Hyperbolic lattice-point counting and modular symbols

Petridis, Yiannis N.; Risager, Morten S.

doi:10.5802/jtnb.698

Petridis, Yiannis N.¹ ; Risager, Morten S.²

¹ Department of Mathematics University College London Gower Street London WC1E 6BT The Graduate Center Mathematics Ph.D. Program 365 Fifth Avenue Room 4208 New York, NY 10016-4309
² Department of Mathematical Sciences University of Aarhus Ny Munkegade Building 530 8000 Aarhus, Denmark Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 2100 Copenhagen Ø, Denmark

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 721-734.

Résumé
Abstract

Soit un sous-groupe $Γ$ de ${SL}_{2} (ℝ)$ cocompact et soit $α$ une forme harmonique réelle (non nulle). Nous étudions le comportement asymptotique de la fonction comptant des points du réseau hyperbolique $Γ$ sous hypothèses imposées par des symboles modulaires $〈γ, α〉$ . Nous montrons que les valeurs normalisées des symboles modulaires, ordonnées selon ce comptage possèdent une répartition gaussienne.

For a cocompact group $Γ$ of ${SL}_{2} (ℝ)$ we fix a real non-zero harmonic $1$ -form $α$ . We study the asymptotics of the hyperbolic lattice-counting problem for $Γ$ under restrictions imposed by the modular symbols $〈γ, α〉$ . We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.

MR Zbl

DOI : 10.5802/jtnb.698

Classification : 11F67, 11F72, 11M36

Affiliations des auteurs :

Petridis, Yiannis N. ¹ ; Risager, Morten S. ²

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Petridis, Yiannis N.; Risager, Morten S. Hyperbolic lattice-point counting and modular symbols. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 721-734. doi : 10.5802/jtnb.698. http://www.numdam.org/articles/10.5802/jtnb.698/

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