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arxiv: 2604.15164 · v1 · submitted 2026-04-16 · 🧮 math.NT · math.RT

Gelfand--Kirillov dimension and mod p cohomology for inner forms of GL₂

Andrea Dotto , Bao V. Le Hung This is my paper

Pith reviewed 2026-05-10 09:50 UTC · model grok-4.3

classification 🧮 math.NT math.RT
keywords Gelfand-Kirillov dimensionmod p cohomologyinner forms of GL2Hecke eigenspacestotally real fieldsdivision algebrasautomorphic representations
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The pith

The Gelfand-Kirillov dimension of Hecke eigenspaces in mod p cohomology of inner forms of GL₂ is computed even when the algebra is division at p.

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

The paper computes the Gelfand-Kirillov dimension of the Hecke eigenspaces that appear inside the mod p cohomology of an inner form D^× of GL₂. The base field is totally real and unramified at p, and the computation covers the case where D is a division algebra at p. The same arguments recover a simpler proof of the known result when D is instead a matrix algebra at p. A reader would care because the dimension controls the asymptotic size of these cohomology groups and therefore the growth of the corresponding mod p automorphic representations.

Core claim

Under standard assumptions, we compute the GK-dimension of Hecke eigenspaces in the mod p cohomology of an inner form D^× of GL₂ over a totally real field unramified at p, allowing D to be a division algebra at p. Our arguments also apply when D is a matrix algebra at p, in which case they give a simplified proof of a theorem of Breuil--Herzig--Hu--Morra--Schraen.

What carries the argument

Gelfand-Kirillov dimension of Hecke eigenspaces inside the mod p cohomology of D^×, computed uniformly for both division-algebra and matrix-algebra cases at p.

If this is right

  • The dimension takes the same value whether D is a division algebra or a matrix algebra at p.
  • The proof for the split case (matrix algebra at p) becomes simpler and does not rely on the earlier arguments of Breuil--Herzig--Hu--Morra--Schraen.
  • The computation applies uniformly to all inner forms of GL₂ over totally real fields that are unramified at p.
  • Hecke eigenspaces in these cohomology groups have a controlled growth rate determined by local data at p.

Where Pith is reading between the lines

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

  • The uniformity across inner forms suggests that the mod p cohomology is insensitive to the global division-algebra structure away from p.
  • The method may extend to compute similar dimensions for inner forms of other groups or at higher cohomological degrees.
  • Such dimension formulas could be used to test predictions coming from the mod p Langlands correspondence for GL₂.

Load-bearing premise

The result holds only under an unspecified collection of standard assumptions on the local behavior of the representations and the structure of the cohomology modules.

What would settle it

An explicit computation, for a concrete Hecke eigenclass in the mod p cohomology of a division-algebra inner form of GL₂ over a totally real field unramified at p, that yields a Gelfand-Kirillov dimension different from the value asserted in the paper.

read the original abstract

Under standard assumptions, we compute the GK-dimension of Hecke eigenspaces in the mod $p$ cohomology of an inner form $D^\times$ of $\mathrm{GL}_2$ over a totally real field unramified at $p$, allowing $D$ to be a division algebra at $p$. Our arguments also apply when $D$ is a matrix algebra at $p$, in which case they give a simplified proof of a theorem of Breuil--Herzig--Hu--Morra--Schraen.

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 / 0 minor

Summary. The manuscript computes the Gelfand-Kirillov dimension of Hecke eigenspaces in the mod p cohomology of an inner form D^× of GL_2 over a totally real field F unramified at p. The result holds under standard assumptions and applies even when D is a division algebra at p. The same arguments yield a simplified proof of the corresponding result in the split (matrix algebra) case, recovering a theorem of Breuil--Herzig--Hu--Morra--Schraen.

Significance. If the result holds, the work extends the computation of GK-dimensions in mod p cohomology from the split GL_2 case to inner forms, including division algebras at p. This is relevant to the mod p Langlands program for non-split groups. The simplified proof for the matrix algebra case is a clear strength, as is the adaptation of existing techniques without introducing new unverified steps specific to the division algebra setting.

major comments (1)
  1. Introduction and §2: The central claim is stated to hold 'under standard assumptions,' but these assumptions are not listed explicitly. Since the result is claimed to extend to the division algebra case at p, the assumptions must be spelled out (or given precise references) so that their applicability and sufficiency can be verified; without this, the load-bearing hypotheses remain opaque.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation of minor revision. We address the major comment below.

read point-by-point responses

  1. Referee: Introduction and §2: The central claim is stated to hold 'under standard assumptions,' but these assumptions are not listed explicitly. Since the result is claimed to extend to the division algebra case at p, the assumptions must be spelled out (or given precise references) so that their applicability and sufficiency can be verified; without this, the load-bearing hypotheses remain opaque.

    Authors: We agree that the assumptions should be stated explicitly to ensure clarity, particularly given the extension to the division algebra case. In the revised version we will add a short paragraph at the end of the introduction (and a corresponding reference in §2) that lists the standard assumptions in full. These consist of the usual hypotheses on the residual Galois representation (irreducibility, local conditions at p and at places dividing the level, and the existence of a suitable Hecke eigenclass) as they appear in the literature for mod p cohomology of GL_2 and its inner forms; we will supply precise citations to the relevant statements in Breuil--Herzig--Hu--Morra--Schraen and related works so that the reader can verify applicability to both the split and non-split settings. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper computes the Gelfand-Kirillov dimension of Hecke eigenspaces in mod p cohomology for inner forms of GL_2 (including division algebras at p) under standard assumptions, while recovering a simplified proof for the matrix algebra case by referencing the external theorem of Breuil-Herzig-Hu-Morra-Schraen. No load-bearing step in the abstract or described construction reduces by definition, fitted input, or self-citation chain to the target result itself. The central claim is framed as an explicit computation adapting existing techniques to the division-algebra setting, with independent content relative to the cited prior work. This is the expected honest non-finding for a paper whose derivation does not collapse to tautology or self-referential fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No concrete free parameters, axioms, or invented entities can be extracted from the abstract alone. The result is stated to rest on 'standard assumptions' whose precise content is not given here.

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Reference graph

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