Density of integral points in the Betti moduli of quasi-projective varieties

TL;DR

This paper demonstrates that integral points are potentially dense in certain character varieties associated with quasi-projective varieties, using reduction to Riemann surfaces and group action dynamics.

Contribution

It establishes potential density of integral points in Betti moduli spaces for quasi-projective varieties, extending understanding of arithmetic properties of local systems.

Findings

Integral points are potentially dense in character varieties.

Reduction to Riemann surfaces simplifies the problem.

Existence of integral points with Zariski-dense orbits under mapping class group.

Abstract

Let $Y$ be a smooth quasi-projective complex variety equipped with a simple normal crossings compactification. We show that integral points are potentially dense in the (relative) character varieties parametrizing $SL_2$-local systems on $Y$ with fixed algebraic integer traces along the boundary components. The proof proceeds by using work of Corlette-Simpson to reduce to the case of Riemann surfaces, where we produce an integral point with Zariski-dense orbit under the mapping class group.