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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) â 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 TeichmÂ¨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(Î½) of the classical modular curves on XD, leading to an explicit description of XD(1).