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

Debargha Banerjee; Srijan Das

arxiv: 2601.07707 · v2 · pith:2MHYIWL4new · submitted 2026-01-12 · 🧮 math.NT

A note on extensions of p-adic representations of GL₂(mathbb{Q}_p)

Debargha Banerjee , Srijan Das This is my paper

Pith reviewed 2026-05-21 15:55 UTC · model grok-4.3

classification 🧮 math.NT

keywords p-adic representationsGL_2(Q_p)Banach space representationslocal Langlands correspondenceextensionsDrinfeld spacesétale cohomologysupercuspidal representations

0 comments

The pith

Extensions between duals of p-adic Banach representations of GL_2(Q_p) are completely classified when the representations arise from generic Galois representations via the local Langlands correspondence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes extension groups in the category of duals of p-adic Banach space representations of GL_2(Q_p). By restricting attention to those representations that come from the p-adic local Langlands correspondence applied to generic Galois representations, the authors give a full classification of the possible extensions. They then use this classification to establish vanishing results for extensions between the duals of reducible representations and the supercuspidal isotypic components inside the étale cohomology of finite-level Drinfeld spaces. A reader would care because the work clarifies how extensions behave in the p-adic setting and links representation theory directly to cohomology computations that appear in the study of p-adic automorphic forms.

Core claim

We compute extension groups in the category of duals of p-adic Banach space representations of GL_2(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 étale cohomology of the finite level Drinfeld spaces.

What carries the argument

The extension groups in the category of duals of p-adic Banach space representations of GL_2(Q_p), classified via the p-adic local Langlands correspondence for generic Galois representations.

Load-bearing premise

The classification and the vanishing statements are proved only after restricting to representations that arise from the p-adic local Langlands correspondence for generic Galois representations.

What would settle it

An explicit computation, for a concrete generic Galois representation, of an extension class in the dual Banach representation category that lies outside the groups listed in the classification, or a non-vanishing extension in the Drinfeld-space cohomology that contradicts the claimed vanishing.

read the original 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 \`etale cohomology of the finite level Drinfeld spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper classifies extensions for duals of Banach representations tied to generic Galois data via p-adic local Langlands and applies this to a vanishing result involving reducible representations in Drinfeld space cohomology.

read the letter

This paper classifies the extension groups completely for duals of p-adic Banach representations of GL_2(Q_p) that come from the p-adic local Langlands correspondence applied to generic Galois representations. It then uses this to prove a vanishing result for extensions involving duals of reducible representations and supercuspidal components in the etale cohomology of Drinfeld spaces. What stands out is the explicit classification in this restricted setting. The authors focus on a class where the Galois side is generic, which likely makes the representation side more manageable for computing Hom and Ext spaces. This kind of concrete result can feed into broader questions about the structure of these categories and how they relate to geometric objects like Drinfeld spaces. The main soft spot is the transition to the vanishing claim. The classification covers the generic case, but the reducible representations in the application might sit outside that. The paper needs to include a clear argument showing why the classification still controls the extensions there, perhaps through some deformation or limit process. Without seeing the full details, it's hard to tell if that step is handled rigorously. The work is aimed at specialists who already know the p-adic Langlands correspondence and the cohomology of Drinfeld towers. A reader with that background can use the classification for further calculations in representation theory or cohomology. I think this deserves a serious referee. The claims are specific enough that experts can verify the computations and the applicability. Recommendation: Send it out for peer review, but flag the need to check the bridge between generic classification and reducible vanishing.

Referee Report

1 major / 0 minor

Summary. The manuscript computes extension groups in the category of duals of p-adic Banach space representations of GL_2(Q_p). Focusing on representations arising from the p-adic local Langlands correspondence for generic Galois representations, it classifies these extensions completely. The results are then applied to prove the vanishing of extensions between the duals of reducible representations and supercuspidal isotypic components of the étale cohomology of finite-level Drinfeld spaces.

Significance. If the classification is rigorous and the application to the vanishing result holds, the work would advance the p-adic Langlands program by providing explicit control over extension groups in the dual category. The complete classification in the generic case, together with its geometric application to Drinfeld-space cohomology, would strengthen connections between representation-theoretic and cohomological aspects of p-adic automorphic forms.

major comments (1)

[Abstract, paragraph 2] Abstract, paragraph 2: the classification of extensions is stated only for duals of representations arising from the p-adic LLC for generic Galois representations. The subsequent vanishing claim concerns extensions between duals of reducible representations and supercuspidal isotypic components. No explicit reduction, compatibility, or genericity check is indicated that would allow the classification to transfer to the reducible case; this bridge is load-bearing for the application and requires a concrete argument or clarification.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. The major comment identifies a need for greater clarity on the connection between the generic classification and the application to reducible representations. We address this point below and will revise the manuscript to make the reduction explicit.

read point-by-point responses

Referee: [Abstract, paragraph 2] Abstract, paragraph 2: the classification of extensions is stated only for duals of representations arising from the p-adic LLC for generic Galois representations. The subsequent vanishing claim concerns extensions between duals of reducible representations and supercuspidal isotypic components. No explicit reduction, compatibility, or genericity check is indicated that would allow the classification to transfer to the reducible case; this bridge is load-bearing for the application and requires a concrete argument or clarification.

Authors: We agree that the abstract does not explicitly outline the reduction step. In the main body of the manuscript we establish the vanishing for reducible duals by relating them to the generic case via the p-adic local Langlands correspondence and the structure of the étale cohomology of the Drinfeld spaces; the generic classification supplies the necessary vanishing of Hom and Ext groups that then implies the result for the reducible/supercuspidal setting. To address the referee's concern we will revise the abstract to mention this compatibility and add a short clarifying paragraph in the introduction that summarizes the reduction argument with a forward reference to the relevant section. revision: yes

Circularity Check

0 steps flagged

No circularity; classification and application steps remain independent in the given text.

full rationale

The abstract describes computing and classifying extension groups for representations arising from the p-adic local Langlands correspondence restricted to generic Galois representations, followed by an application to prove vanishing of certain extensions involving reducible representations and Drinfeld-space cohomology. No equations, self-citations, fitted parameters, or ansatzes are present in the provided text that would allow any claim to reduce by construction to its inputs. The derivation chain therefore exhibits no self-definitional, fitted-input, or self-citation-load-bearing patterns and is treated as self-contained mathematical work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work relies on the existence of the p-adic local Langlands correspondence for generic Galois representations and the setup of the dual category of Banach representations.

pith-pipeline@v0.9.0 · 5609 in / 1152 out tokens · 86552 ms · 2026-05-21T15:55:20.182903+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and initial Peano algebra unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1. Let V be an absolutely irreducible two-dimensional representation of G_Qp over E such that its reduction V is generic. ... Ext^i_G(π^*, Π(V)^*) ≅ E^(3 choose i) for 0≤i≤3
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Corollary 1.2 ... Ext^i_G(Π(ρ_p)^*, H^1_et(M_∞,E(1))[V]) = 0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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[1] [1]

Berger,Représentations modulaires de GL2(Qp)et représentations galoisiennes de dimension2, in L

L. Berger,Représentations modulaires de GL2(Qp)et représentations galoisiennes de dimension2, in L. Berger, C. Breuil, and P. Colmez, editors, Représentationsp-adiques de groupesp-adiques II : Représentations deGL2(Qp)et (φ,Γ)-modules, number 330 in Astérisque, 263–279, Société mathématique de France (2010). 1

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Breuil,Sur quelques représentations modulaires etp-adiques deGL2(Qp)

C. Breuil,Sur quelques représentations modulaires etp-adiques deGL2(Qp). I, Compositio Math.138(2003), no. 2, 165–188. 5

work page 2003

[3] [3]

———,Sur quelques représentations modulaires etp-adiques de GL2(Qp). II, J. Inst. Math. Jussieu2(2003), no. 1, 23–58. 5

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[4] [4]

Chenevier,The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings, in Automorphic forms and Galois representations

G. Chenevier,The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings, in Automorphic forms and Galois representations. Vol. 1, Vol. 414 ofLondon Math. Soc. Lecture Note Ser., 221–285, Cambridge Univ. Press, Cambridge (2014), ISBN 978-1-107-69192-6. 5

work page 2014

[5] [5]

Colmez,Représentations deGL 2(Qp)et(ϕ,Γ)-modules, Astérisque (2010), no

P. Colmez,Représentations deGL 2(Qp)et(ϕ,Γ)-modules, Astérisque (2010), no. 330, 281–509. 1, 3, 4

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[6] [6]

Colmez, G

P. Colmez, G. Dospinescu, and W. Nizioł,Cohomologiep-adique de la tour de Drinfeld: le cas de la dimension 1, J. Amer. Math. Soc.33(2020), no. 2, 311–362. 2, 6, 7

work page 2020

[7] [7]

Pi11(2023) Paper No

———,Factorisation de la cohomologie étalep-adique de la tour de Drinfeld, Forum Math. Pi11(2023) Paper No. e16, 62. 2, 7

work page 2023

[8] [8]

Colmez, G

P. Colmez, G. Dospinescu, and V. Pašk¯ unas,Thep-adic local Langlands correspondence forGL2(Qp), Camb. J. Math. 2(2014), no. 1, 1–47. 2

work page 2014

[9] [9]

———,Irreducible components of deformation spaces: wild 2-adic exercises, Int. Math. Res. Not. IMRN (2015), no. 14, 5333–5356. 3, 6

work page 2015

[10] [10]

V. G. Drinfed,Coverings ofp-adic symmetric domains, Funkcional. Anal. i Priložen.10(1976), no. 2, 29–40. 7

work page 1976

[11] [11]

Emerton,A local-global compatibility conjecture in thep-adic Langlands programme forGL2/Q, Pure Appl

M. Emerton,A local-global compatibility conjecture in thep-adic Langlands programme forGL2/Q, Pure Appl. Math. Q.2(2006), no. 2, Special Issue: In honor of John H. Coates. Part 2, 279–393. 4

work page 2006

[12] [12]

Definition and first properties, Astérisque (2010), no

———,Ordinary parts of admissible representations ofp-adic reductive groups I. Definition and first properties, Astérisque (2010), no. 331, 355–402. 2, 3

work page 2010

[13] [13]

Derived functors, Astérisque (2010), no

———,Ordinary parts of admissible representations ofp-adic reductive groups II. Derived functors, Astérisque (2010), no. 331, 403–459. 1

work page 2010

[14] [14]

Fontaine,Représentations p-adiques des corps locaux

J.-M. Fontaine,Représentations p-adiques des corps locaux. I, in The Grothendieck Festschrift, Vol. II, Vol. 87 of Progr. Math., 249–309, Birkhäuser Boston, Boston, MA (1990), ISBN 0-8176-3428-2. 1, 3

work page 1990

[15] [15]

Périodesp-adiques (Bures-sur-Yvette, 1988)

———,Représentations l-adiques potentiellement semi-stables, in Périodesp-adiques (Bures-sur-Yvette, 1988), 223, 321–347 (1994). Périodesp-adiques (Bures-sur-Yvette, 1988). 7

work page 1988

[16] [16]

Gabriel,Des catégories abéliennes, Bull

P. Gabriel,Des catégories abéliennes, Bull. Soc. Math. France90(1962) 323–448. 4

work page 1962

[17] [17]

Hauseux,Extensions entre séries principalesp-adiques et modulop de G(F ), J

J. Hauseux,Extensions entre séries principalesp-adiques et modulop de G(F ), J. Inst. Math. Jussieu15(2016), no. 2, 225–270. 1

work page 2016

[18] [18]

———,Compléments sur les extensions entre séries principalesp-adiques et modulop de G(F ), Bull. Soc. Math. France145(2017), no. 1, 161–192. 1

work page 2017

[19] [19]

Pašk¯ unas,Extensions for supersingular representations ofGL2(Qp), Astérisque (2010), no

V. Pašk¯ unas,Extensions for supersingular representations ofGL2(Qp), Astérisque (2010), no. 331, 317–353. 1

work page 2010

[20] [20]

———,The image of Colmez’s Montreal functor, Publ. Math. Inst. Hautes Études Sci.118(2013) 1–191. 1, 2, 3, 4, 5, 6

work page 2013

[21] [21]

———,Blocks for modp representations ofGL2(Qp), in Automorphic forms and Galois representations. Vol. 2, Vol. 415 ofLondon Math. Soc. Lecture Note Ser., 231–247, Cambridge Univ. Press, Cambridge (2014), ISBN 978-1-107-69363-0. 4

work page 2014

[22] [22]

6, 1301–1358

———,On 2-dimensional 2-adic Galois representations of local and global fields, Algebra Number Theory10(2016), no. 6, 1301–1358. 1, 5, 6

work page 2016

[23] [23]

Pašk¯ unas and S.-N

V. Pašk¯ unas and S.-N. Tung,Finiteness properties of the category ofmodp representations ofGL2(Qp), Forum Math. Sigma9(2021) Paper No. e80, 39. 6

work page 2021

[24] [24]

C. A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1994). 6 INDIAN INSTITUTE OF SCIENCE EDUCATION AND RESEARCH, PUNE, INDIA 8

work page 1994