Orbifold points on Teichmüller curves and Jacobians with complex multiplication

Mukamel, Ronen E

doi:10.2140/gt.2014.18.779

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Orbifold points on Teichmüller curves and Jacobians with complex multiplication

Ronen E Mukamel

Geometry & Topology 18 (2014) 779–829

DOI: 10.2140/gt.2014.18.779

Abstract

For each integer $D \geq 5$ with $D \equiv 0$ or $1 mod 4$ , 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 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 $W_{D}$ .

Keywords

Teichmueller curves, Jacobians, Hilbert modular surfaces

Mathematical Subject Classification 2010

Primary: 32G15

Secondary: 14K22

References

Publication

Received: 16 February 2012

Revised: 7 March 2013

Accepted: 10 June 2013

Published: 20 March 2014

Proposed: Benson Farb

Seconded: Ronald Stern, Danny Calegari

Authors

Ronen E Mukamel
Department of Mathematics University of Chicago Chicago, IL 60637 USA
http://math.uchicago.edu/~ronen/

Volume 18, issue 2 (2014)