Wiles’ theorem and the arithmetic of elliptic curves H. Darmon September 9, 2007 Contents 1 Prelude: plane conics, Fermat and Gauss 2 2 Elliptic curves and Wiles’ theorem 6 2.1 Wiles’ theorem and L(E/Q,s) .................. 6 2.2 Geometric versions of Wiles’ theorem .............. 9 3 The special values of L(E/Q,s) at s =1 11 3.1 Analytic rank 0 .......................... 11 3.2 Analytic rank 1: the Gross-Zagier formula ........... 12 3.3 Some variants of the Gross-Zagier formula ........... 14 4 The Birch and Swinnerton-Dyer conjecture 18 4.1 Analytic rank 0 .......................... 18 4.1.1 Kolyvagin’s proof ..................... 18 4.1.2 A variant ......................... 19 4.1.3 Kato’s proof ........................ 20 4.2 Analytic rank 1 .......................... 20 1