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Hybrid sup-norm bounds for Hecke–Maass cusp forms

Nicolas Templier (2015)

Journal of the European Mathematical Society

Let f be a Hecke–Maass cusp form of eigenvalue λ and square-free level N . Normalize the hyperbolic measure such that vol ( Y 0 ( N ) ) = 1 and the form f such that f 2 = 1 . It is shown that f ϵ λ 5 24 + ϵ N 1 3 + ϵ for all ϵ > 0 . This generalizes simultaneously the current best bounds in the eigenvalue and level aspects.

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 ( ) 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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