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A note on extensions of $p$-adic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

Debargha Banerjee, Srijan Das

TL;DR

This work computes Ext$^i$ groups in the category of duals of $p$-adic Banach space representations of GL$_2$(Q$_p$), focusing on those arising from the $p$-adic local Langlands correspondence for generic 2-dimensional Galois representations. It achieves a complete Ext classification under a genericity hypothesis and uses Yoneda Ext to overcome the lack of injectives in the Banach setting; a key outcome is the vanishing of Ext between duals of reducible Galois representations and the $V$-isotypic Drinfeld cohomology. These Ext calculations feed into arithmetic geometry via the Drinfeld space’s $p$-adic étale cohomology, showing that no nontrivial extensions occur between certain dual Langlands components and the $V$-isotypic part of cohomology. The results illuminate the compatibility between deformation-theoretic centers (via two-dimensional pseudocharacters) and the block decomposition of Banach representations, linking $p$-adic LLC, Colmez’s Montréal functor, and the geometry of Drinfeld spaces.

Abstract

We compute extension groups in the category of duals of $p$-adic Banach space representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Focusing on representations arising from the $p$-adic local Langlands correspondence for generic Galois representations, we classify these extensions completely. These results are then applied to prove the vanishing of extensions between the duals of reducible representations and supercuspidal isotypic components of the ètale cohomology of the finite level Drinfeld spaces.

A note on extensions of $p$-adic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$

TL;DR

This work computes Ext groups in the category of duals of -adic Banach space representations of GL(Q), focusing on those arising from the -adic local Langlands correspondence for generic 2-dimensional Galois representations. It achieves a complete Ext classification under a genericity hypothesis and uses Yoneda Ext to overcome the lack of injectives in the Banach setting; a key outcome is the vanishing of Ext between duals of reducible Galois representations and the -isotypic Drinfeld cohomology. These Ext calculations feed into arithmetic geometry via the Drinfeld space’s -adic étale cohomology, showing that no nontrivial extensions occur between certain dual Langlands components and the -isotypic part of cohomology. The results illuminate the compatibility between deformation-theoretic centers (via two-dimensional pseudocharacters) and the block decomposition of Banach representations, linking -adic LLC, Colmez’s Montréal functor, and the geometry of Drinfeld spaces.

Abstract

We compute extension groups in the category of duals of -adic Banach space representations of . Focusing on representations arising from the -adic local Langlands correspondence for generic Galois representations, we classify these extensions completely. These results are then applied to prove the vanishing of extensions between the duals of reducible representations and supercuspidal isotypic components of the ètale cohomology of the finite level Drinfeld spaces.
Paper Structure (6 sections, 5 theorems, 19 equations)

This paper contains 6 sections, 5 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. p-adic and mod p local Langlands for GL2(Qp)
  3. Blocks and semisimple mod $p$ LLC
  4. Vanishing Results for Ext^. Groups
  5. Two dimensional pseudocharacters
  6. Rational Cohomology of Drinfeld Space

Key Result

Theorem 1.1

Let $V$ be an absolutely irreducible two-dimensional representation of $\mathrm{G}_{\mathbb{Q}_p}$ over $E$ such that its reduction $\overline{V}$ is generic. Let $\pi$ be an absolutely irreducible unitary $p$-adic Banach space representation of $\mathrm{G}$. If $\mathbf{V}(\pi) \neq V$, then where $\mathbf{V}$ denotes Colmez's Montréal functor. Suppose that $\mathbf{V}(\pi)=V$. Then for every in

Theorems & Definitions (9)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['main thm']}
  • Remark 3.2
  • Theorem 4.1
  • proof : Proof of Corollary \ref{['corollary ordinary']}