Traces des op\'erateurs de Hecke sur les espaces de formes automorphes de $\mathrm{SO}_7$, $\mathrm{SO}_8$ ou $\mathrm{SO}_9$ en niveau $1$ et poids arbitraire

Abstract

In this article, we determine the trace of some Hecke operators on the spaces of level one automorphic forms on the special orthogonal groups of the euclidean lattices $\mathrm{E}_7$, $\mathrm{E}_8$ and $\mathrm{E}_8\oplus \mathrm{A}_1$, with arbitrary weight. Using Arthur's theory, we deduce properties of the Satake parameters of the automorphic representations for the linear groups discovered by Chenevier and Renard. Our results corroborate a conjecture by Bergstr\"om, Faber and van der Geer about the Hasse-Weil zeta function on the moduli spaces of $17$-pointed curves of genus $3$.