Vanishing theorems for Shimura varieties at unipotent level

Abstract

We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite ${\Gamma }_1(p^{\infty })$-level (defined with respect to a Borel subgroup) vanishes above the middle degree, under the assumption that the group of the Shimura datum splits at $p$. This generalizes and strengthens the vanishing result proved in [A. Caraiani et al., Compos. Math. 156 (2020)]. As an application of this vanishing theorem, we prove a result on the codimensions of ordinary completed homology for the same groups, analogous to conjectures of Calegari–Emerton for completed (Borel–Moore) homology.