Construction of torsion cohomology classes for KHT Shimura varieties

Abstract

Let $S{h}_K(G,\mu )$ be a Shimura variety of KHT type, as introduced in Harris-Taylor book, associated to some similitude group $G/\mathbb{Q}$ and a open compact subgroup $K$ of $G(\mathbb{A})$. For any irreducible algebraic ${\overline{\mathbb{Q}}}_l$-representation $\xi$ of $G$, let $V_{\xi }$ be the ${\mathbb{Z}}_l$-local system on $S{h}_K(G,\mu )$. From my paper about p-stabilization, we know that if we allow the local component $K_l$ of $K$ to be small enough, then there must exists some non trivial cohomology classes with coefficient in $V_{\xi }$. The aim of this paper is then to construct explicitly such torsion classes with the control of $K_l$. As an application we obtain the construction of some new automorphic congruences between tempered and non tempered automorphic representations of the same weight and same level at $l$.