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Billiards and Teichmüller curves on Hilbert modular surfaces. (English) Zbl 1030.32012
Let $$\mathcal{M}_{g}$$ be the moduli space of Riemann surfaces of genus $$g$$. Teichmüller curves can be obtained in $$\mathcal{M}_{g}$$ for all $$g$$ taking branched coverings of tori. A Teichmüller curve is said to be primitive if it does not arise from a curve in a moduli space of lower genus via a branched covering construction. Only a finite number of primitive Teichmüller curves have been found in each $$\mathcal{M}_{g}$$. In this paper the author finds an infinite collection of primitive curves in genus two via the families of Jacobians they determine.
Let $$X$$ be a Riemann surface. A Weierstrass form is a holomorphic 1-form $$\omega \in \Omega (X)$$ whose zero divisor is concentrated in a single point. In the case of genus two there are six such forms up to scale, one for each Weierstrass point. Let $$\mathcal{W}_{2}=$${$$X\in \mathcal{M}_{2}:$$ Jac$$(X)$$ admits real multiplication with a Weierstrass eigenform}. The author proves that the locus $$\mathcal{W}_{2}$$ is a countable union of primitive Teichmüller curves. He exploits the connection between billiard tables and Teichmüller curves. We say $$P$$ is a lattice polygon if SL$$(X,\omega)$$ is a lattice in SL$$_{2}(\mathbb{R})$$ or, equivalently, if $$(X,\omega)$$ generates a Teichmüller curve.
The author proves the following result on L-shaped polygons: a L-shaped polygon $$P(a,b)$$ is a lattice polygon iff either $$a$$ and $$b$$ are rational or $$a=x+z\sqrt{d}$$ and $$b=y+z\sqrt{d}$$ for some $$x,y,z\in \mathbb{Q}$$ with $$x+y=1$$ and $$d\geq 0$$ in $$\mathbb{Z}$$. In the second case, the trace field of $$P(a,b)$$ is $$\mathbb{ Q(}\sqrt{d})$$. From this result it is proved that $$P(a,a)$$ is a lattice polygon iff $$a$$ is either rational or $$a=(1\pm \sqrt{d})/2$$ for some $$d\in \mathbb{Q }$$. As a consequence, every quadratic field arises as the trace field of a Teichmüller curve in $$\mathcal{M}_{2}$$ and there are infinitely many primitive curves in $$\mathcal{M}_{2}$$.
The author also gives a direct algorithm to generate elements in SL$$(X,\omega)$$. In particular, he proves that the Teichmüller curve generated by $$P(a,a)$$, $$a=(1\pm \sqrt{d})/2$$ has genus zero for $$d=2,3,4,7,13,17,21,29$$ and $$33$$.

##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010) 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 14G35 Modular and Shimura varieties
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