Calcul des op\'erateurs de Hecke sur les classes d'isomorphisme de r\'eseaux pairs de d\'eterminant 2 en dimension 23 et 25

Abstract

In this article, we compute the Hecke operator $\mathrm{T}_2$, associated to the Kneser $2$-neighbours, defined on the isomorphic classes of even lattices of determinant 2, in dimension 23 and 25. In a previous article, we computed some properties of the Satake parameters of the automorphic representations for the linear groups discovered by Chenevier and Renard. Thanks to these results, we deduce the value of many other Hecke operators. This enables us to compute for every prime $p$ the Kneser graph associated to the $p$-neighbours of lattices in dimension 23 and 25. From our results, we improve Harder's conjecture, and also prove many other congruences involving the Satake parameters of the automorphic representations for the linear groups discovered by Chenevier and Renard.