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Contents xvii

7 The Canonical Height and Norm on an Elliptic Curve . . .......... 137
8 The Canonical Height on Projective Spaces over Global Fields .... 140
7 Galois Cohomology and Isomorphism Classification
of Elliptic Curves over Arbitrary Fields.......................... 143
1 Galois Theory: Theorems of Dedekind and Artin ................ 143
2 Group Actions on Sets and Groups ............................ 146
3 Principal Homogeneous G-Sets and the First Cohomology Set
1
H (G, A) ................................................. 148
4 Long Exact Sequence in G-Cohomology . ...................... 151
5 Some Calculations with Galois Cohomology .................... 153
6 Galois Cohomology Classification of Curves with Given j-Invariant 155

8 Descent and Galois Cohomology ................................ 157
1 Homogeneous Spaces over Elliptic Curves ..................... 157
2 Primitive Descent Formalism ................................. 160
3 Basic Descent Formalism . . .................................. 163
9Elliptic and Hypergeometric Functions .......................... 167
1 Quotients of the Complex Plane by Discrete Subgroups. .......... 167
2 Generalities on Elliptic Functions . ............................ 169
3 The Weierstrass ℘-Function .................................. 171
4 The Differential Equation for ℘(z) ............................ 174
5 Preliminaries on Hypergeometric Functions .................... 179
6 Periods Associated with Elliptic Curves: Elliptic Integrals......... 183

10 Theta Functions ............................................. 189
1 Jacobi q-Parametrization: Application to Real Curves . . .......... 189
2 Introduction to Theta Functions . . . ............................ 193
3 Embeddings of a Torus by Theta Functions ..................... 195
4 Relation Between Theta Functions and Elliptic Functions ......... 197
5 The Tate Curve ............................................ 198
6 Introduction to Tate’s Theory of p-Adic Theta Functions .......... 203

11 Modular Functions ........................................... 209
1 Isomorphism and Isogeny Classification of Complex Tori ......... 209
2 Families of Elliptic Curves with Additional Structures . . .......... 211
3 The Modular Curves X(N), X 1 (N), and X 0 (N) ................... 215
4 Modular Functions ......................................... 220
5 The L-Function of a Modular Form ............................ 222
6 Elementary Properties of Euler Products . ...................... 224
7 Modular Forms for 0 (N), 1 (N),and (N) ................... 227
8 Hecke Operators: New Forms ................................ 229
9 Modular Polynomials and the Modular Equation ................ 230