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

摘要

In the moduli space M mathcal{M} _g of genus-g Riemann surfaces, consider the locus RM_O mathcal{R}{mathcal{M}_{mathcal{O}}} of Riemann surfaces whose Jacobians have real multiplication by the order O mathcal{O} in a totally real number field F of degree g. If g = 3, we compute the closure of RM_O mathcal{R}{mathcal{M}_{mathcal{O}}} in the Deligne–Mumford compactification of M mathcal{M} _g and the closure of the locus of eigenforms over RM_O 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 RM_O mathcal{R}{mathcal{M}_{mathcal{O}}} . Boundary strata of RM_O mathcal{R}{mathcal{M}_{mathcal{O}}} are parameterized by configurations of elements of the field F satisfying a strong geometry of numbers type restriction.