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.