Algebraic models and arithmetic geometry of Teichm\"uller curves in genus two

TL;DR

This paper constructs explicit algebraic models of genus two Teichmüller curves using Hilbert modular forms and explores their arithmetic properties, including reduction behavior and divisors at cusps.

Contribution

It provides the first explicit algebraic models of genus two Teichmüller curves and investigates their arithmetic geometry.

Findings

Explicit algebraic models of genus two Teichmüller curves are constructed.

Evidence of rich arithmetic structure with examples of bad reduction.

Identification of eigenforms for real multiplication on genus two Jacobians.

Abstract

A Teichm\"uller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichm\"uller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formula to identify eigenforms for real multiplication on genus two Jacobians. We also present evidence that Teichm\"uller curves admit a rich arithmetic geometry by exhibiting examples with small primes of bad reduction and notable divisors supported at their cusps.