Arithmetic of curves on moduli of local systems

TL;DR

This paper studies the arithmetic properties of algebraic curves on moduli spaces of rank two local systems on surfaces, providing a structure theorem for morphisms from the affine line and methods to determine integral points.

Contribution

It introduces a structure theorem for morphisms from the affine line into these moduli spaces and offers an effective way to determine integral points on algebraic curves.

Findings

Structure theorem for affine line morphisms

Effective determination of integral points

Insights into arithmetic of local system moduli spaces

Abstract

We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli space. We show that the set of integral points on any nondegenerate algebraic curve on the moduli space can be effectively determined.