Teichmüller curves in genus two: square-tiled surfaces and modular curves (Q6656651)

Teichmüller curves are isometrically embedded copies of hyperbolic surfaces in the moduli space of Riemann surfaces. They arise as the projections of \textit{closed} \(SL(2, \mathbb R)\)-orbits on the space of abelian differentials, and are of interest from a variety of geometric and dynamical perspectives. In genus 2, the \textit{primitive} Teichmüller curves are fully classified, by work of Calta and McMullen. The paper under review makes progress toward classifying the \textit{imprimitive} Teichmüller curves. The author formulates a \textit{parity} conjecture, which would complete this classification, and establishes it in many cases. Precisely, they consider the 1-dimensional subvariety \(W_{d^2} [n] \subset \mathcal M_2\) of Riemann surfaces \(X\) that admit a primitive degree \(d\) holomorphic map \(\pi : X \rightarrow E\) to an elliptic curve \(E\), branched over torsion points of order \(n\). Every imprimitive Teichmüller curve in \(\mathcal M_2\) is a component of one of these \(W_{d^2} [n]\). The parity conjecture states that \(W_{d^2} [n]\) has two components when \(n\) is odd, and one when \(n\) is even, in particular the number of components does not depend on \(d\). The paper proves the parity conjecture for all \(n\), when \(d= 2,3,4,5\), when \(d\) and \(n\) are both prime and \(n>(d^3- d)/4\); and when \(d\) is prime and \(n>C_d\), where \(C_d\) depends on d. The techniques are an impressive blend of geometry, combinatorics, and number theory, and involve significant case-by-case analysis of particular families of translation surfaces.