Foliations of Hilbert Modular Surfaces

Abstract

The Hilbert modular surface (X_D) 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 (X_D(1) \subset X_D). In this paper we show the lamination (X_D(1)) extends to an essentially unique foliation (F_D) of (X_D) by complex geodesics. The geometry of (F_D) is related to Teichm¨uller theory, holomorphic motions, polygonal billiards and Latt`es rational maps. We show every leaf of (F_D) is either closed or dense, and compute its holonomy. We also introduce refinements (T_N(\nu)) of the classical modular curves on (X_D), leading to an explicit description of (X_D(1)).