Modular embeddings of Teichmueller curves

TL;DR

This paper develops a comprehensive theory of twisted modular forms for Fuchsian groups with modular embeddings, explores their arithmetic properties, and applies the theory to Teichmueller curves, revealing new structural and algebraic insights.

Contribution

It introduces a new theory of twisted modular forms, provides explicit examples, and characterizes genus two Teichmueller curves via theta derivatives, advancing understanding of their arithmetic and geometric properties.

Findings

Dimension formulas and growth estimates for twisted modular forms

Explicit Fourier expansions for Teichmueller curves

Modular proof of an Apery-like integrality statement

Abstract

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apery-like integrality statement for solutions of Picard-Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmueller curve in a Hilbert modular surface. In Part III we show that genus two Teichmueller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmueller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmueller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge's compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch's in form, but every detail is different.