Modular Forms: A Computational Approach William A. Stein (with an appendix by Paul E. Gunnells) Department of Mathematics, University of Washington E-mail address: [email protected] Department of Mathematics and Statistics, University of Massachusetts E-mail address: [email protected] 1991 Mathematics Subject Classification. Primary 11; Secondary 11-04 Key words and phrases. abelian varieties, cohomology of arithmetic groups, computation, elliptic curves, Hecke operators, modular curves, modular forms, modular symbols, Manin symbols, number theory Abstract. This is a textbook about algorithms for computing with modular forms. It is nontraditional in that the primary focus is not on underlying theory; instead, it answers the question “how do you explicitly compute spaces of modular forms?” v To my grandmother, Annette Maurer. Contents Preface xi Chapter 1. Modular Forms 1 1.1. Basic Definitions 1 § 1.2. Modular Forms of Level 1 3 § 1.3. Modular Forms of Any Level 4 § 1.4. Remarks on Congruence Subgroups 7 § 1.5. Applications of Modular Forms 9 § 1.6. Exercises 11 § Chapter 2. Modular Forms of Level 1 13 2.1. Examples of Modular Forms of Level 1 13 § 2.2. Structure Theorem for Level 1 Modular Forms 17 § 2.3. The Miller Basis 20 § 2.4. Hecke Operators 22 § 2.5. Computing Hecke Operators 26 § 2.6. Fast Computation of Fourier Coefficients 29 § 2.7. Fast Computation of Bernoulli Numbers 29 § 2.8. Exercises 33 § Chapter 3. Modular Forms of Weight 2 35 3.1. Hecke Operators 36 § 3.2. Modular Symbols 39 § 3.3. Computing with Modular Symbols 41 § vii viii Contents 3.4. Hecke Operators 47 § 3.5. Computing the Boundary Map 51 § 3.6. Computing a Basis for S (Γ (N)) 53 § 2 0 3.7. Computing S (Γ (N)) Using Eigenvectors 58 § 2 0 3.8. Exercises 60 § Chapter 4. Dirichlet Characters 63 4.1. The Definition 64 § 4.2. Representing Dirichlet Characters 64 § 4.3. Evaluation of Dirichlet Characters 67 § 4.4. Conductors of Dirichlet Characters 70 § 4.5. The Kronecker Symbol 72 § 4.6. Restriction, Extension, and Galois Orbits 75 § 4.7. Alternative Representations of Characters 77 § 4.8. Dirichlet Characters in SAGE 78 § 4.9. Exercises 81 § Chapter 5. Eisenstein Series and Bernoulli Numbers 83 5.1. The Eisenstein Subspace 83 § 5.2. Generalized Bernoulli Numbers 83 § 5.3. Explicit Basis for the Eisenstein Subspace 88 § 5.4. Exercises 90 § Chapter 6. Dimension Formulas 91 6.1. Modular Forms for Γ (N) 92 § 0 6.2. Modular Forms for Γ (N) 95 § 1 6.3. Modular Forms with Character 98 § 6.4. Exercises 102 § Chapter 7. Linear Algebra 103 7.1. Echelon Forms of Matrices 103 § 7.2. Rational Reconstruction 105 § 7.3. Echelon Forms over Q 107 § 7.4. Echelon Forms via Matrix Multiplication 110 § 7.5. Decomposing Spaces under the Action of Matrix 114 § 7.6. Exercises 119 § Chapter 8. General Modular Symbols 121 Contents ix 8.1. Modular Symbols 122 § 8.2. Manin Symbols 124 § 8.3. Hecke Operators 128 § 8.4. Cuspidal Modular Symbols 133 § 8.5. Pairing Modular Symbols and Modular Forms 137 § 8.6. Degeneracy Maps 142 § 8.7. Explicitly Computing M (Γ (N)) 144 § k 0 8.8. Explicit Examples 147 § 8.9. Refined Algorithm for the Presentation 154 § 8.10. Applications 155 § 8.11. Exercises 156 § Chapter 9. Computing with Newforms 159 9.1. Dirichlet Character Decomposition 159 § 9.2. Atkin-Lehner-Li Theory 161 § 9.3. Computing Cusp Forms 165 § 9.4. Congruences between Newforms 170 § 9.5. Exercises 176 § Chapter 10. Computing Periods 177 10.1. The Period Map 178 § 10.2. Abelian Varieties Attached to Newforms 178 § 10.3. Extended Modular Symbols 179 § 10.4. Approximating Period Integrals 180 § 10.5. Speeding Convergence Using Atkin-Lehner 183 § 10.6. Computing the Period Mapping 185 § 10.7. All Elliptic Curves of Given Conductor 187 § 10.8. Exercises 190 § Chapter 11. Solutions to Selected Exercises 191 11.1. Chapter 1 191 § 11.2. Chapter 2 193 § 11.3. Chapter 3 194 § 11.4. Chapter 4 196 § 11.5. Chapter 5 197 § 11.6. Chapter 6 197 § 11.7. Chapter 7 198 § x Contents 11.8. Chapter 8 199 § 11.9. Chapter 9 201 § 11.10. Chapter 10 201 § Appendix A. Computing in Higher Rank 203 A.1. Introduction 203 § A.2. Automorphic Forms and Arithmetic Groups 205 § A.3. Combinatorial Models for Group Cohomology 213 § A.4. Hecke Operators and Modular Symbols 225 § A.5. Other Cohomology Groups 232 § A.6. Complements and Open Problems 244 § Bibliography 253 Index 265 Preface This is a graduate-level textbook about algorithms for computing with mod- ular forms. It is nontraditional in that the primary focus is not on underly- ing theory; instead, it answers the question “how do you use a computer to explicitly compute spaces of modular forms?” This book emerged from notes for a course the author taught at Harvard University in 2004, a course at UC San Diego in 2005, and a course at the University of Washington in 2006. The author has spent years trying to find good practical ways to compute with classical modular forms for congruence subgroups of SL2(Z) and has implemented most of these algorithms several times, first in C++ [Ste99b], then in MAGMA [BCP97], and as part of the free open source computer algebra system SAGE (see [Ste06]). Much of this work has involved turning formulas and constructions buried in obscure research papers into precise computational recipes then testing these and eliminating inaccuracies. The author is aware of no other textbooks on computing with modular forms, the closest work being Cremona’s book [Cre97a], which is about computing with elliptic curves, and Cohen’s book [Coh93] about algebraic number theory. In this book we focus on how to compute in practice the spaces Mk(N, ε) of modular forms, where k 2 is an integer and ε is a Dirichlet character ≥ of modulus N (the appendix treats modular forms for higher rank groups). We spend the most effort explaining the general algorithms that appear so far to be the best (in practice!) for such computations. We will not dis- cuss in any detail computing with quaternion algebras, half-integral weight forms, weight 1 forms, forms for noncongruence subgroups or groups other xi xii Preface than GL2, Hilbert and Siegel modular forms, trace formulas, p-adic modular forms, and modular abelian varieties, all of which are topics for additional books. We also rarely analyze the complexity of the algorithms, but instead settle for occasional remarks about their practical efficiency. For most of this book we assume the reader has some prior exposure to modular forms (e.g., [DS05]), though we recall many of the basic defini- tions. We cite standard books for proofs of the fundamental results about modular forms that we will use. The reader should also be familiar with basic algebraic number theory, linear algebra, complex analysis (at the level of [Ahl78]), and algorithms (e.g., know what an algorithm is and what big oh notation means). In some of the examples and applications we assume that the reader knows about elliptic curves at the level of [Sil92]. Chapter 1 is foundational for the rest of this book. It introduces congru- ence subgroups of SL2(Z) and modular forms as functions on the complex upper half plane. We discuss q-expansions, which provide an important computational handle on modular forms. We also study an algorithm for computing with congruence subgroups. The chapter ends with a list of ap- plications of modular forms throughout mathematics. In Chapter 2 we discuss level 1 modular forms in much more detail. In particular, we introduce Eisenstein series and the cusp form ∆ and describe their q-expansions and basic properties. Then we prove a structure theorem for level 1 modular forms and use it to deduce dimension formulas and give an algorithm for explicitly computing a basis. We next introduce Hecke operators on level 1 modular forms, prove several results about them, and deduce multiplicativity of the Ramanujan τ function as an application. We also discuss explicit computation of Hecke operators. In Section 2.6 we make some brief remarks on recent work on asymptotically fast computation of values of τ. Finally, we describe computation of constant terms of Eisenstein series using an analytic algorithm. We generalize many of the constructions in this chapter to higher level in subsequent chapters. In Chapter 3 we turn to modular forms of higher level but restrict for simplicity to weight 2 since much is clearer in this case. (We remove the weight restriction later in Chapter 8.) We describe a geometric way of view- ing cuspidal modular forms as differentials on modular curves, which leads to modular symbols, which are an explicit way to present a certain homol- ogy group. This chapter closes with methods for explicitly computing cusp forms of weight 2 using modular symbols, which we generalize in Chapter 9. In Chapter 4 we introduce Dirichlet characters, which are important both in explicit construction of Eisenstein series (in Chapter 5) and in de- composing spaces of modular forms as direct sums of simpler spaces. The Preface xiii main focus of this chapter is a detailed study of how to explicitly represent and compute with Dirichlet characters. Chapter 5 is about how to explicitly construct the Eisenstein subspace of modular forms. First we define generalized Bernoulli numbers attached to a Dirichlet character and an integer then explain a new analytic algorithm for computing them (which generalizes the algorithm in Chapter 2). Finally we give without proof an explicit description of a basis of Eisenstein series, explain how to compute it, and give some examples.