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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Variations of Hodge structures of a Teichmüller curve
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by Martin Möller PDF
J. Amer. Math. Soc. 19 (2006), 327-344 Request permission

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 $\textrm {SL}_2({\mathbb {R}})$-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.
References
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Additional Information
  • Martin Möller
  • Affiliation: Universität Essen, FB 6 (Mathematik), 45117 Essen, Germany
  • Email: martin.moeller@uni-essen.de
  • Received by editor(s): January 26, 2004
  • Published electronically: December 12, 2005
  • © Copyright 2005 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 19 (2006), 327-344
  • MSC (2000): Primary 32G15; Secondary 14D07
  • DOI: https://doi.org/10.1090/S0894-0347-05-00512-6
  • MathSciNet review: 2188128