Orbifold points on Teichm\"uller curves and Jacobians with complex multiplication

TL;DR

This paper studies orbifold points on Teichmüller curves in genus two, providing a classification, formulas for genus, and explicit descriptions of surfaces at orbifold points, advancing understanding of their geometric and topological structure.

Contribution

It determines the number and types of orbifold points on Weierstrass curves in genus two, completing their topological classification and providing explicit surface descriptions.

Findings

Complete enumeration of orbifold points on Weierstrass curves

Formula for the genus of components of W_D

Identification of genus zero Weierstrass curves

Abstract

For each integer $D \geq 5$ with $D \equiv 0$ or $1 \bmod 4$, the Weierstrass curve $W_D$ is an algebraic curve and a finite volume hyperbolic orbifold which admits an algebraic and isometric immersion into the moduli space of genus two Riemann surfaces. The Weierstrass curves are the main examples of Teichm\"uller curves in genus two. The primary goal of this paper is to determine the number and type of orbifold points on each component of $W_D$. Our enumeration of the orbifold points, together with work of Bainbridge and McMullen, completes the determination of the homeomorphism type of $W_D$ and gives a formula for the genus of its components. We use our formula to give bounds on the genus of $W_D$ and determine the Weierstrass curves of genus zero. We will also give several explicit descriptions of each surface labeled by an orbifold point on $W_D$.