Complete periodicity of Prym eigenforms

Lanneau, Erwan; Nguyen, Duc-Manh

doi:10.24033/asens.2277

Complete periodicity of Prym eigenforms
[Complète périodicité des formes propres Prym]

Lanneau, Erwan ; Nguyen, Duc-Manh

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 1, pp. 87-130.

Résumé
Abstract

Dans cet article nous démontrons plusieurs résultats topologiques sur les formes propres des lieux Prym, formes différentielles abéliennes découvertes par McMullen dans des travaux antérieurs. Nous obtenons une propriété dite de complète périodicité (introduite par Calta), ainsi que de nouvelles familles de surfaces de translation vérifiant la dichotomie topologique de Veech (sans être une surface de Veech) . Comme conséquences nous montrons que l'ensemble limite des groupes de Veech de formes propres de certaines strates en genre $3, 4$ , et 5 est soit vide, soit un point, soit tout le cercle à l'infini. Ceci nous permet de plus de construire de nouveaux exemples de surfaces de translation ayant un groupe de Veech infiniment engendré et de première espèce.

Notre preuve repose sur une nouvelle approche de la notion de feuilletage périodique par les involutions linéaires.

This paper deals with Prym eigenforms which are introduced previously by McMullen.We prove several results on the directional flow on those surfaces, related to complete periodicity (introduced by Calta). More precisely we show that any homological direction is algebraically periodic, and any direction of a regular closed geodesic is a completely periodic direction. As a consequence we draw that the limit set of the Veech group of every Prym eigenform in some Prym loci of genus $3, 4$ , and 5 is either empty, one point, or the full circle at infinity. We also construct new examples of translation surfaces satisfying the topological dichotomy (without being lattice surfaces). As a corollary we obtain new translation surfaces whose Veech group is infinitely generated and of the first kind.

Publié le : 2016-11-12

MR Zbl

DOI : 10.24033/asens.2277

Classification : 37E05; 37D40
Keywords: Real multiplication, Prym locus, translation surface.
Mot clés : Multiplication réelle, lieu Prym, surfaces de translation.

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     author = {Lanneau, Erwan and Nguyen, Duc-Manh},
     title = {Complete periodicity of {Prym} eigenforms},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {87--130},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {1},
     year = {2016},
     doi = {10.24033/asens.2277},
     mrnumber = {3465977},
     zbl = {1338.32013},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2277/}
}

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Lanneau, Erwan; Nguyen, Duc-Manh. Complete periodicity of Prym eigenforms. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 1, pp. 87-130. doi : 10.24033/asens.2277. http://www.numdam.org/articles/10.24033/asens.2277/

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