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

TL;DR

This paper computes traces of Hecke operators on automorphic forms for special orthogonal groups of ranks 7, 8, and 9, linking these to properties of automorphic representations and a conjecture on zeta functions of moduli spaces.

Contribution

It determines Hecke operator traces for level one automorphic forms on specific orthogonal groups with arbitrary weights, using Arthur's theory and connecting to conjectures on zeta functions.

Findings

Trace formulas for Hecke operators on SO(7), SO(8), SO(9)

Properties of Satake parameters for certain automorphic representations

Support for a conjecture on zeta functions of moduli spaces

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 $3$.