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xviii Contents

12 Endomorphisms of Elliptic Curves .............................. 233
1 Isogenies and Division Points for Complex Tori ................. 233
2 Symplectic Pairings on Lattices and Division Points . . . .......... 235
3 Isogenies in the General Case ................................ 237
4 Endomorphisms and Complex Multiplication . . . ................ 241
5 The Tate Module of an Elliptic Curve .......................... 245
6 Endomorphisms and the Tate Module .......................... 246
7 Expansions Near the Origin and the Formal Group ............... 248
13 Elliptic Curves over Finite Fields ............................... 253
1 The Riemann Hypothesis for Elliptic Curves over a Finite Field .... 253
2 Generalities on Zeta Functions of Curves over a Finite Field . . ..... 256
3 Definition of Supersingular Elliptic Curves ..................... 259
4 Number of Supersingular Elliptic Curves . ...................... 263
5 Points of Order p and Supersingular Curves .................... 265
6 The Endomorphism Algebra and Supersingular Curves . .......... 266
7 Summary of Criteria for a Curve To Be Supersingular . . .......... 268
8 Tate’s Description of Homomorphisms. . . ...................... 270
9 Division Polynomial ........................................ 272
14 Elliptic Curves over Local Fields ............................... 275
1 The Canonical p-Adic Filtration on the Points of an Elliptic Curve
over a Local Field . ......................................... 275
2 The N´ eron Minimal Model. .................................. 277
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3 Galois Criterion of Good Reduction of N´ eron–Ogg–Safareviˇ c ..... 280
4 Elliptic Curves over the Real Numbers . . . ...................... 284

15 Elliptic Curves over Global Fields and -Adic Representations ...... 291
1 Minimal Discriminant Normal Cubic Forms
over a Dedekind Ring ....................................... 291
2 Generalities on -Adic Representations ........................ 293
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3 Galois Representations and the N´ eron–Ogg–Safareviˇ c Criterion in
the Global Case ............................................ 296
4 Ramification Properties of -Adic Representations of Number
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Fields: Cebotarev’s Density Theorem .......................... 298
5 Rationality Properties of Frobenius Elements in -Adic
Representations: Variation of ............................... 301
6 Weight Properties of Frobenius Elements in -Adic
Representations: Faltings’ Finiteness Theorem . . ................ 303
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7 Tate’s Conjecture, Safareviˇ c’s Theorem, and Faltings’ Proof . . ..... 305
8 Image of -Adic Representations of Elliptic Curves: Serre’s Open
Image Theorem ............................................ 307