Automorphic congruences between torsion cohomological classes

TL;DR

This paper establishes isomorphisms between the l-torsion parts of cohomology groups associated with congruent local representations, leading to new automorphic congruences for certain similitude groups.

Contribution

It proves isomorphisms of cohomology torsion modules for congruent local systems and constructs precise automorphic congruences for a specific class of groups.

Findings

Isomorphism of l-torsion cohomology groups for congruent representations

Construction of non-tempered automorphic congruences

Application to Shimura varieties and Newton strata

Abstract

For two representations of some local division algebra, congruent modulo $l$, giving rise to two Harris-Taylor local systems on the corresponding Newton strata of the special fiber of a KHT Shimura varieties, we prove that the $l$-torsion of each of their cohomology groups with compact supports are isomorphic, or equivalently the free quotients of each of the cohomology groups are congruent modulo $l$. We then deduce the construction of accurate non tempered automorphic congruences for a similitude group $G/\mathbb Q$ with signature $(1,d-1)$.