Discrete series multiplicities for classical groups over Z and level 1 algebraic cusp forms

6 tables, 62 pages. See http://gaetan.chenevier.perso.math.cnrs.fr/levelone/ or http://otaibi.perso.math.cnrs.fr/levelone/ for an associated homepage containing many tables, auxiliary data, and the source code of our algorithms.International audienceThe aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series in the space of level 1 automorphic forms of a split classical group G over Z, and provide numerical applications in absolute rank ≤ 8. Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of GL n over Q (n arbitrary) whose motivic weight is ≤ 24.In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on GL n , and that we push further on in this paper. We use these vanishing results to obtain an arguably "effortless" computation of the elliptic part of the geometric side of the trace formula of G, for an appropriate test function. Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for Sp 2g (Z): we recover all the previously known cases without relying on any, and go further, by a unified and "effortless" method. Gaëtan Chenevier and Olivier Taïbi are supported by the C.N.R.S. and by the project ANR-14-CE25