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

TL;DR

This paper constructs holomorphic families of Riemann surfaces using Veech groups, characterizes their sections, and applies these results to solve Diophantine equations over function fields, providing bounds on the number of sections.

Contribution

It introduces a new construction of holomorphic families from Veech groups and characterizes their sections, linking complex geometry with Diophantine problems.

Findings

Constructed holomorphic families from Veech groups.

Characterized sections via points on flat surfaces.

Provided upper bounds for the number of sections.

Abstract

In this paper, we construct holomorphic families of Riemann surfaces from Veech groups and characterize their 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 numbers of holomorphic sections of certain holomorphic families of Riemann surfaces.