Vanishing theorems for Shimura varieties at unipotent level.

We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite Γ1(p∞)-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. [ABSTRACT FROM AUTHOR]