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

Technical Summary: Extensions of p-adic Representations of GL₂(Qₚ)

Problem Statement
The paper addresses the computation of extension groups within the category of duals of finite-length admissible p-adic Banach space representations of $G = \text{GL}_2(\mathbb{Q}_p)$. While the p-adic local Langlands correspondence (LLC) establishes a bijection between isomorphism classes of certain Galois representations and Banach space representations, a deeper structural understanding requires analyzing the derived category, specifically the extension groups ($\text{Ext}^i$) between these objects. These extensions are crucial for understanding the compatibility between the deformation theory of Galois representations and that of $G$-representations.

Prior to this work, significant progress had been made in the characteristic $p$ setting (mod $p$ representations) by Emerton, Breuil, Paškunas, and Colmez. However, the literature on extension groups in characteristic 0 (specifically for p-adic Banach space representations) remained sparse. Existing results covered specific cases, such as extensions between unitary continuous and smooth mod $p$ principal series (Hauseux) or between two supercuspidal representations (Paškunas). A gap existed in computing extensions where one representation is the dual of a non-ordinary representation arising from the p-adic LLC.

Methodology
The authors employ the framework of the p-adic local Langlands correspondence as developed by Colmez and Paškunas, utilizing the "Montréal functor" $V$ and its dual variants.

1. Categorical Framework: The study focuses on the category $\text{Ban}^{\text{adm}}_{G,\zeta}(E)$ of unitary admissible $E$-Banach space representations with a central character $\zeta$. Since this category lacks enough injectives, the authors utilize Yoneda's construction to define extension groups $\text{Ext}^i_{G,\zeta}$.
2. Dualities and Functors: The analysis relies heavily on the relationship between a representation $\Pi$ and its continuous dual $\Pi^*$. The authors use the Schikhof dual and Pontryagin dual to relate Banach space representations to modules over the completed group algebra. A key tool is the functor $\check{V}$, which maps dual Banach representations back to Galois representations, satisfying $\check{V}(\Pi^*) \cong V(\Pi)^*(\varepsilon\zeta)$.
3. Block Decomposition: Following Paškunas, the category is decomposed into blocks based on the mod $p$ reduction of the representations. The authors analyze the action of the universal pseudocharacter deformation ring $R^{\text{ps},\psi}_{\text{tr}\rho_B}$ on these blocks.
4. Reduction to Commutative Algebra: For absolutely irreducible representations within a specific block, the endomorphism ring is shown to be the field $E$. The extension groups are then reduced to computing extensions in the category of modules over a complete local ring $A$, which is identified as the completion of the pseudocharacter deformation ring at a maximal ideal.

Key Contributions and Results

• Classification of Extensions (Theorem 1.1): The paper provides a complete classification of extension groups $\text{Ext}^i_{G,\zeta}(\pi^*, \Pi(V)^*)$ under a genericity condition on the reduction of the Galois representation $V$.

  • Vanishing Case: If $\pi$ is an absolutely irreducible admissible unitary Banach representation such that its associated Galois representation $V(\pi)$ is not isomorphic to $V$, then $\text{Ext}^i_{G,\zeta}(\pi^*, \Pi(V)^*) = 0$ for all $i \geq 0$.
  • Non-Vanishing Case: If $V(\pi) \cong V$, the extension groups are non-zero only for $0 \leq i \leq 3$. Specifically, $\text{Ext}^i_{G,\zeta}(\pi^*, \Pi(V)^*) \cong E^{\binom{3}{i}}$. This result relies on the fact that the relevant deformation ring $A$ is a complete regular local ring of dimension 3, allowing the use of the Koszul complex to compute the Ext groups.

• Application to Drinfeld Spaces (Corollary 1.2): The authors apply these vanishing results to the geometric context of the $p$-adic étale cohomology of finite-level Drinfeld spaces.

  • They consider the $V$-isotypic component of the cohomology $H^1_{\text{ét}}(M_\infty, E(1))$, where $V$ is a supercuspidal Galois representation.
  • Using the isomorphism established by Colmez, Dospinescu, and Nizioł, which relates this cohomology to copies of $\Pi(V)^*$, the authors prove that there are no extensions between this cohomology component and the dual of the representation attached to a reducible Galois representation $\rho_p$ (provided $\rho_p$ satisfies specific Hodge-Tate and genericity conditions).

Significance and Claims
The paper claims novelty in computing extensions in the category of duals of finite-length admissible p-adic Banach space representations, specifically when one of the representations is the dual of a non-ordinary representation. The authors state that these results are applied to prove the vanishing of extensions between dual p-adic Banach space representations attached to reducible Galois representations and supercuspidal isotypic components of the $p$-adic étale cohomology of Drinfeld spaces.

The work is presented as a "short note" that fills a specific gap in the characteristic 0 literature, complementing the extensive work done in the mod $p$ setting. The authors emphasize that the results rely on the genericity of the Galois representation's reduction and the specific structure of the deformation rings associated with the blocks of the category. The paper does not propose new experimental applications but rather solidifies the theoretical link between the deformation theory of Galois representations and the representation theory of $p$-adic groups in the context of the local Langlands program.