Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker's method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker's method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.Comment: 20 pages. Some changes, the most important being on Conjecture 3.2, three references added ([Mas75], [MB90] and [Yu94]) and one reference updated [BS12]. Accepted in Bull. Brazil. Mat. So

The p-adic L-functions of modular elliptic curves

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 . . . . . . . . . .