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 \(X_D(1)\).