Torsion classes in the cohomology of KHT Shimura varieties

TL;DR

This paper investigates the rarity of torsion classes in the cohomology of KHT Shimura varieties, establishing a link to automorphic representations and constructing explicit automorphic congruences using completed cohomology.

Contribution

It introduces a method to associate torsion classes with automorphic representations and constructs explicit examples of automorphic congruences in the context of KHT Shimura varieties.

Findings

Associates torsion classes with multiple automorphic representations.

Constructs torsion classes in regular weight using completed cohomology.

Provides explicit examples of automorphic congruences.

Abstract

A particular case of Bergeron-Venkatesh's conjecture predicts that torsion classes in the cohomology of Shimura varieties are rather rare. According to this and for Kottwitz-Harris-Taylor type of Shimura varieties, we first associate to each such torsion class an infinity of irreducible automorphic representations in characteristic zero, which are pairwise non isomorphic and weakly congruent. Then, using completed cohomology, we construct torsion classes in regular weight and then deduce explicit examples of such automorphic congruences.