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On the non-generic part of cohomology of compact unitary Shimura varieties of signature $(1,n)$

Kun Liu

Abstract

In this short note, we prove a result about the non-generic part of the cohomology of certain compact unitary Shimura varieties for good $p$, partially extending a result of Boyer in the case of Harris--Taylor unitary Shimura varieties. Our arguments are different to those of Boyer -- we work in the context of the work of Fargues--Scholze, using ideas introduced by Koshikawa to study the generic part of cohomology.

On the non-generic part of cohomology of compact unitary Shimura varieties of signature $(1,n)$

Abstract

In this short note, we prove a result about the non-generic part of the cohomology of certain compact unitary Shimura varieties for good , partially extending a result of Boyer in the case of Harris--Taylor unitary Shimura varieties. Our arguments are different to those of Boyer -- we work in the context of the work of Fargues--Scholze, using ideas introduced by Koshikawa to study the generic part of cohomology.

Paper Structure

This paper contains 10 sections, 12 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Background: Generic Vanishing
  3. Non-Generic Case
  4. Our Main Results
  5. Overview of proof
  6. Further Remarks
  7. Representation Result
  8. Local Vanishing Result
  9. Proof of Local Vanishing
  10. Mantovan's formula and Conclusion

Key Result

Theorem 1.1

Assume exactly one archimedean place has signature $(1,n)$. Let $p\not= \ell$ and $p>n+1$. Assume $K_p$ is hyperspecial and $p$ satisfies additional split and unramified condition. Let $H_{K_p}$ be the Hecke algebra at $p$ and $\mathfrak{m}\subset H_{K_p}$ be the maximal ideal corresponding to an un

Theorems & Definitions (26)

  • Theorem 1.1: \ref{['main']}
  • Theorem 1.2: \ref{['main2']}
  • Remark 1
  • Theorem 2.1
  • Remark 2
  • proof
  • Lemma 1
  • proof
  • Remark 3
  • Remark 4
  • ...and 16 more