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Finite length for unramified $\mathrm{GL}_2$

Christophe Breuil, Florian Herzig, Yongquan Hu, Stefano Morra, Benjamin Schraen

Abstract

Let $p$ be a prime number and $K$ a finite unramified extension of $\mathbb{Q}_p$. If $p$ is large enough with respect to $[K:\mathbb{Q}_p]$ and under mild genericity assumptions, we prove that the admissible smooth representations of $\mathrm{GL}_2(K)$ that occur in Hecke eigenspaces of the mod $p$ cohomology are of finite length. We also prove many new structural results about these representations of $\mathrm{GL}_2(K)$ and their subquotients.

Finite length for unramified $\mathrm{GL}_2$

Abstract

Let be a prime number and a finite unramified extension of . If is large enough with respect to and under mild genericity assumptions, we prove that the admissible smooth representations of that occur in Hecke eigenspaces of the mod cohomology are of finite length. We also prove many new structural results about these representations of and their subquotients.
Paper Structure (24 sections, 75 theorems, 221 equations)

This paper contains 24 sections, 75 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Some sketches of proofs
  4. Notation and preliminaries
  5. Cohen--Macaulayness of gr_m(pi)
  6. The theorem
  7. Preliminaries on filtered and graded modules
  8. Some homological arguments
  9. The Iwahori representation tau
  10. Proof of the theorem
  11. Verifying assumption \ref{['it:assum-iv']}
  12. Finite length in the split reducible case
  13. Preliminaries
  14. Finite length
  15. Finite length in the nonsplit reducible case
  16. ...and 9 more sections

Key Result

Theorem 1.1.1

Assume that $\overline{r}$ is generic and that $r=1$.

Theorems & Definitions (152)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Theorem 1.1.3
  • Theorem 1.2.1: Theorem \ref{['thm:CMC']}
  • Proposition 1.2.2: § \ref{['sec:verify-assumpt-iv']}
  • Theorem 1.2.3: Proposition \ref{['prop:split-I1']}, Proposition \ref{['prop:nonsplit-I1']}
  • Remark 2.1.1
  • Theorem 2.1.2
  • Remark 2.1.3
  • Remark 2.1.4
  • ...and 142 more