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arxiv: 2605.17836 · v1 · pith:MUIOH6WFnew · submitted 2026-05-18 · 🧮 math.NT · math.RT

Non-admissibility of some universal supersingular representations

Zachary Feng , Heejong Lee , Ray Li , Vaughan McDonald , Nischay Reddy This is my paper

Pith reviewed 2026-05-20 01:14 UTC · model grok-4.3

classification 🧮 math.NT math.RT
keywords universal supersingular representationsnon-admissibilityGL_n representationsSerre weight conjecturesweight cyclingmod p automorphic formsinfinite length representations
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The pith

For n greater than or equal to 3, universal supersingular representations of sufficiently generic weight are non-admissible and have infinite length.

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

This paper establishes that for n at least 3, the universal supersingular representation attached to a weight sigma of GL_n over the residue field is non-admissible and has infinite length under the assumption that sigma is sufficiently generic and meets additional technical conditions. A sympathetic reader would care because this result extends the n equals 2 case to higher dimensions, revealing that these representations are typically much more complicated than finite length modules in the context of p-adic representation theory. The authors achieve this by combining a weight cycling argument with recent advances on the Serre weight conjectures. If the claim is correct, it means that for most weights in higher rank groups, the corresponding supersingular representations do not admit finite length quotients that are admissible.

Core claim

Let K/Q_p be an unramified extension of degree f with residue field k. Let sigma be an irreducible representation of GL_n(k) over the algebraic closure of F_p. For n greater than or equal to 3, we prove that the universal supersingular representation of weight sigma is non-admissible and of infinite length when sigma is sufficiently generic and satisfies certain technical conditions. This generalizes the previous results for n=2 and a non-trivial finite extension K/Q_p. Our method employs a weight cycling argument together with recent progress on the Serre weight conjectures.

What carries the argument

Weight cycling argument combined with recent progress on the Serre weight conjectures to establish non-admissibility and infinite length.

If this is right

  • The universal supersingular representation is non-admissible for n >= 3 under the given conditions on sigma.
  • It has infinite length as a representation.
  • The result applies to any unramified extension of degree f.
  • This provides a generalization from the n=2 case to higher n.

Where Pith is reading between the lines

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

  • This suggests that in higher dimensions the supersingular representations tend to be more complex and infinite in length for generic weights.
  • Similar techniques might apply to other types of representations beyond the supersingular ones.
  • Explicit computations for small values of n and p could provide further evidence for the infinite length property.

Load-bearing premise

The assumption that the weight sigma is sufficiently generic and satisfies certain technical conditions, together with the applicability of recent progress on the Serre weight conjectures.

What would settle it

An explicit construction of an admissible finite length quotient for the universal supersingular representation in the case n=3 with a generic sigma would falsify the main claim.

read the original abstract

Let $K/\mathbf{Q}_p$ be an unramified extension of degree $f$ with residue field $k$. Let $\sigma$ be an irreducible representation of $\mathrm{GL}_n(k)$ over $\overline{\mathbf{F}}_p$. For $n\ge 3$, we prove that the universal supersingular representation of weight $\sigma$ is non-admissible and of infinite length when $\sigma$ is sufficiently generic and satisfies certain technical conditions. This generalizes the previous results for $n=2$ and a non-trivial finite extension $K/\mathbf{Q}_p$. Our method employs a weight cycling argument together with recent progress on the Serre weight conjectures.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript proves that for n ≥ 3, the universal supersingular representation of weight σ (an irreducible representation of GL_n(k) over F_p-bar) attached to an unramified extension K/Q_p of degree f is non-admissible and of infinite length, provided σ is sufficiently generic and satisfies certain technical conditions. The proof proceeds via a weight cycling argument that produces an infinite ascending chain of distinct supersingular representations, combined with recent progress on the Serre weight conjectures for the associated mod p Galois representations; this generalizes the n=2 case.

Significance. If the central claim holds, the result advances the mod p Langlands program by establishing structural properties of supersingular representations in higher rank, with potential implications for admissibility questions and the classification of irreducible representations. The manuscript is credited for successfully generalizing the n=2 results and for integrating weight cycling with recent Serre weight advances to obtain the infinite-length conclusion under the stated genericity hypotheses.

major comments (1)
  1. [§4 (Weight Cycling Argument)] §4 (Weight Cycling Argument): The cycling construction produces an infinite chain only if the invoked Serre weight results supply strictly new weights for generic irreducible σ of GL_n(k) when n≥3. The manuscript must explicitly verify that the genericity and technical conditions on σ match the hypotheses of the cited Serre weight theorems (to exclude periodic orbits or finite subchains) and confirm that the associated Galois representations remain in the required range; without this check the non-admissibility and infinite-length claims do not follow.
minor comments (2)
  1. [Introduction] The precise list of technical conditions on σ is referenced but not restated in the introduction; adding a short enumerated list would improve readability.
  2. [§2] Notation for the residue field k and the extension degree f is introduced early but used inconsistently in some later statements; a uniform convention would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the weight cycling argument. We address the point below and will revise the paper to incorporate the requested explicit verification.

read point-by-point responses

  1. Referee: [§4 (Weight Cycling Argument)] §4 (Weight Cycling Argument): The cycling construction produces an infinite chain only if the invoked Serre weight results supply strictly new weights for generic irreducible σ of GL_n(k) when n≥3. The manuscript must explicitly verify that the genericity and technical conditions on σ match the hypotheses of the cited Serre weight theorems (to exclude periodic orbits or finite subchains) and confirm that the associated Galois representations remain in the required range; without this check the non-admissibility and infinite-length claims do not follow.

    Authors: We appreciate the referee's observation that an explicit verification is needed to guarantee that the weight cycling produces an infinite ascending chain of distinct supersingular representations. While the manuscript states that σ is sufficiently generic and satisfies certain technical conditions, we agree that a direct comparison with the hypotheses of the cited Serre weight theorems is not spelled out in §4. In the revised version we will add a dedicated paragraph (or short subsection) in §4 that (i) recalls the precise genericity hypotheses from the relevant Serre weight results, (ii) confirms that our conditions on σ are at least as strong, and (iii) verifies that the associated mod p Galois representations lie in the range where those theorems guarantee strictly new weights at each cycling step. This will explicitly rule out periodic orbits or finite subchains and thereby complete the argument for non-admissibility and infinite length when n ≥ 3. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external Serre weight progress and weight cycling without self-referential reduction

full rationale

The paper proves non-admissibility and infinite length for universal supersingular representations when n≥3 and σ is sufficiently generic by combining a weight cycling argument with recent progress on the Serre weight conjectures. No equations, definitions, or fitted parameters within the paper reduce to each other by construction, and the central claim is supported by cited external results rather than self-citations or internal fits. The derivation remains self-contained against external benchmarks, with the genericity and technical conditions serving as stated hypotheses rather than outputs derived from the conclusion itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard facts about irreducible representations of GL_n(k) and on external progress on Serre weight conjectures; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • standard math Irreducible representations of GL_n(k) over an algebraically closed field of characteristic p behave according to standard representation theory.
    Invoked when defining the weight sigma and the universal supersingular representation.
  • domain assumption Recent progress on the Serre weight conjectures applies to the weights under consideration.
    The method explicitly employs this progress to run the weight cycling argument.

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