Finiteness results for Teichmüller curves

Möller, Martin

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.

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