Finiteness results for Teichmüller curves

Möller, Martin

Annales de l'Institut Fourier, Volume 58 (2008) no. 1, pp. 63-83.

We show that for each genus there are only finitely many algebraically primitive Teichmüller curves $C$, such that (i) $C$ lies in the hyperelliptic locus and (ii) $C$ is generated by an abelian differential with two zeros of order $g-1$. We prove moreover that for these Teichmüller curves the trace field of the affine group is not only totally real but cyclotomic.

Pour chaque genre $g$ fixé, on montre qu’il n’y a qu’un nombre fini de courbes de Teichmüller $C$ algébriquement primitives telles que (i) $C$ appartient au lieu hyperelliptique et (ii) $C$ est engendrée par une différentielle abélienne avec deux zéros d’ordre $g-1$. On montre en outre que pour ces courbes de Teichmüller le corps de traces du groupe affine n’est pas seulement totalement réel mais cyclotomique.

Classification: 14D07, 32G20
Keywords: Teichmüller curves, cyclotomic field, Neron model
Mot clés : courbes de Teichmüller, corps cyclotomiques, modèle de Neron

[1] Bosch, S.; Lütkebohmert, W.; Raynaud, M. Néron Models, Ergebnisse der Math. 3, Volume 21, Springer-Verlag, 1990 | MR | Zbl

[2] Deligne, P. Un théorème de finitude pour la monodromie, Progress in Math., Volume 67, Birkhäuser, 1987, pp. 1-19 | MR | Zbl

[3] Goren, E. Lectures on Hilbert Modular varieties and Modular forms, CRM Monogr. Series, Volume 14, Amer. Math. Soc., 2002 | MR | Zbl

[4] Gutkin, E.; Judge, C. Affine mappings of translation surfaces, Duke Math. J., Volume 103 (2000), pp. 191-212 | DOI | MR | Zbl

[5] Hubert, P.; Lanneau, E. Veech groups without parabolic elements, Duke Math. J., Volume 133 (2006), pp. 335-346 | DOI | MR | Zbl

[6] Kenyon, R.; Smillie, J. Billiards on rational-angled triangles, Comm. Math. Helv., Volume 75 (2000), pp. 65-108 | DOI | MR | Zbl

[7] Kontsevich, M.; Zorich, A. Connected Components of the Moduli Space of Abelian Differentials with Prescribed Singularities, Invent. Math., Volume 153 (2003), pp. 631-678 | DOI | MR | Zbl

[8] Mann, H. B. On linear relations between roots of unity, Mathematika, Volume 12 (1965), pp. 107-117 | DOI | MR | Zbl

[9] Masur, H. On a class of geodesics in Teichmüller space, Annals of Math., Volume 102 (1975), pp. 205-221 | DOI | MR | Zbl

[10] Masur, Howard; Tabachnikov, Serge Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015-1089 | MR | Zbl

[11] McMullen, C. Billiards and Teichmüller curves on Hilbert modular sufaces, J. Amer. Math. Soc., Volume 16 (2003), pp. 857-885 | DOI | MR | Zbl

[12] McMullen, C. Teichmüller curves in genus two: Discriminant and spin, Math. Ann., Volume 333 (2005), pp. 87-130 | DOI | MR | Zbl

[13] McMullen, C. Teichmüller curves in genus two: The decagon and beyond, J. reine angew. Math., Volume 582 (2005), pp. 173-200 | DOI | MR | Zbl

[14] McMullen, C. Prym varieties and Teichmüller curves, Duke Math. J., Volume 133 (2006), pp. 569-590 | DOI | MR | Zbl

[15] McMullen, C. Teichmüller curves in genus two: Torsion divisors and ratios of sines, Invent. Math., Volume 165 (2006), pp. 651-672 | DOI | MR | Zbl

[16] Möller, M. Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math., Volume 165 (2006), pp. 633-649 | DOI | MR | Zbl

[17] Möller, M. Variations of Hodge structures of Teichmüller curves, J.  Amer. Math. Soc., Volume 19 (2006), pp. 327-344 | DOI | MR | Zbl

[18] Veech, W. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., Volume 97 (1989), pp. 533-583 | DOI | MR | Zbl

[19] Vorobets, Y. B. Plane structures and billiards in rational polygons: the Veech alternative, Russian Math. Surveys, Volume 51 (1996), pp. 779-817 | DOI | MR | Zbl

[20] Ward, C. Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergod. Th. Dyn. Systems, Volume 18 (1998), pp. 1019-1042 | DOI | MR | Zbl