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On the depth of the adjoint representations

Arindam Jana, Amiya Mondal

Abstract

Let $F$ be a non-archimedean local field of odd residual characteristic $p$. The depth of a smooth representation of ${\rm GL}_n(F)$ is an invariant of Local Langlands Correspondence (LLC). The analogous notion on the Galois side of LLC is known as the slope of a local Galois representation. The slope is well related to the Swan conductor for irreducible Galois representations, whereas its behavior is subtle for reducible Galois representations. In this article, we provide an explicit formula for the slope of the adjoint of a Carayol representation of the local Galois group.

On the depth of the adjoint representations

Abstract

Let be a non-archimedean local field of odd residual characteristic . The depth of a smooth representation of is an invariant of Local Langlands Correspondence (LLC). The analogous notion on the Galois side of LLC is known as the slope of a local Galois representation. The slope is well related to the Swan conductor for irreducible Galois representations, whereas its behavior is subtle for reducible Galois representations. In this article, we provide an explicit formula for the slope of the adjoint of a Carayol representation of the local Galois group.
Paper Structure (11 sections, 26 theorems, 108 equations)

This paper contains 11 sections, 26 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Slope of a Galois representation
  3. Herbrand functions
  4. Proof of Theorems \ref{['int thm slope adjoint wild']} and \ref{['int thm slope adjoint any wild']}
  5. Some lemmas
  6. Proof of Theorem \ref{['int thm slope adjoint wild']}
  7. Proof of Theorem \ref{['int thm slope adjoint any wild']}
  8. Proof of Theorems \ref{['int thm slope adjoint tame']} and \ref{['int thm slope carayol']}
  9. Some lemmas
  10. Proof of Theorem \ref{['int thm slope adjoint tame']}
  11. Proof of Theorem \ref{['int thm slope carayol']}

Key Result

Theorem 1.1

Let $F$ be a $p$-adic field of residue characteristic $p\neq 2$. Let $\rho$ be a Carayol representation of $\mathcal{W}_F$ of dimension $mp^r$ where $(p, m)=1$. If $m>1$, then ${\rm sl}(\rho\otimes\rho^\vee)={\rm sl}(\rho)$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • ...and 46 more