Abstract

This article gives a uniform construction of infinitely many primitive Teichmuller curves V⊂Mg for g=2, 3, and 4

Highlights

  • Let D > 1 be an integer congruent to 0 or 1 mod 4, and let OD be the real quadratic order of discriminant D

  • Let XD(1) ⊂ XD denote the locus where Aτ is isomorphic to a polarized product of elliptic curves E1 ×E2

  • Theorem 7.4 Up to the action of ι, FD is the unique extension of the lamination XD(1) to a foliation of XD by complex geodesics

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Summary

Introduction

Let D > 1 be an integer congruent to 0 or 1 mod 4, and let OD be the real quadratic order of discriminant D. The foliated Hilbert modular surface (XD, FD) presents a similar structure, with the fibration p : X → V replaced by the holomorphic foliation AD coming from the level sets of τ1 on XD = H × H. This suggests that one should regard (XD, AD, FD) as a quantum Teichmuller curve, in the same sense that a 3-manifold with a measured foliation can be regarded as a quantum Teichmuller geodesic [Mc3].

Quaternion algebras
Modular curves and surfaces
Laminations
Foliations of Teichmuller space
Genus two
Holomorphic motions
Quasiconformal dynamics
Further results
Full Text
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