Periods of Hilbert modular surfaces

Foliations of Hilbert modular surfaces

The Hilbert modular surface XD is the moduli space of Abelian varietiesA 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 ...

FOLIATIONS OF HILBERT MODULAR SURFACES By CURTIS

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...

Tate Conjectures for Hilbert Modular Surfaces

Billiards and Teichmüller curves on Hilbert modular surfaces

This paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of Riemann surfaces of genus two. These Teichmüller curves lie on Hilbert modular surfaces parameterizing Abelian varieties with real multiplication. Explicit examples, constructed from L-shaped polygons, give billiard tables with optimal dynamical properties.

An arithmetic intersection formula on Hilbert modular surfaces

In this paper, we obtain an explicit arithmetic intersection formula on a Hilbert modular surface between the diagonal embedding of the modular curve and a CM cycle associated to a nonbiquadratic CM quartic field. This confirms a special case of the author’s conjecture with J. Bruinier, and is a generalization of the beautiful factorization formula of Gross and Zagier on singular moduli. As an ...