Shimura and Teichmüller curves

Martin Möller

doi:10.3934/jmd.2011.5.1

Article Contents

2011, Volume 5, Issue 1: 1-32. Doi: 10.3934/jmd.2011.5.1

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Shimura and Teichmüller curves

Martin Möller^1,

1.
Institut für Mathematik, Johann Wolfgang Goethe-Universität Frankfurt, Robert-Mayer-Str. 6–8, 60325 Frankfurt am Main

Received: July 2009

Revised: January 2011

Published: April 2011

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Abstract

We classify curves in the moduli space of curves $M_g$ that are both Shimura and Teichmüller curves: for both $g=3$ and $g=4$ there exists precisely one such curve, for $g=2$ and $g \geq 6$ there are no such curves.
We start with a Hodge-theoretic description of Shimura curves and of Teichmüller curves that reveals similarities and differences of the two classes of curves. The proof of the classification relies on the geometry of square-tiled coverings and on estimating the numerical invariants of these particular fibered surfaces.
Finally, we translate our main result into a classification of Teichmüller curves with totally degenerate Lyapunov spectrum.

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Mathematics Subject Classification: Primary: 14H15; Secondary: 14G35.

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