Foliations of Hilbert modular surfaces

The Hilbert modular surface XD is the moduli space of Abelian varieties A with real multiplication by a quadratic order of discriminant D > 1. The locus where A is a product of elliptic curves determines a finite union of algebraic curves XD(1) ⊂ XD.

In this paper we show the lamination XD(1) extends to an essentially unique foliation FD of XD by complex geodesics. The geometry of FD is related to Teichm¨uller theory, holomorphic motions, polygonal billiards and Lattès rational maps. We show every leaf of FD is either closed or dense, and compute its holonomy. We also introduce refinements TN(ν) of the classical modular curves on XD, leading to an explicit description of XD(1).