Vanishing theorems for Shimura varieties at unipotent level and Galois representations

The Langlands correspondence relates automorphic forms and Galois representations --- for example, the modular form $\eta(z)^2 \eta(11z)^2$ and the Tate module of the elliptic curve $y^2 + y = x^3 - x^2 - 10x - 20$ are related in the sense that they have the same $L$-function. The $p$-adic Langlands program aims to interpolate the Langlands correspondence in $p$-adic families.  In this setting, the role of automorphic forms is played by the completed cohomology groups defined by Emerton.

Calegari and Emerton have conjectured that the completed cohomology vanishes above a certain degree, often denoted $q_0$.  In the case of Shimura varieties of Hodge type, Scholze has proved the conjecture for compactly supported completed cohomology. We give a strengthening of Scholze's result under the additional assumption that the group becomes split over $\mathbf{Q}_p$.  More specifically, we show that the compactly supported cohomology vanishes not just at full infinite level at $p$, but also at unipotent level at $p$.

We also give an application of the above result to Galois representations.  For any totally real or CM field $F$, Scholze has constructed Galois representations associated with torsion classes in the cohomology of locally symmetric spaces for $\mathrm{GL}_n(F)$. We show that, if $F$ splits completely at the relevant prime, then the nilpotent ideal appearing in the construction can be eliminated.

This talk is based on joint work with Ana Caraiani and Christian Johansson and on joint work with Ana Caraiani, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih.