Galois irreducibility implies cohomology freeness for KHT Shimura varieties

TL;DR

This paper proves that the localized cohomology groups of certain KHT Shimura varieties are free when associated Galois representations are irreducible, under specific field extension conditions.

Contribution

It extends previous results by establishing cohomology freeness for a broader class of maximal ideals linked to irreducible Galois representations under new field extension hypotheses.

Findings

Cohomology groups are free for irreducible Galois representations.

The result applies under the condition that the degree of a specific field extension exceeds the variety’s dimension.

Generalizes prior work to include cases with irreducible Galois representations.

Abstract

Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $\mathbb T$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $\mathfrak m$ of $\mathbb T$, appear to be free. In this work, we obtain the same result for $\mathfrak m$ such that its associated Galois $\overline{\mathbb F}_l$-representation $\overline{\rho_{\mathfrak m}}$ is irreducible, under the hypothesis that $[F(\exp(2i\pi/l):F]>d$ where $F$ is the reflex field, $d$ the dimension of the KHT Shimura variety and $l$ the residual characteristic.

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Introduction