Billiards and Hilbert modular surfaces

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 Teichmüller 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 π. Then a billiard ball bouncing off the sides of P moves in only finitely many directions. A typical billiard trajectory in a regular pentagon is shown in Figure 1.

Figure 1. Billiards in a regular pentagon.

By gluing together copies of (P, dz) reflected through its sides, one obtains a compact Riemann surface X equipped with a holomorphic 1-form ω. Billiard trajectories in P then correspond to geodesics on the surface (X, |ω|). The metric |ω| is flat apart from isolated singularities coming from the vertices of P.

The space of all pairs (X, ω) forms a bundle ΩM_{g} → M_{g} over the moduli space of Riemann surface of genus g, and admits a natural action of SL_{2}(ℝ). Upon projection to M_{g}, the SL_{2}(ℝ)-orbit of a given point (X, ω) determines a ‘complex geodesic’

f : ℍ → M_{g},

i.e. a holomorphic and isometric immersion of the hyperbolic plane into moduli space.

Usually the image of f is dense in M_{g}. On rare occasions, however, f may cover an algebraic curve V ⊂ M_{g}. This happens exactly when the stabilizer SL(X, ω) of (X, ω) is a lattice in SL_{2}(ℝ). In this case V is a Teichmüller curve and P is a lattice polygon.

Using renormalization and Teichmüller theory, Veech showed that billiards in a lattice polygon is dynamically optimal:

• every billiard trajectory is either periodic or uniformly distributed, and

• the number of classes of closed trajectories of length ≤ L is asymptotic to c(P) · L^{2} .

It is classical that a square is a lattice polygon, with SL(X, ω) = SL_{2}(ℤ). Only a handful of additional examples are known: these include the regular n-gons [V], certain triangles [Wa], [KS], [Pu], and other polygons derived from these. Similarly, until recently only finitely many ‘primitive’ Teichmüller curves were known in each M_{g}. (A Teichmüller curve is primitive if it does not arise from a curve in a moduli space of lower genus via a branched covering construction.)

In [Mc1] we give a synthetic construction of infinitely many primitive Teichmüller curves V ⊂ M_{2}.

We begin by observing that if (X, ω) ∈ ΩM_{2} generates a primitive Teichmüller curve V, then the Jacobian of X admits real multiplication by the trace field K ≅ ℚ(\sqrt{d}) of SL(X, ω). This observation shows V lies on a certain Hilbert modular surface: we have

V ⊂ Σ ≅ (ℍ × ℍ)/Γ ⊂ \overline{M_{2}},

where Γ is commensurable to SL_{2}(O_{K}), and Σ parameterizes those X admitting real multiplication by a given order in K.

Let us say ω is a Weierstrass form if its zero divisor is concentrated at a single point. By imposing this additional condition, we reduce from surfaces to curves and obtain:

Theorem 1 The locus

W_{2} = {X : Jac(X) admits real multiplication with

a Weierstrass eigenform} ⊂ M_{2}

is a union of infinitely many primitive Teichmüller curves.

As a concrete application, consider the L-shaped billiard table P (a) with sides of lengths 1 and a as in Figure 2. Then we find:

Theorem 2 P (a) is a lattice polygon if and only if a = (1 + \sqrt{d})/2 for some d ∈ ℚ_{+}.

Figure 2. An L-shaped billiard table determines a Riemann surface of genus two.

The geometry of the Teichmüller curve V ⊂ Σ ⊂ M_{2} generated by P (a) is also interesting. After passing to the universal cover, the lift

Ṽ ⊂ ℍ × ℍ = Σ̃

becomes the graph of a holomorphic function F : ℍ → ℍ, intertwining the action of Γ = SL(X, ω) ⊂ SL_{2}(K) with its Galois conjugate Γ′. When d = 5, F is the ‘pentagon-star’ map shown in Figure 3; it can be constructed by mapping an ideal pentagon to an ideal star, and then analytically continuing to the whole of ℍ by Schwarz reflection. (In particular F is transcendental, and V is not a Shimura curve on Σ.)

Figure 3. A holomorphic map from an ideal pentagon to an ideal star.

To place these results in context, recall that Ratner has establish a powerful classification theorem for the orbits and invariant measures of unipotent flows on homogeneous spaces [Rat]. One can hope to find a similar structure for the action of SL_{2}(ℝ) on ΩM_{g}. We conclude by reporting on recent progress on classifying invariant measures and orbit closures in ΩM_{2}.

The papers [Mc1], [Mc2] contain more details and references, and are avail-

able at http://math.harvard.edu/~ctm/papers.

References

[KS] R. Kenyon and J. Smillie. Billiards on rational-angled triangles. Comment. Math. Helv. 75(2000), 65–108.

[Mc1] C. McMullen. Billiards and Teichmüller curves on Hilbert modular surfaces. To appear, J. Amer. Math. Soc.

[Mc2] C. McMullen. Teichmüller geodesics of infinite complexity. To appear, Acta Math.

[Pu] J.-C. Puchta. On triangular billiards. Comment. Math. Helv. 76(2001), 501–505.

[Rat] M. Ratner. Interactions between ergodic theory, Lie groups and number theory. In Proceedings of the International Congress of Mathematicians (Zürich, 1994), pages 156–182. Birkhaüser, Basel, 1995.

[V] W. Veech. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. math. 97(1989), 553–583.

[Wa] C. C. Ward. Calculation of Fuchsian groups associated to billiards in a rational triangle. Ergod. Th. & Dynam. Sys. 18(1998), 1019–1042.

Mathematics Department, Harvard University, Cambridge, MA