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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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