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Some vanishing results for the rational completed cohomology of Shimura varieties

Kai-Wen Lan, Lue Pan

Abstract

Based on an almost Kodaira-type vanishing result in mixed characteristics of Bhatt, we show that, in the locally analytic completed cohomology of a general Shimura variety, sufficiently regular infinitesimal weights can only show up in the middle degree.

Some vanishing results for the rational completed cohomology of Shimura varieties

Abstract

Based on an almost Kodaira-type vanishing result in mixed characteristics of Bhatt, we show that, in the locally analytic completed cohomology of a general Shimura variety, sufficiently regular infinitesimal weights can only show up in the middle degree.

Paper Structure

This paper contains 34 sections, 49 theorems, 172 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Main result
  3. Overview of the proof
  4. An almost Kodaira vanishing theorem of Bhatt
  5. Bhatt's mixed characteristic Kodaira vanishing result
  6. Reformulation via Kummer étale site
  7. Locally symmetric varieties and automorphic vector bundles
  8. General setup
  9. Arithmetic quotients and double coset spaces
  10. Locally symmetric varieties and their compactifications
  11. Compatibility with central isogenies
  12. Automorphic vector bundles and their canonical extensions
  13. Automorphic line bundles of positive parallel weights
  14. Geometric Sen theory for towers of Shimura varieties
  15. Shimura varieties and their compactifications
  16. ...and 19 more sections

Key Result

Theorem 1.1.2

Let $\mathcal{I}$ be as above. Then $\mathcal{I}^n$ annihilates (i.e., acts by zero on ) $\tilde{H}_C^{< d, \text{\rm la}}$, for some $n > 0$. In particular, if $[\lambda]: Z(U(\mathfrak{g})) \to C$ is sufficiently regular, then the $[\lambda]$-isotypic part $\tilde{H}_{C, [\lambda]}^{< d, \text{\rm

Theorems & Definitions (135)

  • Theorem 1.1.2
  • Remark 1.1.3
  • Remark 1.1.4
  • Remark 1.1.5
  • Remark 1.1.6
  • Theorem 2.1.1: Bhatt
  • proof
  • Corollary 2.1.2
  • proof
  • Remark 2.1.7
  • ...and 125 more