Vanishing theorems for Shimura varieties at unipotent level

TL;DR

This paper proves a vanishing theorem for the compactly supported cohomology of certain Shimura varieties at infinite unipotent level, under specific splitting conditions, with applications to homology codimension conjectures.

Contribution

It generalizes previous vanishing results to Hodge type Shimura varieties with unipotent level structure and applies this to homology codimension conjectures.

Findings

Vanishing of cohomology above middle degree for specified Shimura varieties.

Establishment of codimension bounds for ordinary completed homology.

Extension of prior results to a broader class of Shimura varieties.

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.