Counting integral points of bounded height on varieties with large fundamental group

TL;DR

This paper improves bounds on the number of integral points of bounded height on varieties with large fundamental groups, refining previous results by Ellenberg, Lawrence, and Venkatesh.

Contribution

It establishes subpolynomial growth of integral points on varieties with large fundamental groups, advancing understanding of Diophantine geometry in this context.

Findings

Proves subpolynomial growth of integral points

Requires large fundamental group condition

Refines previous bounds in the literature

Abstract

The present note is devoted to an amendment to a recent paper of Ellenberg, Lawrence and Venkatesh. Roughly speaking, the main result here states the subpolynomial growth of the number of integral points with bounded height of a variety over a number field whose fundamental group is large. Such an improvement, i.e. requiring large fundamental group as opposed to the existence of a geometric variation of pure Hodge structures, was already asked in op.cit..