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

TL;DR

This paper computes Hecke operators on classes of even lattices with determinant 2 in dimensions 23 and 25, advancing understanding of automorphic representations and related conjectures.

Contribution

It provides explicit calculations of Hecke operators and Kneser graphs in high dimensions, improving conjectures and revealing new congruences in automorphic representation theory.

Findings

Computed Hecke operator T_2 for specific lattices

Constructed Kneser graphs for p-neighbours in dimensions 23 and 25

Enhanced understanding of Satake parameters and proved new congruences

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.