Orbifold points on Teichmüller curves and Jacobians with complex multiplication

Abstract

For each integer $D\ge 5$ with $D\equiv 0$ or $1mod4$, 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 Teichmüller 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 Bainbridge [Geom. Topol. 11 (2007) 1887–2073] and McMullen [Math. Ann. 333 (2005) 87–130], 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 $W_D$.