Summary: Billiards and Hilbert modular surfaces
MPI Arbeitstagung 2003
Curtis T. McMullen
In this talk we discuss a connection between billiards in polygons and
algebraic curves in the moduli space of Riemann surfaces. In genus two, we
find these Teichm¨uller curves lie on Hilbert modular surfaces parameterizing
Abelian varieties with real multiplication. Explicit examples give L-shaped
billiard tables with optimal dynamical properties.
Let P C be a polygon whose angles are rational multiplies of . Then
a billiard ball bouncing off the sides of P moves in only finitely many direc-
tions. A typical billiard trajectory in a regular pentagon is shown in Figure
Figure 1. Billiards in a regular pentagon.
By gluing together copies of (P, dz) reflected through its sides, one ob-
tains a compact Riemann surface X equipped with a holomorphic 1-form
. Billiard trajectories in P then correspond to geodesics on the surface
(X, ||). The metric || is flat apart from isolated singularities coming from
the vertices of P.
The space of all pairs (X, ) forms a bundle Mg Mg over the moduli
space of Riemann surface of genus g, and admits a natural action of SL2(R).