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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Neeb, Karl-Hermann / Noguchi, Junjiro / Shahidi, Freydoon / Sogge, Christopher D. / Wienhard, Anna

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Rank 88 out of 312 in category Mathematics and 124 out of 254 in Applied Mathematics in the 2015 Thomson Reuters Journal Citation Report/Science Edition

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Impact per Publication (IPP) 2015: 0.606

Mathematical Citation Quotient (MCQ) 2015: 0.66

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Le module de continuité des valeurs au bord des fonctions propres et des formes automorphes

Jean-Pierre Otal1

1Laboratoire Émile Picard, Université Paul Sabatier, 118, route de Narbonne, Toulouse, France.

Citation Information: Forum Mathematicum. Volume 22, Issue 5, Pages 1009–1032, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum.2010.055, February 2010

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Let Γ be a Fuchsian group acting on the hyperbolic plane ℍ. Any eigenfunction of the Laplacian, any automorphic form on an hyperbolic surface ℍ/Γ induce a distribution on the boundary ℍ. This distribution is the derivative of a certain order of a fonction F on ℍ : the derivative of order 1 in the case of bounded eigenfunctions, the derivative of order in the case of cuspidal automorphic forms of weight k. For cuspidal eigenfunctions (resp. for cuspidal automorphic forms) the optimal Hölder exponent of F at a point ξℍ can be computed exactly when ℍ/Γ has finite volume : this exponent depends only on the eigenvalue of the function (resp. on the weight of the automorphic form) and on the fact that the ray be recurrent or not ; in particular F is not differentiable at any point, except possibly at the cusps, depending on the eigenvalue (resp. on the weight). For regular but non-cuspidal automorphic forms, there is a continuous family of possibilities for the modulus of continuity. We study in details the case of automorphic forms of weight (like the Jacobi theta function, whose associate function F on ℍ has imaginary part the Riemann function : using properties of the geodesic flow such as the Khintchine-Sullivan theorem, we show that at almost all ξ a modulus of continuity is , for any є > 0.

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