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 "Shimura varieties at level ${\Gamma }_1(p^{\infty })$ and Galois representations". 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.

38 pages, minor revisions to improve exposition

References in corpus (1)

• Relative p-adic Hodge theory, II: Imperfect period rings

Cited by in corpus (1)

• Serre weights for $GS{p}_4$ over totally real fields