Year: 2024
Type: article
Abstract: This work contributes to the classification of Teichmüller curves in the moduli space ᏹ 2 of Riemann surfaces of genus 2. While the classification of primitive Teichmüller curves in ᏹ 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 Œn ᏹ 2 be the one-dimensional subvariety consisting of those X 2 ᏹ 2 that admit a primitive degree d holomorphic map W X !E to an elliptic curve E, branched over torsion points of order n.It is known that every imprimitive Teichmüller curve in ᏹ 2 is a component of some W d 2 Œn.The parity conjecture states that (with minor exceptions) W d 2 Œn has two components when n is odd, and one when n is even.In particular, the number of components of W d 2 Œn does not depend on d .We establish the parity conjecture in the following three cases: (1) for all n when d 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 / D H= .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 Ä 5.
Source: Geometry & Topology
Author: Eduard Duryev
Institution: Harvard University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W2946072229