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

Abstract

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.

We apply this computation to give evidence for the conjecture that there are only finitely many algebraically primitive Teichmüller curves in $ \mathcal{M} $₃. In particular, we prove that there are only finitely many algebraically primitive Teichmüller curves generated by a 1-form having two zeros of order 3 and 1. We also present the results of a computer search for algebraically primitive Teichmüller curves generated by a 1-form having a single zero.