Siegel's theorem via the Lawrence-Venkatesh method

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 $\ge 2$ 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.

Comments welcome

References in corpus (1)

• Uniformity of fibres of period mappings and the $S$-unit equation