Finiteness results for Teichmüller curves
[Résultats de finitude pour les courbes de Teichmüller]
Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 63-83.

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.

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.

DOI : https://doi.org/10.5802/aif.2344
Classification : 14D07,  32G20
Mots clés : courbes de Teichmüller, corps cyclotomiques, modèle de Neron
@article{AIF_2008__58_1_63_0,
     author = {M\"oller, Martin},
     title = {Finiteness results for Teichm\"uller curves},
     journal = {Annales de l'Institut Fourier},
     pages = {63--83},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {1},
     year = {2008},
     doi = {10.5802/aif.2344},
     mrnumber = {2401216},
     zbl = {1140.14010},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2344/}
}
Möller, Martin. Finiteness results for Teichmüller curves. Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 63-83. doi : 10.5802/aif.2344. http://archive.numdam.org/articles/10.5802/aif.2344/

[1] Bosch, S.; Lütkebohmert, W.; Raynaud, M. Néron Models, Ergebnisse der Math. 3, Volume 21, Springer-Verlag, 1990 | MR 1045822 | Zbl 0705.14001

[2] Deligne, P. Un théorème de finitude pour la monodromie, Progress in Math., Volume 67, Birkhäuser, 1987, pp. 1-19 | MR 900821 | Zbl 0656.14010

[3] Goren, E. Lectures on Hilbert Modular varieties and Modular forms, CRM Monogr. Series, Volume 14, Amer. Math. Soc., 2002 | MR 1863355 | Zbl 0986.11037

[4] Gutkin, E.; Judge, C. Affine mappings of translation surfaces, Duke Math. J., Volume 103 (2000), pp. 191-212 | Article | MR 1760625 | Zbl 0965.30019

[5] Hubert, P.; Lanneau, E. Veech groups without parabolic elements, Duke Math. J., Volume 133 (2006), pp. 335-346 | Article | MR 2225696 | Zbl 1101.30044

[6] Kenyon, R.; Smillie, J. Billiards on rational-angled triangles, Comm. Math. Helv., Volume 75 (2000), pp. 65-108 | Article | MR 1760496 | Zbl 0967.37019

[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 | Article | MR 2000471 | Zbl 1087.32010

[8] Mann, H. B. On linear relations between roots of unity, Mathematika, Volume 12 (1965), pp. 107-117 | Article | MR 191892 | Zbl 0138.03102

[9] Masur, H. On a class of geodesics in Teichmüller space, Annals of Math., Volume 102 (1975), pp. 205-221 | Article | MR 385173 | Zbl 0322.32010

[10] Masur, Howard; Tabachnikov, Serge Rational billiards and flat structures, Handbook of dynamical systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 1015-1089 | MR 1928530 | Zbl 1057.37034

[11] McMullen, C. Billiards and Teichmüller curves on Hilbert modular sufaces, J. Amer. Math. Soc., Volume 16 (2003), pp. 857-885 | Article | MR 1992827 | Zbl 1030.32012

[12] McMullen, C. Teichmüller curves in genus two: Discriminant and spin, Math. Ann., Volume 333 (2005), pp. 87-130 | Article | MR 2169830 | Zbl 1086.14024

[13] McMullen, C. Teichmüller curves in genus two: The decagon and beyond, J. reine angew. Math., Volume 582 (2005), pp. 173-200 | Article | MR 2139715 | Zbl 1073.32004

[14] McMullen, C. Prym varieties and Teichmüller curves, Duke Math. J., Volume 133 (2006), pp. 569-590 | Article | MR 2228463 | Zbl 1099.14018

[15] McMullen, C. Teichmüller curves in genus two: Torsion divisors and ratios of sines, Invent. Math., Volume 165 (2006), pp. 651-672 | Article | MR 2242630 | Zbl 1103.14014

[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 | Article | MR 2242629 | Zbl 1111.14019

[17] Möller, M. Variations of Hodge structures of Teichmüller curves, J.  Amer. Math. Soc., Volume 19 (2006), pp. 327-344 | Article | MR 2188128 | Zbl 1090.32004

[18] Veech, W. Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math., Volume 97 (1989), pp. 533-583 | Article | MR 1005006 | Zbl 0676.32006

[19] Vorobets, Y. B. Plane structures and billiards in rational polygons: the Veech alternative, Russian Math. Surveys, Volume 51 (1996), pp. 779-817 | Article | MR 1436653 | Zbl 0897.58029

[20] Ward, C. Calculation of Fuchsian groups associated to billiards in a rational triangle, Ergod. Th. Dyn. Systems, Volume 18 (1998), pp. 1019-1042 | Article | MR 1645350 | Zbl 0915.58059