Dependency on the Group in Automorphic Sobolev Inequalities

In [1] and [2], Bernstein and Reznikov have introduced a new way of estimating the coefficients in the spectral expansion of φ2, where φ is a Maass cusp of norm 1 on a quotient Y = Γ\H of the Poincaré upper half-plane with finite volume. The question of obtaining the precise exponential decay of those coefficients had been posed by Selberg, and first solved by Good [5] (for holomorphic forms) and Sarnak [13] (for Maass forms). Bernstein and Reznikov obtain in fact the right polynomial growth conjectured by Sarnak: let (φi) be an orthonormal basis of the space of cusp forms on Y , eigenfunctions of the Laplace operator with eigenvalue