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We show that the compactly supported cohomology of Shimura varieties of Hodgetype of infinite $\Gamma_1(p^\infty)$-level (defined with respect to a Borelsubgroup) vanishes above the middle degree, under the assumption that the groupof the Shimura datum splits at $p$. This generalizes and strengthens thevanishing result proved in "Shimura varieties at level $\Gamma_1(p^\infty)$ andGalois representations". As an application of this vanishing theorem, we provea result on the codimensions of ordinary completed homology for the samegroups, analogous to conjectures of Calegari--Emerton for completed(Borel--Moore) homology.

Vanishing theorems for Shimura varieties at unipotent level