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Variations of Hodge structures of a Teichmüller curve

Author: Martin Möller
Journal: J. Amer. Math. Soc. 19 (2006), 327-344
MSC (2000): Primary 32G15; Secondary 14D07
Published electronically: December 12, 2005
MathSciNet review: 2188128
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Abstract: Teichmüller curves are geodesic discs in Teichmüller space that project to an algebraic curve in the moduli space $ M_g$. We show that for all $ g \geq 2$ Teichmüller curves map to the locus of real multiplication in the moduli space of abelian varieties. Observe that McMullen has shown that precisely for $ g=2$ the locus of real multiplication is stable under the $ {\rm SL}_2({\mathbb{R}})$-action on the tautological bundle $ \Omega M_g$.

We also show that Teichmüller curves are defined over number fields and we provide a completely algebraic description of Teichmüller curves in terms of Higgs bundles. As a consequence we show that the absolute Galois group acts on the set of Teichmüller curves.

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Additional Information

Martin Möller
Affiliation: Universität Essen, FB 6 (Mathematik), 45117 Essen, Germany

Keywords: Teichm\"uller curve, real multiplication, maximal Higgs local subsystem
Received by editor(s): January 26, 2004
Published electronically: December 12, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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