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Ordinary local representations and $\Ext$ groups

Debargha Banerjee, Srijan Das

TL;DR

The paper proves vanishing results for Ext (and Hom) involving the $p$-adic Langlands correspondence for $GL_2(\mathbb{Q}_p)$ when the local Galois representation is ordinary, by bridging local and global objects through Shimura and Drinfeld towers. A central tool is the Colmez–Dospinescu–Niziol factorization of the Drinfeld tower’s cohomology, which isolates the Galois-isotypic components and expresses them in terms of $\Pi(\rho_p)'$ and related modules. The results show that ordinary local Langlands representations do not occur in the finite-level Shimura or Drinfeld cohomology, and they establish vanishing (and in some cases finiteness) results for Ext groups in the local setting, using a combination of p-adic and mod $p$ Langlands compatibility, block theory for $p$-adic Banach representations, and cohomological uniformization techniques. These findings advance the understanding of where $p$-adic Langlands representations can appear in arithmetic geometry, and they connect local representation theory with global automorphic objects via robust factorization theorems. The work thus strengthens the philosophy that ordinary $p$-adic Langlands representations tend not to appear in the “supersingular” parts of cohomology and clarifies their role in the global–local panorama.

Abstract

We can associate an admissible unitary representation $Π(ρ_p)$ of $\GL_2(\Q_p)$ with every local Galois representation $ρ_p$ by the $p$-adic local Langlands correspondence. If $ρ_p$ is ordinary, we prove local and global vanishing results for $\Ext$ functors with respect to these representations.

Ordinary local representations and $\Ext$ groups

TL;DR

The paper proves vanishing results for Ext (and Hom) involving the -adic Langlands correspondence for when the local Galois representation is ordinary, by bridging local and global objects through Shimura and Drinfeld towers. A central tool is the Colmez–Dospinescu–Niziol factorization of the Drinfeld tower’s cohomology, which isolates the Galois-isotypic components and expresses them in terms of and related modules. The results show that ordinary local Langlands representations do not occur in the finite-level Shimura or Drinfeld cohomology, and they establish vanishing (and in some cases finiteness) results for Ext groups in the local setting, using a combination of p-adic and mod Langlands compatibility, block theory for -adic Banach representations, and cohomological uniformization techniques. These findings advance the understanding of where -adic Langlands representations can appear in arithmetic geometry, and they connect local representation theory with global automorphic objects via robust factorization theorems. The work thus strengthens the philosophy that ordinary -adic Langlands representations tend not to appear in the “supersingular” parts of cohomology and clarifies their role in the global–local panorama.

Abstract

We can associate an admissible unitary representation of with every local Galois representation by the -adic local Langlands correspondence. If is ordinary, we prove local and global vanishing results for functors with respect to these representations.
Paper Structure (9 sections, 2 theorems, 36 equations)

This paper contains 9 sections, 2 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Shimura curves
  3. Drinfeld tower for $\mathbb{Q}_p$:
  4. $p$-adic and mod $p$ local Langlands for ${\mathrm{GL}}_2(\mathbb{Q}_p)$
  5. The Banach space $\Pi(\rho_{f,p})$
  6. Compatibility of $p$-adic and mod $p$ local Langlands
  7. Factorization theorem
  8. Local Galois representations and cohomologies of Shimura curves
  9. Local Vanishing Results of $\mathrm{Ext}$

Key Result

Theorem 1.1

Let $\rho: G_{\mathbb{Q}} \rightarrow {\mathrm{GL}}_2(E)$ be a pro-modular Galois representation with the corresponding local representation $\rho_p \simeq \otimes \eta$ with $\star \neq 0$, $\eta_1, \eta_2 :\mathbb{Q}_p^{\times} \rightarrow \mathcal{O}_E^{\times}$ integral characters and $\eta:G_{

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 3.2
  • proof
  • proof : Proof of Global Vanishing in Theorem \ref{['Mainthmordinary']}
  • proof
  • proof
  • proof : Proof of Local Vanishing in Theorem \ref{['Mainthmordinary']}
  • Remark 5.3
  • Remark 5.4