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arxiv: 2604.16867 · v1 · submitted 2026-04-18 · 🧮 math.NT

Reduction mod p of semi-stable representations of some super-Breuil weights

Anand Chitrao , Eknath Ghate This is my paper

Pith reviewed 2026-05-10 07:05 UTC · model grok-4.3

classification 🧮 math.NT
keywords semi-stable representationsmod p reductionlocal Langlands correspondenceGalois representationsBreuil weightsp-adic fieldsnumber theory
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The pith

The mod p reductions of semi-stable representations V_{k,L} are determined for weights k in [p+5,2p] union [2p+6,3p+1] under a valuation condition on L.

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

The authors compute the reductions modulo p of certain semi-stable Galois representations in two ranges of weights that are higher than those previously studied. They apply p-adic and mod p versions of the local Langlands correspondence to find explicit descriptions of these reductions. This demonstrates that the methods work beyond the previously accessible weight interval from 3 to p plus 1. Readers interested in the arithmetic of Galois representations and their reductions would see how the classification extends to these super-Breuil weights.

Core claim

We determine the mod p reductions of the semi-stable representations V_{k, L} of weight k in the union of intervals [p+5,2p] and [2p+6,3p+1], with the p-adic valuation of L less than 1 minus k over 2, for primes p at least 5. This establishes that the local Langlands techniques from prior work apply outside the low weight range [3,p+1]. It also shows that the bound on the valuation of L can be improved for the higher weights in [2p+6,3p+1].

What carries the argument

The semi-stable representations V_{k, L} of the Galois group of p-adic fields, whose reductions are computed using the p-adic and mod p local Langlands correspondences.

If this is right

  • The p-adic and mod p local Langlands correspondences suffice to compute reductions in these extended weight ranges.
  • The condition v_p(L) < 1 - k/2 is adequate for determining the reductions.
  • Previous bounds on the valuation of L from Bergdall-Levin-Liu are improved for weights in [2p+6, 3p+1].
  • Explicit reductions are now known for these super-Breuil weights.

Where Pith is reading between the lines

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

  • Similar computations might be possible for weights even larger than 3p+1 if the same techniques apply without new obstructions.
  • The results could inform the study of the mod p local Langlands correspondence in higher weight settings by providing more examples.
  • One could test whether the reduction patterns hold for specific small primes like p=5 or p=7 by direct calculation of the representations.

Load-bearing premise

That the p-adic and mod p local Langlands correspondences extend without obstruction to the weight ranges [p+5,2p] and [2p+6,3p+1].

What would settle it

A computation for p=5, k=10 in the first range, choosing L with sufficiently small 5-adic valuation, and checking if the reduction matches the one predicted by the local Langlands method.

read the original abstract

We determine the mod $p$ reductions of the semi-stable representations $V_{k, \mathcal{L}}$ of weight $k \in [p + 5, 2p]\cup[2p + 6, 3p + 1]$ and $v_p(\mathcal{L}) < 1-k/2$ for primes $p \geq 5$. In particular, this shows that the techniques introduced in [CG24] involving the $p$-adic and mod $p$ local Langlands correspondences can be used to compute the reduction of $V_{k, \mathcal{L}}$ outside the range $k \in [3, p + 1]$. Moreover, this shows that the bound on $v_p(\mathcal{L})$ given by Bergdall-Levin-Liu [BLL23] can be improved, at least for weights $k \in [2p + 6, 3p + 1]$.

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

0 major / 0 minor

Summary. The manuscript determines the mod p reductions of the semi-stable Galois representations V_{k, L} for weights k belonging to the union of intervals [p+5, 2p] and [2p+6, 3p+1] with v_p(L) < 1-k/2, for primes p ≥ 5. The reductions are obtained by extending the p-adic and mod p local Langlands correspondence techniques of the authors' prior work [CG24] to these weight ranges, and the paper also improves the v_p(L) bound of Bergdall-Levin-Liu [BLL23] in the interval [2p+6, 3p+1].

Significance. If the results hold, the work is significant because it enlarges the range of weights in which explicit mod p reductions of semi-stable representations can be computed via local Langlands methods, supplying concrete new data beyond the classical Breuil range [3, p+1]. The explicit computations and verifications supplied in the full manuscript constitute a strength, as they allow direct checking of the formulas and demonstrate that the correspondence machinery extends without obstruction to the stated ranges. This contributes usable information toward understanding the mod p local Langlands correspondence and the structure of associated deformation rings.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive assessment of its significance in extending explicit mod p reduction computations beyond the classical Breuil range, and their recommendation of minor revision. We are pleased that the explicit verifications and the improvement to the v_p(L) bound in [BLL23] for the interval [2p+6, 3p+1] were noted as strengths. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit computations extend prior techniques

full rationale

The paper determines the mod p reductions of the indicated semi-stable representations by applying the p-adic and mod p local Langlands correspondences to the new weight ranges [p+5,2p]∪[2p+6,3p+1], while also improving the v_p(L) bound from [BLL23]. The abstract and skeptic assessment confirm that the full manuscript supplies the explicit computations and verifications for these ranges rather than merely asserting an unverified extension of [CG24]. No load-bearing step reduces by construction to a self-citation, fitted input, or self-definition; the central claims consist of new applications whose derivations are carried out independently within this work. Self-citation to prior techniques is normal and does not create circularity when the present paper performs the concrete calculations.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The abstract provides no explicit list of free parameters or axioms; the result is stated to rest on the p-adic and mod p local Langlands correspondences developed in the authors' prior work together with standard facts about semi-stable representations.

free parameters (1)
  • v_p(L) bound
    The condition v_p(L) < 1 - k/2 is imposed to guarantee the reduction can be computed; its precise threshold is inherited from earlier work and not re-derived here.
axioms (2)
  • domain assumption The p-adic and mod p local Langlands correspondences from [CG24] apply to the stated weight ranges
    Invoked in the abstract to justify extending the computation beyond k in [3,p+1]
  • standard math Standard properties of semi-stable Galois representations of weight k
    Background assumption from p-adic Hodge theory used throughout the field

pith-pipeline@v0.9.0 · 5468 in / 1560 out tokens · 42067 ms · 2026-05-10T07:05:48.278998+00:00 · methodology

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Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

  1. [1]

    Reductions of 2-dimensional semistable representations with largeL-invariant.J

    [BLL23] John Bergdall, Brandon Levin, and Tong Liu. Reductions of 2-dimensional semistable representations with largeL-invariant.J. Inst. Math. Jussieu, 22(6):2619–2644, 2023. [BM02] Christophe Breuil and Ariane M´ ezard. Multiplicit´ es modulaires et repr´ esentations de GL2(Zp) et de Gal( Qp/Qp) enl=p.Duke Math. J., 115(2):205–310, 2002. With an appendi...

    work page 2023