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

Eduard Duryev

Geometry & Topology 28 (2024) 3973–4056
Abstract

This work contributes to the classification of Teichmüller curves in the moduli space 2 of Riemann surfaces of genus 2. While the classification of primitive Teichmüller curves in 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 Wd2[n] 2 be the one-dimensional subvariety consisting of those X 2 that admit a primitive degree d holomorphic map π: X E to an elliptic curve E, branched over torsion points of order n. It is known that every imprimitive Teichmüller curve in 2 is a component of some Wd2[n]. The parity conjecture states that (with minor exceptions) Wd2[n] has two components when n is odd, and one when n is even. In particular, the number of components of Wd2[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 > (d3 d)4; and (3) when d is prime and n > Cd, where Cd is an implicit constant that depends on d.

In the course of the proof we will see that the modular curve X(d) = Γ(d)¯ is itself a square-tiled surface equipped with a natural action of SL 2Z. 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 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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