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

TL;DR

This paper explores the elliptic analogue of Baker's method to derive bounds on S-integral points on elliptic curves, connecting it to the Birch and Swinnerton-Dyer conjecture and implications for abc-type inequalities.

Contribution

It introduces an elliptic Baker's method-based bound that depends on the Birch and Swinnerton-Dyer conjecture, linking elliptic curves to abc-type inequalities.

Findings

Derived a new bound for S-integral points using elliptic Baker's method.

Connected elliptic bounds to the Birch and Swinnerton-Dyer conjecture.

Explored implications for abc-type inequalities over number fields.

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.