## The Deligne-Mumford compactification of the real multiplication locus and Teichmüller curves in genus 3.(English)Zbl 1250.14014

The extensive paper under review investigates the real multiplication locus $$\mathcal{RM}$$ in the moduli space $$\mathcal{M}_g$$ of Riemann surfaces of genus $$g$$. In particular, it is concerned with the boundary of $$\mathcal{RM}$$ in the Deligne-Mumford compactification $$\bar{\mathcal{M}}_g$$.
For $$g=3$$, the authors give a complete description of the corresponding stable curves. They also identify those stable curves equipped with holomorphic 1-forms which are in the boundary of the eigenform locus in the bundle $\Omega\mathcal{M}_g\to\mathcal{M}_g$ of holomorphic 1-forms (the case $$g=2$$ has been treated in [Geom. Topol. 11, 1887–2073 (2007; Zbl 1131.32007)]).
For genus $$g > 3$$, the authors prove quite strong restrictions on the stable curves in the boundary of $$\mathcal{RM}$$. Many questions about Riemann surfaces with real multiplication can thus be reduced to concrete problems in algebraic geometry and number theory by passing to the boundary of $$\mathcal{M}_g$$.
The boundary classification is used to prove, for instance, finiteness results for Teichmüller curves in $$\mathcal{M}_3$$ and the non-invariance of the eigenform locus under the action of GL$$^+_2(\mathbb{R})$$ on $$\Omega\mathcal{M}_3$$. Relations to Hilbert modular varieties are also discussed.

### MSC:

 14G35 Modular and Shimura varieties 14D22 Fine and coarse moduli spaces 14H10 Families, moduli of curves (algebraic) 14H15 Families, moduli of curves (analytic) 14H55 Riemann surfaces; Weierstrass points; gap sequences 30F30 Differentials on Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Zbl 1131.32007
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### References:

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