Upper bound for the height of S-integral points on elliptic curves

TL;DR

This paper derives explicit upper bounds for the height of S-integral points on elliptic curves, depending on number field and Mordell-Weil group parameters, using elliptic logarithm techniques.

Contribution

It provides a new explicit upper bound for S-integral points on elliptic curves, combining number field invariants and elliptic logarithm methods.

Findings

Explicit height bounds in terms of S, degree, rank, regulator, and basis height.

Application of elliptic logarithm lower bounds to derive these bounds.

Enhancement over previous bounds by explicit dependence on field and group parameters.

Abstract

We establish new upper bounds for the height of the S-integral points of an elliptic curve. This bound is explicitly given in terms of the set S of places of the number field K involved, but also in terms of the degree of K, as well as the rank, the regulator and the height of a basis of the Mordell-Weil group of the curve. The proof uses the elliptic analogue of Baker's method, based on lower bounds for linear forms in elliptic logarithms.