Siegel's theorem via the Lawrence-Venkatesh method

TL;DR

This paper applies the Lawrence-Venkatesh geometric method to reprove Siegel's theorem, demonstrating the finiteness of $S$-integral points on elliptic curves through the construction of specific abelian-by-finite families.

Contribution

It adapts the Lawrence-Venkatesh approach to elliptic curves, providing a new proof of Siegel's theorem using geometric families.

Findings

Finiteness of $S$-integral points on elliptic curves proven

Constructed abelian-by-finite family on a punctured elliptic curve

Reproves Siegel's theorem via geometric methods

Abstract

In the recent paper arXiv:1807.02721, B. Lawrence and A. Venkatesh develop a method of proving finiteness theorems in arithmetic geometry by studying the geometry of families over a base variety. Their results include a new proof of both the $S$-unit theorem and Faltings' theorem, obtained by constructing and studying suitable abelian-by-finite families over $\mathbb{P}^1\setminus\{0,1,\infty\}$ and over an arbitrary curve of genus $\geq 2$ respectively. In this paper, we apply this strategy to reprove Siegel's theorem: we construct an abelian-by-finite family on a punctured elliptic curve to prove finiteness of $S$-integral points on elliptic curves.