# Foliations of Hilbert Modular Surfaces

 Title: Foliations of Hilbert Modular Surfaces Author: McMullen, Curtis T. Citation: McMullen, Curtis T. 2007. Foliations of Hilbert modular surfaces. American Journal of Mathematics 129(1): 183-215. Full Text & Related Files: McMullen_FoliationsHilbert.pdf (366.5Kb; PDF) 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 $$X_D(1)$$. Published Version: http://www.math.jhu.edu/~ajm/ Other Sources: http://www.math.harvard.edu/~ctm/papers/index.html Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3446012 Downloads of this work: