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Cuspidal cohomology for $GL(n)$ over a number field

Nasit Darshan, A. Raghuram

Abstract

The main result of this article proves the nonvanishing of cuspidal cohomology for $GL(n)$ over a number field which is Galois over its maximal totally real subfield. The proof uses the internal structure of a strongly-pure weight that can possibly support cuspidal cohomology and the foundational work of Borel, Labesse, and Schwermer.

Cuspidal cohomology for $GL(n)$ over a number field

Abstract

The main result of this article proves the nonvanishing of cuspidal cohomology for over a number field which is Galois over its maximal totally real subfield. The proof uses the internal structure of a strongly-pure weight that can possibly support cuspidal cohomology and the foundational work of Borel, Labesse, and Schwermer.
Paper Structure (49 sections, 28 theorems, 125 equations)

This paper contains 49 sections, 28 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on the cuspidal cohomology of ${ \rm GL}(n)$
  3. The base field
  4. Locally symmetric spaces
  5. Sheaves on locally symmetric spaces
  6. The field of coefficients $E$
  7. Characters of the torus $T$
  8. The sheaf $\widetilde{\mathcal{M}}_{\lambda, E}$
  9. Cohomology of $\widetilde{\mathcal{M}}_{\lambda, E}$
  10. Cuspidal cohomology
  11. Strongly pure weights
  12. Strongly-pure weights over $\mathbb{C}$
  13. Strongly-pure weights over $E$
  14. Inner structure of strongly-pure weights
  15. The nonvanishing problem of cuspidal cohomology for ${ \rm GL}(n)$
  16. ...and 34 more sections

Key Result

Theorem 1.1

Let $F$ be a number field which is Galois over its maximal totally real subfield. Let $G = \mathrm{Res}_{F/\mathbb{Q}}({ \rm GL}_n/F)$; rest of the notations as above. Suppose $\lambda \in X^+(\mathrm{Res}_{F/\mathbb{Q}}(T_0) \times E)$ is a strongly-pure weight, then we have nonvanishing of cuspid for every embedding $\iota : E \to \mathbb{C}$.

Theorems & Definitions (43)

  • Theorem 1.1: Theorem \ref{['thm:main-gln']}
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Conjecture 3.4
  • Proposition 4.1
  • proof
  • Proposition 4.3
  • proof
  • Theorem 5.1: Thm. 10.4 of Borel--Labesse--Schwermer borel-labesse-schwermer
  • ...and 33 more