2016/03/17 by Kolja Hept, Hept, Kolja
Mathematics · #Geometric and Algebraic Topology #Algebraic Geometry and Number Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1603.05527
In this paper, we classify three-dimensional complex Abelian varieties isogenous to a product A1 × A2, where one of the factors admits real multiplication by a real quadratic order OD of discriminant D. We show that the moduli space XD(3) of these varieties essentially is the disjoint union of certain Hilbert modular varieties X_\mathfraka, each component depending on the choice of an ideal \mathfraka of OD. We give an explicit construction of these varieties. We show that the boundary of the eigenform locus for pseudo-real multiplication by an order O in ℚ(√(D)) ⊕ ℚ over geometric genus zero stable curves is contained in the union of subvarieties defined by equations involving cross-ratios of projective coordinates. Moreover, restricted to certain topological types of stable curves relevant to the classification of primitive Teichmüller curves, we show that the boundary of the eigenform locus coincides with the subspace cut out by these cross-ratio equations. We compute these equations for the example of genus three Prym Teichmüller curves.