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

Abstract

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 ξ ∈ ∂ ℍ 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 oξ 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.