Journal
PUBLICATIONS MATHEMATIQUES DE L IHES
Volume 131, Issue 1, Pages 261-323Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s10240-020-00115-z
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The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation 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.
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