Differential equations associated with nonarithmetic Fuchsian groups

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.