MathSciDoc: An Archive for Mathematician ∫

Complex Variables and Complex AnalysisGeometric Analysis and Geometric Topologymathscidoc: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.
@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.