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

Duryev, Eduard

doi:10.2140/gt.2024.28.3973

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

Eduard Duryev

Geometry & Topology 28 (2024) 3973–4056

DOI: 10.2140/gt.2024.28.3973

Abstract

This work contributes to the classification of Teichmüller curves in the moduli space $ℳ_{2}$ of Riemann surfaces of genus 2. While the classification of primitive Teichmüller 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 one-dimensional subvariety consisting of those $X \in ℳ_{2}$ that admit a primitive degree $d$ holomorphic map $π : X \to E$ to an elliptic curve $E$ , branched over torsion points of order $n$ . It is known that every imprimitive Teichmüller curve in $ℳ_{2}$ 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 $X (d) = \bar{ℍ ∕ Γ (d)}$ is itself a square-tiled surface equipped with a natural action of ${SL}_{2} Z$ . 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 $d \leq 5$ .

Keywords

translation surface, Teichmüller curve, square-tiled surface, modular curve, illumination, pagoda, elliptic covers, absolute period leaf, rel leaf

Mathematical Subject Classification 2010

Primary: 05B45, 32G15, 51H30, 52C20, 57M12

Secondary: 14H45, 14H52, 14H55

References

Publication

Received: 21 June 2019

Revised: 8 March 2023

Accepted: 6 May 2023

Published: 27 December 2024

Proposed: David Gabai

Seconded: David Fisher, Mladen Bestvina

Authors

Eduard Duryev
Department of Mathematics Harvard University Cambridge, MA United States

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Volume 28, issue 9 (2024)