Diophantine Analysis on Moduli of Local Systems

Abstract

We develop a Diophantine analysis on moduli of special linear rank two local systems over surfaces with prescribed boundary traces. We first show that such a moduli space is a log Calabi-Yau variety if the surface has nonempty boundary, and relate this property to a symmetry of generating series for combinatorial counts of essential multicurves on surfaces. We establish the finiteness of “class numbers” for integral orbits of mapping class group dynamics on the moduli space, generalizing a classical Diophantine work of Markoff. We also derive effective finiteness results for integral points of algebraic curves on the moduli space, as well as a structure theorem for morphisms from the affine line into the moduli space. As part of our work, we establish boundedness theorems for archimedean and nonarchimedean systoles of local systems. Finally, we give a complete classification of the finite orbits of the mapping class group on the moduli space, for surfaces of positive genus.