no. 1
Finiteness results for Teichmüller curves
[Résultats de finitude pour les courbes de Teichmüller]
Möller, Martin1
1 Universität Essen FB 6 (Mathematik) 45117 Essen (Germany)
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 : 10.5802/aif.2344
Classification : 14D07, 32G20
Keywords: Teichmüller curves, cyclotomic field, Neron model
Mot clés : courbes de Teichmüller, corps cyclotomiques, modèle de Neron

Möller, Martin 1

1 Universität Essen FB 6 (Mathematik) 45117 Essen (Germany) 
@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.2344/}
}
TY  - JOUR
AU  - Möller, Martin
TI  - Finiteness results for Teichmüller curves
JO  - Annales de l'Institut Fourier
PY  - 2008
SP  - 63
EP  - 83
VL  - 58
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2344/
DO  - 10.5802/aif.2344
LA  - en
ID  - AIF_2008__58_1_63_0
ER  - 
%0 Journal Article
%A Möller, Martin
%T Finiteness results for Teichmüller curves
%J Annales de l'Institut Fourier
%D 2008
%P 63-83
%V 58
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2344/
%R 10.5802/aif.2344
%G en
%F AIF_2008__58_1_63_0
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. https://aif.centre-mersenne.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 | 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

Cité par Sources :