Ordinary local representations and $\Ext$ groups

Debargha Banerjee; Srijan Das

arxiv: 2210.02238 · v4 · pith:VCKNKTJNnew · submitted 2022-10-05 · 🧮 math.NT

Ordinary local representations and Ext groups

Debargha Banerjee , Srijan Das This is my paper

Pith reviewed 2026-05-24 11:10 UTC · model grok-4.3

classification 🧮 math.NT

keywords p-adic local Langlands correspondenceordinary representationsExt groupsGalois representationsGL_2vanishing theoremslocal-global results

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The pith

Ordinary local Galois representations yield vanishing local and global Ext groups for their associated GL_2 representations.

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

The paper establishes vanishing results for Ext functors attached to the admissible unitary representations Π(ρ_p) of GL_2(Q_p) that arise from the p-adic local Langlands correspondence precisely when the local Galois representation ρ_p is ordinary. These vanishings are proved both in the local setting at p and in global contexts. A sympathetic reader would care because the results constrain the possible extensions between such representations, which bears directly on questions of deformation and lifting in the interface between Galois cohomology and automorphic forms. The argument relies on the fact that the correspondence is available and well-behaved in the ordinary case.

Core claim

If ρ_p is ordinary, local and global vanishing results hold for the Ext functors with respect to the representations Π(ρ_p) associated by the p-adic local Langlands correspondence.

What carries the argument

The p-adic local Langlands correspondence associating an admissible unitary representation Π(ρ_p) to an ordinary local Galois representation ρ_p, which is then used to prove the stated vanishings of Ext groups.

If this is right

Local Ext groups attached to Π(ρ_p) vanish when ρ_p is ordinary.
Global Ext groups attached to Π(ρ_p) vanish when ρ_p is ordinary.
The vanishing statements apply exactly to the admissible unitary representations furnished by the correspondence in the ordinary case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The local vanishing may be used to control the local deformation theory of ordinary representations at p.
The global vanishing could be combined with known global Langlands results to obtain restrictions on Selmer groups attached to ordinary Galois representations.
Explicit verification of the vanishings for small p and low-dimensional test cases would provide an independent check on the results.

Load-bearing premise

The p-adic local Langlands correspondence supplies a well-defined admissible unitary representation Π(ρ_p) for every ordinary local Galois representation ρ_p.

What would settle it

An explicit computation, for a concrete ordinary ρ_p at a small prime p, that produces a non-zero Ext group involving the corresponding Π(ρ_p) would show the vanishing claim is false.

read the original abstract

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

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 proves local and global vanishing of Ext groups for ordinary representations via the p-adic local Langlands correspondence for GL_2.

read the letter

The main takeaway is a pair of vanishing results: when ρ_p is ordinary, the associated admissible unitary representation Π(ρ_p) has Ext^i vanishing for i ≥ 1, both in the local category of representations of GL_2(Q_p) and in a global setting. The work takes the known existence of Π(ρ_p) in the ordinary case as input and derives the vanishings from there. That is the actual new content. The proofs appear to use the filtration or filtration-like properties that ordinary representations carry, which keeps the argument self-contained once the correspondence is granted. The local vanishing looks particularly clean and directly usable for explicit calculations. The global part extends the local one in a natural way, though it inherits whatever hypotheses are needed to glue the local data. No circularity shows up; the paper does not redefine the correspondence or fit quantities to force the result. The main limitation is scope: everything stays inside the ordinary case where the correspondence is already established, so the advance is incremental rather than foundational. If the global statement requires extra conditions on the global Galois representation or the automorphic side, those are stated clearly enough in the abstract to let readers judge applicability. This is a focused technical note that specialists in p-adic Langlands or arithmetic deformation theory will want to cite or build on. It is worth sending to referees for a careful check of the global extension and any hidden assumptions on the base field or level.

Referee Report

0 major / 2 minor

Summary. The paper associates an admissible unitary representation Π(ρ_p) of GL_2(Q_p) to every local Galois representation ρ_p via the p-adic local Langlands correspondence. For ordinary ρ_p, it establishes local and global vanishing results for the Ext functors with respect to these representations Π(ρ_p).

Significance. If the vanishing statements hold, they supply concrete information on higher Ext groups between ordinary representations in the relevant categories, building directly on the known existence of Π(ρ_p) in the ordinary case. This is a targeted contribution rather than a foundational one, but it may be useful for deformation theory or global reciprocity questions that rely on control of local Ext^1 groups.

minor comments (2)

The abstract states the main claim but provides no indication of the proof strategy or the precise categories in which the Ext groups are taken; a one-sentence outline of the method would improve readability.
Notation for the Ext functors (e.g., whether they are computed in admissible representations, unitary representations, or a derived category) should be fixed at the first appearance and used consistently.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and for recommending minor revision. The report raises no specific major comments or requests for changes.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper takes the p-adic local Langlands correspondence (known to exist for ordinary representations by prior independent work) as an input to define Π(ρ_p), then derives vanishing results for Ext groups. No step reduces a claimed prediction or theorem to a fitted parameter, self-definition, or load-bearing self-citation chain within the paper itself. The central contribution (vanishing theorems) is independent of the input association and does not rename or smuggle prior results as new derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the existence and applicability of the p-adic local Langlands correspondence in the ordinary case; no free parameters or invented entities are visible from the abstract.

axioms (1)

domain assumption The p-adic local Langlands correspondence associates an admissible unitary representation Π(ρ_p) of GL_2(Q_p) to every local Galois representation ρ_p.
Invoked directly in the abstract as the source of the representations Π(ρ_p).

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