Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture
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.
Keywords
Cite
@article{arxiv.1203.3865,
title = {Elliptic logarithms, diophantine approximation and the Birch and Swinnerton-Dyer conjecture},
author = {Vincent Bosser and Andrea Surroca},
journal= {arXiv preprint arXiv:1203.3865},
year = {2013}
}
Comments
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. Soc