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Hyperbolic lattice-point counting and modular symbols

Yiannis N. Petridis; Morten S. Risager — 2009

Journal de Théorie des Nombres de Bordeaux

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.

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