本书是Springer《数学研究生丛书》第228卷，内容主要包括：椭圆曲线、复环面和代数曲，模曲线 、黎曼曲面和代数曲线，Hecke算子和Athkin—Lehner 理论，Hecke特征形式及它们的算术性质，模曲线的雅可比行列式和Hecke特征形式的阿贝尔簇，椭圆曲线、模曲线模P及Eichler—Shimura关系，椭圆曲线和Hecke特征形式的Galois表示。学习本书不需要代数数论及代数几何的背景知识，适用于高年级本科生和一年级研究生，全书配有相关习题。

Preface

1 Modular Forms, Elliptic Curves, and Modular Curves

  1.1 First definitions and examples

  1.2 Congruence subgroups

  1.3 Complex tori

  1.4 Complex tori as elliptic curves

  1.5 Modular curves and moduli spaces

2 Modular Curves as Riemann Surfaces

  2.1 Topology

  2.2 Charts

  2.3 Elliptic points

  2.4 Cusps

  2.5 Modular curves and Modularity

3 Dimension Formulas

  3.1 The genus

  3.2 Automorphic forms

  3.3 Meromorphic differentials

  3.4 Divisors and the Riemann-Roch Theorem

  3.5 Dimension formulas for even k

  3.6 Dimension formulas for odd k

  3.7 More on elliptic points

  3.8 More on cusps

  3.9 More dimension formulas

4 Eisenstein Series

  4.1 Eisenstein series for SL2(Z)

  4.2 Eisenstein series for F(N) when k≥ 3

  4.3 Dirichlet characters, Gauss sums, and eigenspaces

  4.4 Gamma, zeta, and L-functions

  4.5 Eisenstein series for the eigenspaces when k ≥ 3

  4.6 Eisenstein series of weight 2

  4.7 Bernoulli numbers and the Hurwitz zeta function

  4.8 Eisenstein series of weight 1

  4.9 The Fourier transform and the Mellin transform.

  4.10 Nonholomorphic Eisenstein series

  4.11 Modular forms via theta functions

5 Hecke Operators

  5.1 The double coset operator

  5.2 The <d>and Tp operators

  5.3 The <n> and Tn operators

  5.4 The Petersson inner product

  5.5 Adjoints of the Hecke Operators

  5.6 Oldforms and Newforms

  5.7 The Main Lemma

  5.8 Eigenforms

  5.9 The connection with L-functions

  5.10 Functional equations

  5.11 Eisenstein series again

6 Jacobians and Abelian Varieties

  6.1 Integration, homology, the Jacobian, and Modularity

  6.2 Maps between Jacobians

  6.3 Modular Jacobians and Hecke operators

  6.4 Algebraic numbers and algebraic integers

  6.5 Algebraic eigenvalues

  6.6 Eigenforms, Abelian varieties, and Modularity

7 Modular Curves as Algebraic Curves

  7.1 Elliptic curves as algebraic curves

  7.2 Algebraic curves and their function fields

  7.3 Divisors on curves

  7.4 The Weil pairing algebraically

  7.5 Function fields over C

  7.6 Function fields over Q

  7.7 Modular curves as algebraic curves and Modularity

  7.8 Isogenies algebraically

  7.9 Hecke operators algebraically

8 The Eichler-Shimura Relation and L-functlons

  8.1 Elliptic curves in arbitrary characteristic

  8.2 Algebraic curves in arbitrary characteristic

  8.3 Elliptic curves over Q and their reductions

  8.4 Elliptic curves over Q and their reductions

  8.5 Reduction of algebraic curves and maps

  8.6 Modular curves in characteristic p: Igusa's Theorem

  8.7 The Eichler-Shimura Relation

  8.8 Fourier coefficients, L-functions, and Modularity

9 Galois Representations

  9.1 Galois number fields

  9.2 The l-adic integers and the l-adic numbers

  9.3 Galois representations

  9.4 Galois representations and elliptic curves

  9.5 Galois representations and modular forms

  9.6 Galois representations and Modularity

Hints and Answers to the Exercises

List of Symbols

Index

Rreferences