Veech holomorphic families of Riemann surfaces, holomorphic sections, and Diophantine problems

Abstract

In this paper, we construct holomorphic families of Riemann surfaces from Veech groups and characterize their holomorphic sections by some points of corresponding flat surfaces. The construction gives us concrete solutions for some Diophantine equations over function fields. Moreover, we give upper bounds of the number of holomorphic sections of certain holomorphic families of Riemann surfaces.