CONTENTS

Chapter I. Introduction 1 Chapter II. Modular symbol algorithms 7 2.1 Modular Symbols and Homology 7

2.1.1 The upper half-plane, the modular group and cusp forms 7 2.1.2 The duality between cusp forms and homology 9 2.1.3 Real structure 11 2.1.4 Modular symbol formalism 12 2.1.5 Rational structure and the Manin-Drinfeld Theorem 12 2.1.6 Triangulations and homology 13

2.2 M-symbols and Γ_{0}(N) 15 2.3 Conversion between modular symbols and M-symbols 18 2.4 Action of Hecke and other operators 18 2.5 Working in H^{+}(N) 23 2.6 Modular forms and modular elliptic curves 24 2.7 Splitting off one-dimensional eigenspaces 25 2.8 L(f, s) and the evaluation of L(f, 1)/Ω(f) 29 2.9 Computing Fourier coefficients 31 2.10 Computing periods I 33 2.11 Computing periods II: Indirect method 37 2.12 Computing periods III: Evaluation of the sums 41 2.13 Computing L^{(r)}(f, 1) 42 2.14 Obtaining equations for the curves 45 2.15 Computing the degree of a modular parametrization 46

2.15.1 Modular Parametrizations 47 2.15.2 Coset representatives and Fundamental Domains 48 2.15.3 Implementation for Γ_{0}(N) 50

Appendix to Chapter II. Examples 52 Example 1. N = 11 52 Example 2. N = 33 57 Example 3. N = 37 58 Example 4. N = 49 60

Chapter III. Elliptic curve algorithms 62 3.1 Terminology and notation 62 3.2 The Kraus–Laska–Connell algorithm and Tate’s algorithm 64 3.3 The Mordell–Weil group I: finding torsion points 68 3.4 Heights and the height pairing 71 3.5 The Mordell–Weil group II: generators 75 3.6 The Mordell–Weil group III: the rank 78 3.7 The period lattice 97 3.8 Finding isogenous curves 98 3.9 Twists and complex multiplication 101 Chapter IV. The tables 104

Table 1. Elliptic curves 109 Table 2. Mordell–Weil generators 255 Table 3. Hecke eigenvalues 264 Table 4. Birch–Swinnerton-Dyer data 313 Table 5. Parametrization degrees 362

Bibliography 374