3/2

−2( p − 1) B p−1 (χ4 p )



∞ 

σ p−2 (χ4 p , χ1 ; n)q n .

(4.20)

n=1

The desired result follows from multiplying (4.19) with (4.20) and simplifying.



Next, we discuss the asymptotic behavior of the results given in Theorem 4.5.1. Remark 4.5.2 Let f (z) be any one of the eta quotients studied in Theorem 4.5.1. In each case we have k > 2 and |B f (m)| = 1 > ζ (k − 1) − 1, A( f ) for each m coprime to 4 p. Therefore by Theorem 2.7.3 in each case we have lim

n→∞

a( f ; N n + m) = 1, e( f ; N n + m)

for all 0 < m < 4 p except when 2 | m or p | m.

4.6

Eta Quotients and Newforms

Some eta quotients are known to be newforms, see [42] for the complete list. In [46] some weight 2 newforms are given as linear combinations of eta quotients, and in [4] some weight 2 newforms are given as linear combinations of eta quotients and Eisenstein series. In this section, we give further examples of newforms that can be written as linear combinations of eta quotients and Eisenstein series. As most previous research was focused on weight 2

132

4 Application: Eta Quotients

newforms in this section we opted to search for similar results in higher weight newforms. To produce our examples in Example 4.6.1 we only search modular form spaces with weight greater than 2 and whose cusp subspace dimension is 1. However, such examples are possible for spaces whose cusp space has a dimension bigger than 1. This is discussed at the end of this section in Example 4.6.2. We find the results given in Example 4.6.1 by running the following SAGE command EQS=FindAllEtaQ(k,N,chi) chars=EisBaseFind(k,N,chi) for i in range(len(EQS)): etaq=EQS[i] BB=((EtaQExpand(N,etaq,prec)-EisPartofEtaQ(N,etaq\ ,bftout=0,prec=prec,chars=chars))).series(q,prec) if BB.coefficient(q,1)!=0: print(etaq) print(BB) print(EisPartofEtaQ(N,etaq,bftout=1\ ,prec=0,chars=chars))

for modular form spaces with cusp subspace dimension 1 that are listed in Lemma 2.1.3 that has weight greater than 2. We present some of the output in Example 4.6.1 below. We note that in most cases there are many other linear combinations of eta quotients and Eisenstein series that represent the same newform. In the statement below we choose linear combinations where the coefficient of the eta quotient is 1, if this is not possible, we choose the examples that have some kind of symmetry either in exponents of the eta quotient or in the coefficients of the Eisenstein series. Example 4.6.1 Recall that N(k, N , χ; z) denotes a normalized newform in Mk (0 (N ), χ ). Then we have: N(3, 7, χ−7 ; z) = −8η7 [−1, 7](z) +

∞ 

σ2 (χ−7 , χ1 ; n)q n ,

n=1

N(3, 8, χ−8 ; z) =

3 1 η8 [−6, 9, 9, −6](z) + 4 2

∞ 

(σ2 (χ1 , χ−8 ; n) − 8σ2 (χ−8 , χ1 ; n)) q n ,

n=1

N(3, 12, χ−3 ; z) = η12 [3, 3, −1, −3, 3, 1](z) + 3

∞ 

(σ2 (χ−3 , χ1 ; n/2)

n=1

−8σ2 (χ−3 , χ1 ; n/4)) q n ,

4.6

Eta Quotients and Newforms

133

N(3, 16, χ−4 ; z) = η16 [4, 2, −4, 4, 0](z) + 4

∞ 

(σ2 (χ−4 , χ1 ; n/2)

n=1

−8σ2 (χ−4 , χ1 ; n/4)) q n ,

N(4, 5, χ1 ; z) = −13η5 [−2, 10](z) +

∞ 

(σ3 (χ1 , χ1 ; n) − σ3 (χ1 , χ1 ; n/5)) q n ,

n=1

N(4, 6, χ1 ; z) = 5η6 [6, −6, −2, 10](z) +

∞ 

(σ3 (χ1 , χ1 ; n) − 16σ3 (χ1 , χ1 ; n/2)

n=1

−σ3 (χ1 , χ1 ; n/3) + 16σ3 (χ1 , χ1 ; n/6)) q n , N(4, 8, χ1 ; z) = η8 [4, 6, −6, 4](z) + 4

∞ 

(σ3 (χ1 , χ1 ; n/2) − 17σ3 (χ1 , χ1 ; n/4)

n=1

+16σ3 (χ1 , χ1 ; n/8)) q n , ∞

N(4, 9, χ1 ; z) = η9 [9, −4, 3](z) +

1 (σ3 (χ1 , χ1 ; n) − 82σ3 (χ1 , χ1 ; n/3) 2 n=1

+81σ3 (χ1 , χ1 ; n/9) − σ3 (χ−3 , χ−3 ; n)) q n , N(6, 3, χ1 ; z) = −39η3 [−6, 18](z) +

∞ 

(σ5 (χ1 , χ1 ; n) − σ5 (χ1 , χ1 ; n/3)) q n ,

n=1

N(6, 4, χ1 ; z) = 2η4 [−16, 36, −8](z) −

∞ 

(σ5 (χ1 , χ1 ; n) − σ5 (χ1 , χ1 ; n/2)) q n ,

n=1

N(8, 2, χ1 ; z) = −136η2 [−16, 32](z) +

∞ 

(σ7 (χ1 , χ1 ; n) − σ7 (χ1 , χ1 ; n/2)) q n .

n=1

Now we describe how one can find such examples for modular form spaces whose cusp subspace dimension is greater than 1 via an example. Example 4.6.2 The dimension of S4 (0 (12), χ1 ) is 3. By [39, Newform orbit 12.4.a.a] the q-series expansion of the normalized newform of this space is given by

134

4 Application: Eta Quotients

N(4, 12, χ1 ; z) = q + 3q 3 − 18q 5 + 8q 7 + 9q 9 + 36q 11 − 10q 13 − 54q 15 + 18q 17 − 100q 19 + 24q 21 + 72q 23 + 199q 25 + 27q 27 − 234q 29 − 16q 31 + 108q 33 − 144q 35 − 226q 37 − 30q 39 + 90q 41 + 452q 43 − 162q 45 + 432q 47 − 279q 49 + 54q 51 + 414q 53 − 648q 55 − 300q 57 − 684q 59 + 422q 61 + 72q 63 + 180q 65 + 332q 67 + 216q 69 − 360q 71 + 26q 73 + 597q 75 + 288q 77 + 512q 79 + 81q 81 − 1188q 83 − 324q 85 − 702q 87 − 630q 89 − 80q 91 − 48q 93 + 1800q 95 − 1054q 97 + 324q 99 + O(q 100 ). We aim to write this as linear combinations of the Eisenstein series and eta quotients. Next we find all eta quotients in M4 (0 (12), χ1 ) by running the SAGE command N=12 G=DirichletGroup(N) k=4 chi=G[0] EQS=FindAllEtaQ(k,N,chi)

Then we find the q-series expansion of Eisenstein series in M4 (0 (12), χ1 ) by running the SAGE commands: prec=9 eps=G[0] psi=G[0] f1=Eis(k,eps,psi,1,prec) f2=Eis(k,eps,psi,2,prec) f3=Eis(k,eps,psi,3,prec) f4=Eis(k,eps,psi,4,prec) f5=Eis(k,eps,psi,6,prec) f6=Eis(k,eps,psi,12,prec)

At this point, we would like to find out if N(4, 12, χ1 ; z) =

 t|12

at E(χ1 , χ1 ; t z) +

∗ 

η12

(4.21)

j=1

for some eta quotients. First, by the Sturm bound (Theorem 2.2.4), if the q-series expansion of left hand side and right hand side of (4.21) agree up to and including the coefficients of q 9 , then the equality will hold. Now we use the SAGE function given on page 28 and write a search algorithm to see if there are any suitable eta quotients. Running the SAGE commands

4.7

Bases for Cusp Form Spaces

135

q=var('q') Nf=q + 3*qˆ3 - 18*qˆ(5) + 8*qˆ(7) + 9*qˆ(9) for i1 in range(len(EQS)): f7=EtaQExpand(N,EQS[i1],prec) for i2 in range(i1+1,len(EQS)): f8=EtaQExpand(N,EQS[i2],prec) forms=[f1,f2,f3,f4,f5,f6,f7,f8] AA=ModIdFinder(forms,Nf.series(q,prec+1),prec) if AA and 240*AA[0]==1: print(EQS[i1],EQS[i2]) print(AA)

we obtain more than 2000 suitable pairs of eta quotients with a1 = 1/240. Among them we choose and present the following result: 1 3 5 E(χ1 , χ1 ; z) − E(χ1 , χ1 ; 2z) − E(χ1 , χ1 ; 3z) 240 80 48 1 11 1 − E(χ1 , χ1 ; 4z) E(χ1 , χ1 ; 6z) + E(χ1 , χ1 ; 12z) 6 80 6 − 48η12 [3, 2, −1, −7, −10, 21](z) + 48η12 [3, −4, −1, 5, −4, 9](z).

N(4, 12, χ1 ; z) =

Using this and the q-series expansion of E(χ1 , χ1 ; t z) for primes p greater than 3 we have a(N(4, 12, χ1 ; z); p) = ( p 3 + 1) + 48(−a(η12 [3, 2, −1, −7, −10, 21]; p) + a(η12 [3, −4, −1, 5, −4, 9]; p)). Therefore we obtain a curious equivalence relation modulo 48: a(N(4, 12, χ1 ; z); p) ≡ p 3 + 1 (mod 48). This finishes our discussion on relationships between newforms, Eisenstein series and eta quotients. In the next section, we write bases of some cusp form spaces in terms of eta quotients.

4.7

Bases for Cusp Form Spaces

If a modular form space is generated by eta quotients, then many interesting results and connections follow. Therefore it is always of interest to find a basis in terms of eta quotients. By the results of Rouse and Webb [50] the characterization of the modular form spaces that can be generated by eta quotients is known. As a result, it is known that not all spaces can be

136

4 Application: Eta Quotients

generated by eta quotients. In this section, we give the bases for level 12 and level 16 cusp form spaces. We additionally outline our strategy for finding these bases by demonstrating it for the level 16 case. Theorem 4.7.1 ([9, Theorem 5.1.3]) Let C j,k (z) :=

η6k−3 j−15 (z)ηk−2 j+2 (6z)η3 j−3 (12z) , η3k−2 j−10 (2z)η2k− j−5 (3z)η j−1 (4z)

and D j,k (z) :=

η6k−3 j−11 (z)ηk−2 j (6z)η3 j−2 (12z) . η3k−2 j−6 (2z)η2k− j−5 (3z)η j−2 (4z)

Then the set {C j,k (z) : 1 ≤ j ≤ 2k − 5} constitutes a basis for Sk (0 (12), χ1 ) for all even k greater than 2 and Sk (0 (12), χ−3 ) for all odd k greater than 1; the set {D j,k (z) : 1 ≤ j ≤ 2k − 4} constitutes a basis for Sk (0 (12), χ12 ) for all even k greater than 2 and Sk (0 (12), χ−4 ) for all odd k greater than 1. Next, we give bases of level 16 cusp form spaces. The base for Sk (0 (16), χ1 ) is from the results of [16], the rest are new. Theorem 4.7.2 Let E j,k (z) :=

η4k−2 j−10 (z)η6 (4z)η2 j−2 (16z) , η2k− j−5 (2z)η j−1 (8z)

and F j,k (z) :=

η4k−2 j−8 (z)η(4z)η2 j−2 (16z) . η2k− j−6 (2z)η j−3 (8z)

Then the set {E j,k (z) : 1 ≤ j ≤ 2k − 5} constitutes a basis for Sk (0 (16), χ1 ) for all even k greater than 2 and Sk (0 (16), χ−4 ) for all odd k greater than 1; the set

4.7

Bases for Cusp Form Spaces

137

{F j,k (z) : 1 ≤ j ≤ 2k − 4} constitutes a basis for Sk (0 (16), χ8 ) for all even k greater than 2 and Sk (0 (16), χ−8 ) for all odd k greater than 1. Remark 4.7.1 Below we explain how we discover these examples. Similar arguments may yield bases for other spaces. We first use the dimension formulas given in [53, Proposition 6.1] to compute that  0 if k = 2, dim(Sk (0 (16), χ1 )) = 2k − 5 if k(even) > 2. Therefore to give a basis for Sk (0 (16), χ1 ) with k > 2, we need 2k − 5 linearly independent eta quotients in the space. By Lemma 4.1.3 we have ⎞⎛ ⎞ ⎛ ⎞ ⎛ r1 v1 16 8 4 2 1 ⎜ 4 8 4 2 1 ⎟⎜r ⎟ ⎜v ⎟ 2⎟ 2⎟ ⎜ ⎜ ⎟ ⎜ 1 ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ 1 2 4 2 1 ⎟ ⎜ r 4 ⎟ = ⎜ v4 ⎟ . ⎟⎜ ⎟ ⎜ ⎟ 24 ⎜ ⎝ 1 2 4 8 4 ⎠ ⎝ r 8 ⎠ ⎝ v8 ⎠ 1 2 4 8 16 r16 v16 Solving this system for rt (t | 16) we obtain ⎞⎛ ⎞ ⎛ ⎞ ⎛ v1 r1 2 −2 0 0 0 ⎜−1 5 −4 0 0 ⎟ ⎜ v ⎟ ⎜ r ⎟ ⎟⎜ 2 ⎟ ⎜ 2 ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎜ 0 −2 10 −2 0 ⎟ ⎜ v4 ⎟ = ⎜ r4 ⎟ . ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎝ 0 0 −4 5 −1⎠ ⎝ v8 ⎠ ⎝ r8 ⎠ 0 0 0 −2 2 v16 r16

(4.22)

Similar to Remark 4.5.1, once we find a set of suitable orders v1/c , it uniquely determines the eta quotient. Next, we give some properties of orders and write our problem in terms of them. First, we note that in a set of eta quotients if the order of each eta quotient at 1/16 is different then the set of eta quotients is linearly independent. If η16 ∈ Sk (0 (16), χ1 ), then by (2.1) we have v1/c,16 (η16 ) ∈ N

(4.23)

for all c | 16, and by (4.6) we have v1/1,16 (η16 ) + v1/2,16 (η16 ) + 2v1/4,16 (η16 ) + v1/8,16 (η16 ) + v1/16,16 (η16 ) = 2k. (4.24) So our problem is to determine 2k − 5 many sets of {v1/c : c | 16} that would make each rt (t | 16) an integer where t |rt | is a perfect square; satisfy (4.23); satisfy (4.24); and where

138

4 Application: Eta Quotients

v1/16 are different in each set. We next try fixing three of the orders and let the remaining two vary. We fix v1/2 = 1, v1/4 = 1, v1/8 = 1, and therefore by (4.24) we need v1/1 v1/16 to be equal to 2k − 4 where v1/16 are different in each set. Thus, for the varying ones, if we let v1/16 = j, where j ranges from 1 to 2k − 5, then we must let v1/1 = 2k − j − 4. By (4.22) with this choice each rt ∈ Z (t | 16), and thus this choice satisfies all requirements. In this example, there is a very nice interplay between orders and the dimension. This allows a straightforward analysis. For example, in the case of level 20, the analysis requires more work. One needs to analyze v1/20 in each congruence class modulo 5, see [10, Sect. 6]. Proof The proofs are direct applications of Theorem 4.1.2 and we leave them as exercises.  This finishes our discussion of applications of modular forms to eta quotients. In the next chapter we discuss a variety of other applications of modular forms.

5

Various Applications

5.1

The Convolution Sums

Glaisher [31] and independently Ramanujan [47] have given the following formula for the convolution of the sum of divisor functions  a+b=n, a,b∈N

σ1 (χ1 , χ1 ; a)σ1 (χ1 , χ1 ; b) =

5 1 − 6n σ3 (χ1 , χ1 ; n) + σ1 (χ1 , χ1 ; n). 12 12

There has been a plethora of work on slight variations of convolution of the sum of divisor functions. See [12] and its references for related work. In [47] Ramanujan has additionally shown that  a+b=n, a,b∈N

σ2k1 −1 (χ1 , χ1 ; a)σ2k2 −1 (χ1 , χ1 ; b) = −

k B2k1 (χ1 )B2k2 (χ1 ) σ2k−1 (χ1 , χ1 ; n) 4k1 k2 B2k (χ1 )

+ O(n 2(2k−1)/3 ), where k1 and k2 are positive integers with k1 + k2 = k > 2. Further in [16] a character analog of this formula was proven. In this section, we use the techniques developed in this book to consider convolutions of the sum of divisor functions in a more general setting. Theorem 5.1.1 Let 1 , ψ1 , 2 , ψ2 be primitive Dirichlet characters with conductors L 1 , M1 , L 2 and M2 , respectively. Let k1 and k2 be integers that are greater than or equal to 2 Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-3-031-32629-5_5.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 Z. S. Aygin, Introduction to Applications of Modular Forms, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-32629-5_5

139

140

5 Various Applications

where 1 ψ1 (−1) = (−1)k1 and 2 ψ2 (−1) = (−1)k2 . Assume that neither (k1 , 1 , ψ1 ) nor (k2 , 2 , ψ2 ) is equal to (2, χ1 , χ1 ). Then for all positive integers d1 and d2 we have   E(E k1 (1 , ψ1 ; d1 z)E k2 (2 , ψ2 ; d2 z); z) = a(, ψ, t)E k (, ψ; t z), (,ψ)∈E(N ,χ) t|N /L M

(5.1) where k = k1 + k2 , N = lcm(d1 L 1 M1 , d2 L 2 M2 ), χ = 1 ψ1 2 ψ2 (mod N ), a(, ψ, t) :=

 p|N

pk



p k − ψ( p) c|N /L M

R,ψ (t, c)S,ψ (N , t, c)CcM,ψ ,

and CcM,ψ :=

R1 ,ψ1 (cM, d1 M1 )R2 ,ψ2 (cM, s M2 ) φ(cM)

cM 

ψψ1 ψ2 (a).

a=1, gcd(a,cM)=1

Remark 5.1.1 To keep the statement of this theorem simple we have excluded the Eisenstein series for k = 1 and the Eisenstein series for (k, , ψ) = (2, χ1 , χ1 ). However, both cases can be handled by using the modularity conditions and the constant terms given by Theorem 2.4.3. To illustrate this, in Example 5.1.4, we give an example for a case involving weight 1 Eisenstein series. Proof By Theorem 2.2.1 we have E k1 (1 , ψ1 ; d1 z)E k2 (2 , ψ2 ; d2 z) ∈ Mk (0 (N ), χ ), and by Theorems 2.4.1 and 2.2.1 we have Ca/c (E k1 (1 , ψ1 ; z)E k2 (2 , ψ2 ; d1 z)) = ψ1 ψ2 (a)R1 ,ψ1 (c, d1 M1 )R2 ,ψ2 (c, d2 M2 ). Now the statement follows from Theorem 2.6.1.



If we isolate the convolution sums using the identity above we obtain a formula for convolution sums. Corollary 5.1.1 With the assumptions of the Theorem 5.1.1 we have  σk1 −1 (1 , ψ1 ; a)σk2 −1 (2 , ψ2 ; b) d1 a+d2 b=n, a,b∈N

5.1 The Convolution Sums

=

 (,ψ)∈E(N ,χ)

141

⎛ ⎝



⎞ C1 (, ψ, t)σk−1 (, ψ; n/t)⎠

t|N /L M

+ C2 σk2 −1 (2 , ψ2 ; n/d2 ) + C3 σk1 −1 (1 , ψ1 ; n/d1 ) + O(n k/2 ), where C1 (, ψ, t) := a(, ψ, t)

Nk (, ψ) , Nk1 (1 , ψ1 )Nk2 (2 , ψ2 )

C2 := −

1 (0) , Nk1 (1 , ψ1 )

C3 := −

2 (0) . Nk2 (2 , ψ2 )

and

Proof First we expand the products of the Eisenstein series on the left side of (5.1). Then we compare the coefficients of q n for n ≥ 1. The result follows by isolating the desired term.  In the SAGE function below we automate the process of computing C1 (, ψ, t), C2 and C3 of Corollary 5.1.1. SAGE function to compute C1 (, ψ, t), C2 and C3 Inputs: • eps1, psi1, eps2, psi2: The elements of DirichletGroup(N) which correspond to the Dirichlet characters 1 , ψ1 , 2 , ψ2 , respectively. • k1, k2, d1, d2: The positive integers k1 , k2 , d1 and d2 , respectively. • chars: A value of 0 or the character pairs (, ψ) ∈ E(N , χ ) given as a list. This is an optional input with a default value of 0. This is useful when working with multiple modular forms in the same space, as EisBaseFind(k,N,chi) can be time consuming to run each time.

def ConvolutionSum(k1,eps1,psi1,d1,k2,eps2,psi2,d2,chars=0): eps1=eps1.primitive_character() eps2=eps2.primitive_character() psi1=psi1.primitive_character() psi2=psi2.primitive_character() G1=DirichletGroup(1)

142

5 Various Applications if eps1(-1)*psi1(-1)!=(-1)ˆk1: return 'First character pair and the weight is not\ compatible!' if eps2(-1)*psi2(-1)!=(-1)ˆk2: return 'Second character pair and the weight is not\ compatible!' if eps1==G1[0] and psi1==G1[0] and k1==2: return '(2,chi_1,chi_1) is not allowed!' if eps2==G1[0] and psi2==G1[0] and k2==2: return '(2,chi_1,chi_1) is not allowed!' L1,L2=eps1.conductor(),eps2.conductor() M1,M2=psi1.conductor(),psi2.conductor() EisNorml1=EisNormalCoeff(k1,eps1,psi1) EisNorml2=EisNormalCoeff(k2,eps2,psi2) k=k1+k2 N=lcm(L1*M1*d1,L2*M2*d2) G=DirichletGroup(N) chi=eps1.extend(N)*psi1.extend(N)*eps2.extend(N)\ *psi2.extend(N) if chars==0: chars=EisBaseFind(k,N,chi) for char in chars: eps,psi=char[0],char[1] L,M=eps.conductor(),psi.conductor() Constants={} for c in divisors(N/L/M): val=0 for a in range(1,c*M+1): if gcd(a,c*M)==1: val+=QQbar(psi(a)*psi1.bar()(a)\ *psi2.bar()(a)).radical_expression()\ *CalR(k1,eps1,psi1,c*M,d1*M1)\ *CalR(k2,eps2,psi2,c*M,d2*M2) Constants.update({c*M: val/euler_phi(c*M)}) EisNl=EisNormalCoeff(k,eps,psi) for t in divisors(N/L/M): print(k-1,eps,psi,t,QQbar(bft(k,N,eps\ ,psi,t,Constants)/EisNorml1/EisNorml2)\ .radical_expression()) print(k1-1,eps1,psi1,d1,-eps1(0)/EisNorml1) print(k2-1,eps2,psi2,d2,-eps2(0)/EisNorml2)

Output: None, prints k − 1, , ψ, t and the coefficient of σk−1 (, ψ; n/t) of all the terms in Corollary 5.1.1.

5.1 The Convolution Sums

143

Example 5.1.1 (Ramanujan) For even k, we have Nk (χ1 , χ1 ) =

−2k , Bk (χ1 )

and a(χ1 , χ1 , 1) = 1. Therefore, using Corollary 5.1.1, for all even positive integers k1 and k2 that are both greater than 2 we obtain Ramanujan’s formula  σk1 −1 (χ1 , χ1 ; a)σk2 −1 (χ1 , χ1 ; b) a+b=n, a,b∈N

= C1 (χ1 , χ1 , 1)σk−1 (χ1 , χ1 ; n) + C2 σk2 −1 (χ1 , χ1 ; n) + C3 σk1 −1 (χ1 , χ1 ; n) + O(n k/2 ), where C1 (χ1 , χ1 , t) =

−k Bk1 (χ1 )Bk2 (χ1 ) , 2k1 k2 Bk (χ1 )

C2 =

Bk1 (χ1 ) , 2k1

C3 =

Bk2 (χ1 ) . 2k2

and

Example 5.1.2 To obtain a formula for the convolution sum  σ2 (χ1 , χ−3 ; a)σ2 (χ1 , χ−3 ; b), a+b=n, a,b∈N

we run the code G=DirichletGroup(12) eps1=G[0] psi1=G[2] eps2=G[0] psi2=G[2] ConvolutionSum(3,eps1,psi1,1,3,eps2,psi2,1)

and the output is

144

5 Various Applications

5 Dirichlet character modulo 1 of conductor 1\ Dirichlet character modulo 1 of conductor 1 \ 1 1/117 5 Dirichlet character modulo 1 of conductor 1\ Dirichlet character modulo 1 of conductor 1 \ 3 -81/13 2 Dirichlet character modulo 1 of conductor 1\ Dirichlet character modulo 3 of conductor 3 mapping 2 |--> -1\ 1 1/9 2 Dirichlet character modulo 1 of conductor 1\ Dirichlet character modulo 3 of conductor 3 mapping 2 |--> -1\ 1 1/9

Using this output and Corollary 5.1.1 we write the formula 

σ2 (χ1 , χ−3 ; a)σ2 (χ1 , χ−3 ; b) =

a+b=n, a,b∈N

1 81 σ5 (χ1 , χ1 ; n) − σ5 (χ1 , χ1 ; n/3) 117 13

1 1 + σ2 (χ1 , χ−3 ; n) + σ2 (χ1 , χ−3 ; n) + O(n 3 ). 9 9 Example 5.1.3 Let

ρ(n) =

⎧ ⎪ ⎪0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨1 i ⎪ ⎪ ⎪ ⎪ −i ⎪ ⎪ ⎪ ⎩−1

if n ≡ 0 (mod 5), if n ≡ 1 (mod 5), if n ≡ 2 (mod 5), if n ≡ 3 (mod 5), if n ≡ 4 (mod 5),

as in Example 4.4.4. To find a formula for the convolution sum  σ1 (χ5 , χ5 ; a)σ2 (χ1 , ρ; b), a+b=n, a,b∈N

we run the code G=DirichletGroup(5) eps1=G[2] psi1=G[2] eps2=G[0] psi2=G[1] ConvolutionSum(2,eps1,psi1,1,3,eps2,psi2,1)

5.1 The Convolution Sums

145

and from the output and Corollary 5.1.1 we write the formula 

σ1 (χ5 , χ5 ; a)σ2 (χ1 , ρ; b) =

a+b=n, a,b∈N

43 + 74i 2+i σ4 (χ5 , ρ 3 ; n) + σ2 (χ1 , ρ; n) 1465 5

+ O(n 5/2 ). Example 5.1.4 Let p be a prime congruent to 3 modulo 4. Then by Theorem 2.2.1 we have E 1 (χ1 , χ− p ; z)2 ∈ M2 (0 ( p), χ1 ) and by Theorem 2.4.3 we have −i√ p C1/c (E 1 (χ1 , χ− p ; z)) =

p

if c = 1, if c = p.

1

Therefore, by Theorem 2.2.1 we have C1/c (E 1 (χ1 , χ− p ; z) ) = 2

−1 p

if c = 1,

1

if c = p.

And hence by Theorem 2.6.1 we have E 1 (χ1 , χ− p ; z)E 1 (χ1 , χ− p ; z) = −

1 p E 2 (χ1 , χ1 ; z) + E 2 (χ1 , χ1 ; pz) p−1 p−1

+ C p (z)

(5.2)

for some cusp form in C p (z) ∈ S2 (0 ( p), χ1 ). Multiplying the Eisenstein series and comparing the coefficients of q n in (5.2) for all n ≥ 1 we obtain the formula  a+b=n, a,b∈N

σ0 (χ1 , χ− p ; a)σ0 (χ1 , χ− p ; b) =

24 24 p σ1 (χ1 , χ1 ; n) − σ1 (χ1 , χ1 ; n/ p) p−1 p−1 − B1 (χ− p )σ0 (χ1 , χ− p ; n) + O(n).

Compared to the terms σ1 (χ1 , χ1 ; n) the error O(n) is not small. Therefore, this formula may not be an asymptotic one. However, for p = 11 and p = 19 we have dim(S2 (0 ( p), χ1 )) = 1. And therefore by Remark 2.3.1 we conclude that C p (z) of (5.2) is a newform for p = 11 and p = 19. If we normalize the terms we obtain

146

5 Various Applications

N(2, 11, χ1 ; z) =

5 (E 1 (χ1 , χ−11 ; z)E 1 (χ1 , χ−11 ; z) 8

1 11 + E 2 (χ1 , χ1 ; z) − E 2 (χ1 , χ1 ; 11z) 10 10

N(2, 19, χ1 ; z) =

3 (E 1 (χ1 , χ−19 ; z)E 1 (χ1 , χ−19 ; z) 8

1 19 + E 2 (χ1 , χ1 ; z) − E 2 (χ1 , χ1 ; 19z) . 18 18

and

5.2

Klein Forms

In this section, we study Klein forms which are defined as follows. Definition 5.2.1 Let s and t be rational numbers, not both integers. Then the Klein form K (s, t; z) is defined by the infinite product K (s, t; z) := eπ it(s−1) q s(s−1)/2 (1 − e2π it q s )

∞  (1 − e2π it q n+s )(1 − e−2πit q n−s ) . (1 − q n )2

n=1

Klein forms are not modular forms, however, certain products and linear combinations are. Klein forms appear in various contexts like q-analogs of continued fractions [34], crank generating function [41]. In this section, our goal is to develop tools that should be useful in working with combinations of Klein forms that are modular forms. We start with some useful transformation properties which are due to Kubert and Lang [37], and below we give a refined version from [29, Proposition 2.1]. Proposition 5.2.1 Let s and t be rational numbers, not both integers. Then the following statements hold. (a) We have K (−s, −t; z) = −K (s, t; z). (b) For integers u and v we have K (s + u, t + v; z) = (−1)uv+u+v e−πi(ut−vs) K (s, t; z). (c) If M =

 ab ∈ 0 (1), then we have cd K (s, t; M(z)) = (cz + d)−1 K (sa + tc, sb + td; z).

Next, we compute the constant term of the reciprocals of Klein forms.

5.2

Klein Forms

147

Lemma 5.2.1 Let s and t be rational numbers, not both integers. Then we have ⎧ ⎨0 if s ∈ / Z, 1 π i(s+1)(t+1) lim = e ⎩− z→i∞ K (s, t; z) if s ∈ Z. 1 − e2π it Proof If s is not an integer, then by definition the order of vanishing of than 0. And thus the result follows. If we let s = 0, we have

1 K (s,t;z)

is greater

∞  (1 − q n )2 1 eπ it = K (0, t; z) 1 − e2π it (1 − e2π it q n )(1 − e−2πit q n ) n=1

eπ it = (1 + O(q)) . 1 − e2π it

(5.3)

Furthermore, by Proposition 5.2.1 for all integers s we have 1 1 1 = = (−1)s eπist . K (s, t; z) K (0 + s, t + 0; z) K (0, t; z) Hence the result follows using (5.3) in this.



Next we study the order of vanishing and the constant terms of Klein forms at each cusp. We note that x denotes the fractional part of x, that is, x = x − x. Proposition 5.2.2 Let s and t be rational numbers, not both integers. Then we have

 N 1 = va/c,N sa + tc (1 − sa + tc ). K (s, t; z) gcd(c2 , N ) Additionally, we have ⎧ ⎨0 if sa + tc ∈ / Z, (cz + d)−1 π i(sa+tc+1)(sb+td+1) lim = ⎩− e z→i∞ K (s, t; M(z)) if sa + tc ∈ Z. 1 − e2π i(sb+td) Proof For each M =

 ab ∈ 0 (1), we have cd

1 1 1 = =  . −1 K (s, t; M(z)) (cz + d) K (sa + tc, sb + td; z) K s, t; az+b cz+d

148

5 Various Applications

Therefore the first part follows from the definition of vanishing order and the definition of Klein forms. The second part is deduced from Proposition 5.2.1 together with Lemma 5.2.1.  In [29, Theorem 2.8] Eum, Koo and Shin have determined a family of quotients of Klein forms which are modular forms. Next, we give their statement and compute its constant term. Theorem 5.2.1 For N an integer greater than 1 we have N −1 

K (0, j/N ; z)−12/ gcd(N −1,12) ∈ Mk (0 (N ), χ1 ),

j=1

12(N − 1) . The constant terms of it at a/c with c | N are given by gcd(N − 1, 12) ⎧ ⎛ ⎞ ⎪0 if c = N , ⎪ N −1 ⎨  N −1 −12/ gcd(N −1,12) πi j( j+1)/N ⎠=  K (0, j/N ; z) Ca/c ⎝ e ⎪ (−1) j if c = N . ⎪ j=1 ⎩ 1 − e2π i j/N

where k =

j=1

Proof The first part is from [29, Theorem 2.8]. The second part is determined by using Proposition 5.2.2.  Next we restrict N to be a prime and use Theorem 5.2.1 in Theorem 2.6.1 to find the following formula. Corollary 5.2.1 Let p be a prime and k = p  ∞  j=1 n=1

=

12 . Then we have gcd( p − 1, 12)

(1 − q n )2 (1 − e2π i j/ p q n )(1 − e−2πi j/ p q n )

−1 pk E k (χ1 , χ1 ; z) + k E k (χ1 , χ1 ; pz) + C(z) −1 p −1

pk

for some cusp form C(z) in Mk (0 ( p), χ1 ). The previous results and the results we derive in this section can be used to study the modularity of Klein forms. Some linear combinations of the form

5.3 The Character Analog of Dedekind’s Eta Function N −1  s=1

149

cs K (0, −s/N ; z)

are known to be a modular form of weight 1 for certain cs ∈ C. We encourage the interested reader to consider these types of linear combinations of reciprocals of Klein forms and determine modularity criteria for them. Additionally, similar to eta quotients, it should be interesting to study the modularity of quotients of Klein form of the form  K (s, t; dz)rd . d|N

In the next section we define and study the character analog of Dedekind’s eta function. These functions are finite products of Klein forms, and the results of this section will be helpful in the next one.

5.3

The Character Analog of Dedekind’s Eta Function

In this section, we introduce the character analog of Dedekind’s eta function. Definition 5.3.1 Let D > 1 be a quadratic discriminant. Then the character analog of Dedekind’s eta function is defined by the infinite product R D (z) := q B2 (χ D )/4

∞ 

(1 − q n )χ D (n) .

n=1

From the definition, it is clear that R D (z) has no zeros or poles in the upper half plane of complex numbers, see also [32, Theorem 1.1]. We recall that Dedekind’s eta function is given by η(z) = q 1/24

∞ 

(1 − q n ).

n=1

This can be rewritten as η(z) = q B2 (χ1 )/4

∞ 

(1 − q n )χ1 (n) .

n=1

From this we see that the Dedekind’s eta function is the primitive principal character version of R D (z). Hence the term character analog of Dedekind’s eta function is coined. However, we exclude the D = 1 case from the definition as the weight of the transformation formula of the D = 1 case differ from that of the cases D > 1.

150

5 Various Applications

The following SAGE function gives the q-series expansion of R D (z) up to a given precision. SAGE function to compute q-series of R D (z) Inputs: • D: The positive quadratic discriminant D. • prec: A positive integer, for the number of terms to compute in the q-series expansion.

def RD(D,prec): q=var('q') t=var('t') B=2*(sum((kronecker(D,a)*t*exp(a*t)/(exp(D*t)-1))\ for a in range(1,D+1)).series(t,3)).coefficient(t,2) return qˆ(B/4)*(prod((1-qˆn)ˆkronecker(D,n)\ for n in range(1,prec+2))).series(q,prec+1)

Output: The q-series expansion of R D (z) up to and including the term q pr ec+B2 (χ D )/4 .

In [48] Ramanujan and in [49] Rogers have independently shown that the function R5 (z) has the continued fraction expansion R5 (z) =

q 1/5 q

1+ 1+

.

q2 q3 1+ q4 1+ ···

Additionally, Ramanujan has uncovered the following identities that relate eta quotients and the character analog of Dedekind’s eta function: η(z/5) 1 − R5 (z) − 1 = , R5 (z) η(5z) 1 R55 (z) and

− R55 (z) − 11 =

η6 (z) η6 (5z)

5.3 The Character Analog of Dedekind’s Eta Function

151

η2 (z) 1 − R13 (z) − 3 = 2 . R13 (z) η (13z) Inspired by these identities and more recent studies like [14, 32, 34] we work on the reciprocal sums of the form 1 R tD (z)

± R tD (z)

and provide tools that should be useful in integrating character analogs of eta quotients in studying modular forms. First, we give useful transformation properties concerning R D (z). This is a generalization of [14, Lemma 4]. Lemma 5.3.1 (a) Let D = 5. Then for all positive integers t the transformation formulas 1 R55t (M z)

+ (−1)t R55t (M z) =

1 R55t (z)

+ (−1)t R55t (z)

and 

1 R55t (M z)

− (−1)

t

R55t (M z)

= χ5 (d)

1 R55t (z)

 − (−1)

t

R55t (z)

hold for all M ∈ 0 (D). (b) Let D = 8. Then for all positive integers t the transformation formulas 1 1 + R82t (M z) = 2t + R82t (z) R82t (M z) R8 (z) and 1 − R82t (M z) = χ8 (d) R82t (M z)



1 − R82t (z) R82t (z)



hold for all M ∈ 0 (D). (c) Let D > 8 be a prime quadratic discriminant (that is, D( = 5) is a prime congruent to 1 modulo 4). Then for all positive integers t the transformation formulas 

1 1 t t t t − (−1) − (−1) R (M z) = χ (d) R (z) D D D R tD (M z) R tD (z) and 1 1 + (−1)t R tD (M z) = t + (−1)t R tD (z) R tD (M z) R D (z) hold for all M ∈ 0 (D).

152

5 Various Applications

(d) Let D > 8 be a nonprime quadratic discriminant. Then for all positive integers t the transformation formulas 1 1 + R tD (M z) = t + R tD (z) R tD (M z) R D (z) and 1 − R tD (M z) = χ D (d) t R D (M z)



1 − R tD (z) t R D (z)

hold for all M ∈ 0 (D). Proof By [32, Theorem 1.1], for D > 1 we have R D (M(z)) = L(M)χ (d)R D (z)χ D (d) , where

L(M) =

⎧ ⎪ ⎪ ⎨a fifth root of unity

if D = 5,

a second root of unity ⎪ ⎪ ⎩1

if D = 8, otherwise,

and χ=

χD

if D is a prime,

χ1

if D is not a prime. 

The result follows from this after a careful case analysis. Next, we give the order of vanishing of R D (z) at each cusp. Lemma 5.3.2 If N is a multiple of D, then for all cusps a/c we have va/c,N (R D ) =

N gcd(c, D)2 gcd(c2 , N )D



χ D ( j)

0< j 2t|va/c,N (R D )| for each cusp a/c ∈ R(0 (N )), then   1 2t + R8 (z) ∈ Sk (0 (N ), χ ); f (z) R82t (z) and  f (z)

 1 2t − R8 (z) ∈ Sk (0 (N ), χ χ8 ). R82t (z)

(c) If D | N is a prime greater than 8 and congruent to 1 modulo 4; and if f satisfies va/c,N ( f ) > t|va/c,N (R D )| for each cusp a/c, then  f (z)

1 t t + (−1) R (z) ∈ Sk (0 (N ), χ ); D R tD (z)

and  f (z)

1 t t − (−1) R (z) ∈ Sk (0 (N ), χ χ D ). D R tD (z)

(d) If D | N is a nonprime positive fundamental discriminant greater than 8 and if f satisfies va/c,N ( f ) > t|va/c,N (R D )| for each cusp a/c, then  f (z)

1 t + R (z) ∈ Sk (0 (N ), χ ); D R tD (z)

and  f (z)

1 t − R (z) ∈ Sk (0 (N ), χ χ D ). D R tD (z)

Proof By Lemma 5.3.2 and the fact that each f (z) is a modular form, the desired transformation property in each case is satisfied.

156

5 Various Applications

Given that f (z) is a cusp form, it has no poles in the upper half plane of complex numbers. Additionally R D (z) has no zeros  or poles in the  upper half plane of complex numbers. 1 t Therefore, the reciprocal sum R t (z) ± R D (z) does not have any poles in the upper half D   plane of complex numbers. Hence the product f (z) R t 1(z) ± R tD (z) is holomorphic on D the upper half plane of complex numbers. By Lemma 5.3.2 we have 

  1 t ± R (z) = −t va/c,N (R D ) . va/c,N D t R D (z) Hence the condition on orders of  vanishing of f (z) in each statement, guarantees that the  1 t function f (z) R t (z) ± R D (z) vanishes at each cusp. Therefore in each case D    f (z) R t 1(z) ± R tD (z) is a cusp form in the spaces given. D

Theorem 5.4.1 can be used to produce cusp forms from the known ones. Next, we illustrate this idea by giving a basis for the spaces S4 (0 (17), χ1 ) and S4 (0 (17), χ17 ). Example 5.4.1 ([14, Lemma 6]) The dimension of S4 (0 (17), χ1 ) is 4 and there is only 1 eta quotient in it, namely η4 (z)η4 (17z). The dimension of S4 (0 (17), χ17 ) is also 4 and there are only 2 eta quotients in it, namely η(z)η7 (17z) and η7 (z)η(17z). Therefore we cannot find a basis for these spaces in terms of level 17 eta quotients. Using Theorem 5.4.1 we find that the forms 

1 η4 (z)η4 (17z) − R17 (z) , R17 (z) 

1 η(z)η7 (17z) + R17 (z) , R17 (z)   1 7 2 η(z)η (17z) − R17 (z) 2 (z) R17 are in S4 (0 (17), χ1 ); and the forms  η4 (z)η4 (17z)

1 + R17 (z) , R17 (z)

5.5

Elliptic Curves

157



1 η(z)η (17z) − R17 (z) , R17 (z) 

2 1 η(z)η7 (17z) − R17 (z) R17 (z) 7

are in S4 (0 (17), χ17 ). Next, we compare q-series coefficients of these cusp forms and conclude that the set of cusp forms  



1 1 η4 (z)η4 (17z), η4 (z)η4 (17z) − R17 (z) , η(z)η7 (17z) + R17 (z) , R17 (z) R17 (z)   1 2 η(z)η7 (17z) (z) − R17 2 R17 (z) is linearly independent, hence it constitutes a basis for the space S4 (0 (17), χ1 ), and similarly we find that the set of cusp forms  

1 7 7 7 η (z)η(17z), η(z)η (17z), η(z)η (17z) − R17 (z) , R17 (z)  

2 1 η(z)η7 (17z) − R17 (z) R17 (z) is linearly independent, hence it constitutes a basis for the space S4 (0 (17), χ17 ). This finishes or discussion of the character analogs of Dedekind’s eta function. Next we discuss some applications of modular forms to elliptic curves.

5.5

Elliptic Curves

In this section, we discuss applications of modular forms to elliptic curves. The connection between modular forms and elliptic curves has been suggested by Taniyama, and later proven by Wiles for a large class of elliptic curves. This famously established a proof for Fermat’s Last Theorem. A proof of the connection in its most general form was given by Breuil, Conrad, Diamond, and Taylor [24]. This connection is known as the Modularity Theorem. One of the aspects of the modularity theorem is that it gives a relationship between the q-series coefficients of weight 2 newforms and the number of solutions to an elliptic curve modulo p. Diamond and Shurman in [27] use the term Modularity Theorem, version a p for this connection and below we state it as it was given in [27, Theorem 8.8.1].

158

5 Various Applications

Theorem 5.5.1 (Modularity Theorem, version a p ) Let E be an elliptic curve over Q with conductor N with the Weierstrass equation given by E : y 2 + a1 x y + a3 y = x 3 + a2 x 2 + a4 x + a6 , where a1 , a2 , a3 , a4 and a6 are integers. Let E(Z p ) = {∞} ∪ {(x, y) ∈ Z p × Z p |y 2 + a1 x y + a3 y = x 3 + a2 x 2 + a4 x + a6 }, where Z p denotes the finite field of characteristic p. Then for some normalized newform f E (z) ∈ S2 (0 (N ), χ1 ), the number of solutions to an elliptic curve modulo p is given by #E(Z p ) = p + 1 − a( f E ; n) for each prime p  N . The table of Weierstrass equations of elliptic curves of conductors N is given in LMFDB [39, Elliptic Curves over Q]. In the examples below we use the Modularity Theorem, the elliptic curve data tables from LMFDB [39, Elliptic Curves over Q] and previous results from this book to write #E(Z p ) in terms of the sum of divisor function and other arithmetic functions. The subscripts in the elliptic curves below are the labels given by LMFDB [39]. Example 5.5.1 (Conductor 11) There is a unique isogeny class of elliptic curves of conductor 11. We choose E 11.a1 : y 2 + y = x 3 − x 2 − 7820x − 263580, as given in LMFDB [39, Elliptic curve with LMFDB label 11.a1] as the representative of its class. On the other hand by Example 5.1.4 we have N(2, 11, χ1 ; z) =

5 (E 1 (χ1 , χ−11 ; z)E 1 (χ1 , χ−11 ; z) 8

1 11 + E 2 (χ1 , χ1 ; z) − E 2 (χ1 , χ1 ; 11z) . 10 10

As this is the unique normalized newform in S2 ((11), χ1 ), by Theorem 5.5.1, we have #E 11.a1 (Z p ) =

5  5(χ−11 ( p) + 1) p+1 σ0 (χ1 , χ−11 ; i)σ0 (χ1 , χ−11 ; j) + − , 2 2 2 i+ j= p, i, j∈N

for all primes p = 11. Using this we deduce that for all primes p ≡ −1 (mod 10), we have #E 11.a1 (Z p ) ≡ 0 (mod 5).

5.5

Elliptic Curves

159

Example 5.5.2 (Conductors 19, 20, 24, 32, 36) Let E 19.a1 : y 2 + y = x 3 + x 2 − 769x − 8470, E 20.a1 : y 2 = x 3 + x 2 − 41x − 116, E 24.a1 : y 2 = x 3 − x 2 − 384x − 2772, E 32.a1 : y 2 = x 3 − 11x − 14, and E 36.a1 : y 2 = x 3 − 135x − 594. By using arguments similar to Example 5.5.1 and the newforms given in Examples 5.1.4 and 3.4.2 we obtain #E 19.a1 (Z p ) =

3  3(χ−19 ( p) + 1) p+1 σ0 (χ1 , χ−19 ; a)σ0 (χ1 , χ−19 ; b) + + 2 2 2 a+b= p, a,b∈N

for all primes p = 19; 3 p+1 #E 20.a1 (Z p ) = r (1, 1, 5, 5; p) + 8 2 for all primes p = 2 or 5; #E 24.a1 (Z p ) =

1 r (1, 1, 6, 6; p) 2

for all primes p = 2 or 3; #E 32.a1 (Z p ) =

1 1 − χ−4 ( p) r (1, 1, 8, 8; p) + ( p + 1) 2 2

for all primes p = 2; 3 p+1 #E 36.a1 (Z p ) = r (1, 1, 9, 9; p) + 8 2 for all primes p = 2 or 3. This finishes our discussion of applications of modular forms to elliptic curves.

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Index

B Bernoulli number, 11

C Character analog of Dedekind’s eta function, 149 Conductor, 4 Constant term of f at the cusp r , 18 Cusp form, 18 Cusp part of f (z), 19 Cusps, 14

D Dedekind’s eta function, 105 Dirichlet character, 1 nonreal, 3 real, 3 Dirichlet L-function, 11

E Eisenstein form, 18 Eisenstein part of f (z), 19 Eisenstein series, 18 Elliptic point, 14 Eta quotient, 108 Even character, 3

F Full modular group, 13 G Gauss sum, 9, 68 H Hecke bound, 19 K Klein form, 146 Kronecker symbol, 1 L Linear fractional transformation, 13 The L-functions and modular forms database (LMFDB), 30 M Modular form, 18 Modularity Theorem, 157 Modular subgroup, 14 N Newform, 30 Normalized newforms, 30

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 Z. S. Aygin, Introduction to Applications of Modular Forms, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-32629-5

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166 O Odd character, 3 Oldform, 29 Order of vanishing of f at r , 17 P Positive definite matrix, 65 Positive definite quadratic form, 65 Primitive Dirichlet character corresponding to χ, 4 Primitive Dirichlet character of conductor N , 4 Primitive principal character, 4 Principal character, 2 Q q-series expansion, 17 Quadratic discriminant, 1

Index S Separable quadratic form, 71 Space of newforms, 30 Space of oldforms, 29 Sturm bound, 27

T Theta function, 66 Transformation property for 0 (N ) and χ with weight k, 15

W Weight k Eisenstein series associated to  and ψ, 31

Index of Notation

A( f ), 59 A N (c), 14 B f (n), 59 n B(t)| ∞ n=0 an q , 122 Bk (χ ), 11 C j,k (z), 136 D(N ), 1 Do (N ), 4 D j,k (z), 136 E(Z p ), 158 E( f ; z), 18 E k (0 (N ), χ ), 18 E k (, ψ; t z), 24, 31 E kold (0 (N ), χ ), 30 E j,k (z), 136 F j,k (z), 136 K (s, t; z), 146 L(χ , k), 11 M(z), 13 Mk (0 (N ), χ ), 18 Mkold (0 (N ), χ ), 29 Nk (, ψ), 31 R(0 (N )), 14 R D (z), 149 S( f ; z), 19 S2 (s), 98 S4 (s), 98 Sk (0 (N ), χ ), 18 Skold (0 (N ), χ ), 30 Sknew (0 (N ), χ ), 30

(z), 30 0 (N ), 14 βt , 107 χ (L; a), 4 χo (a), 4 χd (a), 2 δt , 107  d ,2 p L

 = ψ(mod N ), 3   = ψ(mod N ), 3 η(z), 105 κ,   a 16 n K , 77

N0 + 21 , 115 B(R, α), 73 Cr ( f ), 18 Cc,ρ ( f ), 36 D(α1 , . . . , α2k ), 78 E(N , χ ), 5 Ek (, ψ; z), 21 ER (N , χd ), 84 G(α, β), 78 G(χ ), 9 Ga/c (Q), 66, 68 Ga/c (D), 80 R,ψ (A, B), 32 S,ψ (N , A, B), 43 N (k, N , χ ; z), 30 e(d), 10

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 Z. S. Aygin, Introduction to Applications of Modular Forms, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-32629-5

167

168 r LM (n), 38 r M (n), 38 ν p (n), 38 ω, 31 χ, 7 φ(N ), 3 σk−1 (, ψ; n), 31 τ (n), 76 τ16 (n), 76 τ30 (n), 76 θ (Q; z), 65 θ (α1 , . . . , α2k ; z), 90 ζ (k), 38 Z p , 158 a( f ; n), 19 a f (, ψ, t), 43

Index of Notation ar ( f ; n), 17 b f (, ψ, t), 44 e( f ; n), 19 f Q (X ), 65 f D , 78 h t , 107, 112 n t , 107 qc,N , 16 r (Q; n), 65 r (α1 , . . . , α2k ; n), 90 s( f ; n), 19 tq , 4 vr ,N ( f ), 17 w(M), 106 SL 2 (Z), 13

Index of SAGE Functions

BernoulliKS, 86 CalRR, 87 CalR, 54 CalSR, 87 CalS, 55 ChEtaOrder, 153 CharPairsFind, 6 ConvolutionSum, 141 DiagonalFirstTerm, 81 EisBaseFind, 33 EisNormalCoeff, 42 EisPartAsymptotic, 62 EisPartofDiagForm, 89 EisPartofEtaQ, 117 EisPartofQF, 69 EisPart, 57 Eis, 42, 43 EtaExpand, 106

EtaQConstantTerm, 114 EtaQExpand, 109 FindAllEtaQ, 115 G, 79 ModIdFinder, 28 OrderEtaQ, 111 QFConstantTerm, 68 RD, 150 aft, 55 bftR, 88 bft, 56 isEtaQModular, 113 isQD, 87 isQFModular, 66 sigmaR, 86 sigma, 41 vp, 54

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 Z. S. Aygin, Introduction to Applications of Modular Forms, Synthesis Lectures on Mathematics & Statistics, https://doi.org/10.1007/978-3-031-32629-5

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