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arXiv · 1203.3865

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

Vincent Bosser·Andrea Surroca
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Abstract

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.

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Vincent Bosser, Andrea Surroca. 2013-01-06. Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture. https://arxiv.org/abs/1203.3865

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