Arithmetic of curves on moduli of local systems

Junho Peter Whang

Published 2018-03-13Version 1

We study the Diophantine geometry of algebraic curves on relative moduli of special linear rank two local systems over surfaces. We prove that the set of integral points on any nondegenerately embedded algebraic curve can be effectively determined. Under natural hypotheses on the embedding in relation to mapping class group dynamics of the moduli space, the set of all imaginary quadratic integral points on the curve is shown to be finite. Our ingredients include a boundedness result for nonarchimedean systoles of local systems and Baker's theory. We also derive a structure theorem for morphisms from the affine line into the moduli space.