Integral Points on Elliptic Curves and Modularity

TL;DR

This paper proves the finiteness of S-integral points on certain elliptic curves using the Chabauty-Kim method and modularity techniques, extending previous results to non-CM cases.

Contribution

It introduces a novel approach combining modularity and the Chabauty-Kim method to establish finiteness results for S-integral points on elliptic curves without complex multiplication.

Findings

Finiteness of S-integral points on non-CM elliptic curves proven

Modularity techniques used to show vanishing of Selmer groups

Extension of Kim's results to broader class of elliptic curves

Abstract

In this paper we prove the finiteness of the set of S-integral points of a punctured rational elliptic curve without complex multiplication using the Chabauty-Kim method. This extends previous results of Kim in the complex multiplication case. The key input of our approach is the use of modularity techniques to prove the vanishing of certain Selmer groups involved in the Chabauty-Kim method.