The Chabauty--Coleman method and p-adic linear forms in logarithms

TL;DR

This paper applies $p$-adic transcendence theory to enhance the Chabauty-Coleman method, providing bounds on rational points and exploring implications of Wieferich statistics conjectures for curves of small rank.

Contribution

It introduces new bounds on $p$-adic distances between rational points and links conjectures on Wieferich statistics to height bounds in the Chabauty-Coleman framework.

Findings

Lower bounds on $p$-adic distances between rational points.

Connection between Wieferich statistics conjecture and height bounds.

Application of transcendence theory to problems in the Chabauty-Coleman method.

Abstract

Results in $p$-adic transcendence theory are applied to two problems in the Chabauty-Coleman method. The first is a question of McCallum and Poonen regarding repeated roots of Coleman integrals. The second is to give lower bounds on the $p$-adic distance between rational points in terms of the heights of a set of Mordell-Weil generators of the Jacobian. We also explain how, in some cases, a conjecture on the 'Wieferich statistics' of Jacobians of curves implies a bound on the height of rational points of curves of small rank, in terms of the usual invariants of the curve and the height of Mordell-Weil generators of its Jacobian. The proof uses the Chabauty-Coleman method, together with effective methods in transcendence theory. We also discuss generalisations to the Chabauty-Kim method.