Vanishing theorems for Shimura varieties at unipotent level
Ana Caraiani, Daniel R. Gulotta, Christian Johansson
TL;DR
This work proves a vanishing theorem for the compactly supported cohomology of Shimura varieties of Hodge type at unipotent, infinite p-adic level under a p-splitting assumption, showing vanishing above the middle degree. The authors develop a robust p-adic geometric framework using diamonds and perfectoid spaces, leveraging the Hodge–Tate period map and Bruhat stratification to reduce cohomology questions to vanishing statements on the Hodge–Tate fibers. A key contribution is the construction of quotients and inverse limits at infinite level, plus a fiber-wise analysis that yields vanishing results and enables a Poincaré duality spectral sequence for ordinary completed cohomology. As an application, they obtain lower bounds on codimensions for ordinary completed (Borel–Moore) homology, aligning with conjectures of Calegari–Emerton and providing new control over ordinary parts in the p-adic Langlands program. The results extend prior vanishing phenomena from quasi-split cases to general Hodge-type Shimura varieties with a unified, diamond-based approach, and establish new structural results for ordinary parts and completed homology.
Abstract
We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite $Γ_1(p^\infty)$-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 $Γ_1(p^\infty)$ 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.