The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3

Matt Bainbridge Department of Mathematics, Indiana University Martin Möller Institut für Mathematik, Goethe-Universität Frankfurt

Complex Variables and Complex Analysis Geometric Analysis and Geometric Topology mathscidoc:1701.08002

Acta Mathematica, 208, (1), 1-92, 2009.12
In the moduli space $$ \mathcal{M} $$ _{$g$}of genus-$g$Riemann surfaces, consider the locus $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ of Riemann surfaces whose Jacobians have real multiplication by the order $$ \mathcal{O} $$ in a totally real number field$F$of degree$g$. If$g$= 3, we compute the closure of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of $$ \mathcal{M} $$ _{$g$}and the closure of the locus of eigenforms over $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ in the Deligne–Mumford compactification of the moduli space of holomorphic 1-forms. For higher genera, we give strong necessary conditions for a stable curve to be in the boundary of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ . Boundary strata of $$ \mathcal{R}{\mathcal{M}_{\mathcal{O}}} $$ are parameterized by configurations of elements of the field$F$satisfying a strong geometry of numbers type restriction.
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@inproceedings{matt2009the,
  title={The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3},
  author={Matt Bainbridge, and Martin Möller},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203357921698750},
  booktitle={Acta Mathematica},
  volume={208},
  number={1},
  pages={1-92},
  year={2009},
}
Matt Bainbridge, and Martin Möller. The Deligne–Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3. 2009. Vol. 208. In Acta Mathematica. pp.1-92. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203357921698750.
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