The p-adic L-functions of modular elliptic curves Massimo Bertolini ∗ Henri Darmon † September 5, 2007 Contents 1 Elliptic curves and modular forms 5 1.1 Elliptic curves ........................... 5 1.2 The Birch and Swinnerton-Dyer conjecture ........... 7 1.3 Modularity ............................ 10 2 Cyclotomic p-adic L-functions 12 2.1 The Mazur–Swinnerton-Dyer p-adic L-function ......... 13 2.2 The Mazur-Tate-Teitelbaum conjecture ............. 16 2.3 Results on the Mazur-Tate-Teitelbaum conjecture ....... 17 3 Schneider’s approach 18 3.1 Rigid analysis ........................... 18 3.2 Shimura Curves .......................... 20 3.3 Modular forms .......................... 23 3.4 Schneider’s distribution ...................... 25 3.5 The Jacquet-Langlands correspondence ............. 27 3.6 Schneider’s p-adic L-function ................... 30 * Partially funded by MURST, GNSAGA (CNR) and the European Community † Supported by an NSERC research grant and by CICMA. 1