Teichm\"uller curves in genus two: Square-tiled surfaces and modular curves

TL;DR

This paper advances the classification of Teichmüller curves in genus two, focusing on imprimitive cases related to square-tiled surfaces and modular curves, and proves the parity conjecture in several cases.

Contribution

It proves the parity conjecture for certain cases of imprimitive Teichmüller curves in genus two, linking the problem to the classification of finite orbits of a modular curve's SL(2,Z) action.

Findings

Parity conjecture proven for d=2,3,4,5 for all n.

Parity conjecture proven for prime d and large n.

Modular curve X(d) is a square-tiled surface with SL(2,Z) action.

Abstract

This work is a contribution to the classification of Teichm\"uller curves in the moduli space $\M_2$ of Riemann surfaces of genus 2. While the classification of primitive Teichm\"uller curves in $\M_2$ is complete, the classification of the imprimitive curves, which is related to branched torus covers and square-tiled surfaces, remains open. Conjecturally, the classification is completed as follows. Let $W_{d^2}[n] \subset \M_2$ be the 1-dimensional subvariety consisting of those $X \in \M_2$ that admit a primitive degree $d$ holomorphic map $\pi: X \to E$ to an elliptic curve $E$, branched over torsion points of order $n$. It is known that every imprimitive Teichm\"uller curve in $\M_2$ is a component of some $W_{d^2}[n]$. The {\em parity conjecture} states that (with minor exceptions) $W_{d^2}[n]$ has two components when $n$ is odd, and one when $n$ is even. In particular, the number of components of $W_{d^2}[n]$ does not depend on $d$. In this work we establish the parity conjecture in the following three cases: (1) for all $n$ when $d=2,3,4,5$; (2) when $d$ and $n$ are prime and $n > (d^3-d)/4$; and (3) when $d$ is prime and $n > C_d$, where $C_d$ is an implicit constant that depends on $d$. In the course of the proof we will see that the modular curve $X(d) = \overline{\Hyp \big/ \Gamma(d)}$ is itself a square-tiled surface equipped with a natural action of $\SLZ$. The parity conjecture is equivalent to the classification of the finite orbits of this action. It is also closely related to the following {\em illumination conjecture}: light sources at the cusps of the modular curve illuminate all of $X(d)$, except possibly some vertices of the square-tiling. Our results show that the illumination conjecture is true for $d \le 5$.