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

Vincent Bosser (1) , Andrea Surroca (2)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 IMJ - Institut de Mathématiques de Jussieu

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.

Mots clés: integral points on elliptic curves; quantitative Siegel's theorem; elliptic logarithms; Birch and Swinnerton-Dyer conjecture; abc-conjecture

Domaines: Théorie des nombres [math.NT]

Vincent Bosser, Andrea Surroca. Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture. Boletim da Sociedade Brasileira de Matemática / Bulletin of the Brazilian Mathematical Society, 2014, 45 (1), ⟨10.1007/s00574-014-0038-x⟩. ⟨hal-02151923⟩