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

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 $W_D$.