Foliations of Hilbert modular surfaces

By Curtis T. McMullen. 2007

59 citations recorded by the source corpus.

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).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/504545afd12bde55dee6dae01c051f6e4da73a51

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In Geometry and Topology

  1. Curtis T. McMullen
  2. Martin Möller
  3. A. Wright
  4. Martin Moeller
  5. Simion Filip
  6. Matt Bainbridge
  7. Erwan Lanneau
  8. Serge Cantat

In Mathematical Physics

  1. Curtis T. McMullen
  2. Martin Moeller
  3. Serge Cantat
  4. Franccois Berteloot
  5. De-Qi Zhang
  6. Jinpeng An
  7. Shou-wu Zhang
  8. Sai-Kee Yeung

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