Differential equations associated with nonarithmetic Fuchsian groups

TL;DR

This paper studies special rank 2 differential operators linked to nonarithmetic Fuchsian groups, revealing their properties, solutions, and connections to Teichmüller curves, challenging existing conjectures.

Contribution

It introduces globally nilpotent differential operators associated with nonarithmetic Fuchsian groups and explores their solutions and moduli, providing counterexamples to previous conjectures.

Findings

Differential operators have S-integral solutions.

Counterexamples to Chudnovsky--Chudnovsky and Dwork conjectures.

Determined fields of moduli for genus 2 Teichmüller curves.

Abstract

We describe globally nilpotent differential operators of rank 2 defined over a number field whose monodromy group is a nonarithmetic Fuchsian group. We show that these differential operators have an S-integral solution. These differential operators are naturally associated with Teichmueller curves in genus 2. They are counterexamples to conjectures by Chudnovsky--Chudnovsky and Dwork. We also determine the field of moduli of primitive Teichmueller curves in genus 2, and an explicit equation in some cases.