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Contents xix

16 L-Function of an Elliptic Curve and Its Analytic Continuation ...... 309
1 Remarks on Analytic Methods in Arithmetic .................... 309
2 Zeta Functions of Curves over Q .............................. 310
3 Hasse–Weil L-Function and the Functional Equation . . . .......... 312
4 Classical Abelian L-Functions and Their Functional Equations..... 315
5 Gr¨ ossencharacters and Hecke L-Functions ...................... 318
6 Deuring’s Theorem on the L-Function of an Elliptic Curve with
Complex Multiplication . . . .................................. 321
7 Eichler–Shimura Theory. . . .................................. 322
8 The Modular Curve Conjecture . . . ............................ 324
17 Remarks on the Birch and Swinnerton–Dyer Conjecture ........... 325
1 The Conjecture Relating Rank and Order of Zero ................ 325
2 Rank Conjecture for Curves with Complex Multiplication I, by
Coates and Wiles . . . ........................................ 326
3 Rank Conjecture for Curves with Complex Multiplication II, by
Greenberg and Rohrlich . . . .................................. 327
4 Rank Conjecture for Modular Curves by Gross and Zagier . . . ..... 328
5 Goldfeld’s Work on the Class Number Problem and Its Relation to
the Birch and Swinnerton–Dyer Conjecture ..................... 328
6 The Conjecture of Birch and Swinnerton–Dyer on the Leading Term 329
7 Heegner Points and the Derivative of the L-function at s = 1, after
Gross and Zagier ........................................... 330
8 Remarks On Postscript: October 1986 . . . ...................... 331

18 Remarks on the Modular Elliptic Curves Conjecture and
Fermat’s Last Theorem ....................................... 333
1 Semistable Curves and Tate Modules .......................... 334
2 The Frey Curve and the Reduction of Fermat Equation to Modular
Elliptic Curves over Q ...................................... 335
3 Modular Elliptic Curves and the Hecke Algebra . ................ 336
4 Hecke Algebras and Tate Modules of Modular Elliptic Curves ..... 338
5 Special Properties of mod 3 Representations .................... 339
6 Deformation Theory and -Adic Representations ................ 339
7 Properties of the Universal Deformation Ring . . . ................ 341
8 Remarks on the Proof of the Opposite Inequality ................ 342
9 Survey of the Nonsemistable Case of the Modular Curve Conjecture 342

19 Higher Dimensional Analogs of Elliptic Curves:
Calabi–Yau Varieties ......................................... 345
1 Smooth Manifolds: Real Differential Geometry . ................ 347
2 Complex Analytic Manifolds: Complex Differential Geometry..... 349
3 K¨ ahler Manifolds........................................... 352
4 Connections, Curvature, and Holonomy . . ...................... 356
5 Projective Spaces, Characteristic Classes, and Curvature .......... 361