Teichmüller curves in genus two : square-tiled surfaces and modular curves

Duryev, Eduard¹

¹ Department of Mathematics, Harvard University, Cambridge, MA, United States

Geometry & topology, Tome 28 (2024) no. 9, pp. 3973-4056.

Résumé

This work contributes to the classification of Teichmüller curves in the moduli space ${\mathcal{ℳ}}_2$ of Riemann surfaces of genus 2. While the classification of primitive Teichmüller curves in ${\mathcal{ℳ}}_2$ 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 $W_{d^2}[n]\subset {\mathcal{ℳ}}_2$ be the one-dimensional subvariety consisting of those $X\in {\mathcal{ℳ}}_2$ 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 Teichmüller curve in ${\mathcal{ℳ}}_2$ is a component of some $W_{d^2}[n]$. The 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 $d$.

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 $X(d)=\overline{ℍ∕\Gamma (d)}$ is itself a square-tiled surface equipped with a natural action of ${\mathrm{SL}}_{2}Z$. The parity conjecture is equivalent to the classification of the finite orbits of this action. It is also closely related to the following 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 $d\le 5$.

Détail

DOI : 10.2140/gt.2024.28.3973

Classification : 05B45, 32G15, 51H30, 52C20, 57M12, 14H45, 14H52, 14H55
Keywords: translation surface, Teichmüller curve, square-tiled surface, modular curve, illumination, pagoda, elliptic covers, absolute period leaf, rel leaf

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