pith. sign in

arxiv: 2607.00930 · v1 · pith:QTMT7M77new · submitted 2026-07-01 · 🧮 math.NT

Reductions Of Crystalline Representations Of Fractional Slope <p-1

Shalini Bhattacharya , Eknath Ghate , Ravitheja Vangala This is my paper

Pith reviewed 2026-07-02 06:57 UTC · model grok-4.3

classification 🧮 math.NT
keywords crystalline representationsmod p reductionGalois representationslocal Langlands correspondencefractional slopessemi-simplificationirreducibility
0
0 comments X

The pith

The mod p semi-simplification of crystalline Galois representations with fractional slope less than p-1 has an explicit shape for large weights.

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

The paper examines two-dimensional crystalline representations V_{k,a_p} of Gal(Q_p) where the slope is the p-adic valuation of a_p. When this slope is a positive fraction below p-1, the authors apply the mod p local Langlands correspondence to determine the precise semi-simple form of the reduction ar{V}_{k,a_p} once the weight k is large enough relative to the slope. The description holds except when certain Jordan-Hölder factors of dimension p-1 appear or when the slope condition for bad weight classes modulo p fails. They further establish that the reduction is irreducible for all even weights under the stricter bound slope less than p-2.

Core claim

Using the mod p local Langlands correspondence, the semi-simplification ar{V}_{k,a_p} takes an explicit form determined by the fractional slope and the weight k for all sufficiently large k, provided Jordan-Hölder factors of dimension p-1 do not intervene when k is odd and the slope satisfies the stated bound relative to the congruence class of k-2 modulo p-1 for bad classes; this yields irreducibility of ar{V}_{k,a_p} for even k whenever the slope is less than p-2.

What carries the argument

The mod p local Langlands correspondence, which translates the crystalline representation data into explicit information about the semi-simplification of its mod p reduction.

If this is right

  • The semi-simplification is completely determined by the slope fraction and the weight once k exceeds a slope-dependent bound.
  • Irreducibility holds for every even weight when the slope is below p-2.
  • Additional criteria resolve the shape in some cases where Jordan-Hölder factors of dimension p-1 do appear.
  • The result applies uniformly across all positive fractional slopes less than p-1 under the listed restrictions.

Where Pith is reading between the lines

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

  • The explicit shape may allow direct comparison with reductions arising from global automorphic forms of the same weight and level.
  • The method could be tested on small primes by direct computation of filtered phi-modules to verify the predicted forms without the local Langlands step.
  • Removing the slope bound on bad congruence classes would require handling additional cases where the representative exceeds the slope.

Load-bearing premise

That the slope lies below the representative in [1,p-1] of the class of k-2 mod (p-1) for bad congruence classes of k mod p, and that no Jordan-Hölder factors of dimension p-1 intervene when k is odd.

What would settle it

Compute the reduction ar{V}_{k,a_p} explicitly for p=5, slope 1/3, and a large even weight k satisfying the slope bound, and check whether the result is irreducible as predicted.

Figures

Figures reproduced from arXiv: 2607.00930 by Eknath Ghate, Ravitheja Vangala, Shalini Bhattacharya.

Figure 5.1
Figure 5.1. Figure 5.1: Contribution of JH factors when a = 2i − 2n − 1 and r ≡ a − i + n mod p. The symbol × at position (t, T) indicates that, for this t, the image of indG KZ(V (i−T) r /V (i−T +1) r ) vanishes in Θ¯ k,ap . Thus for a given value of t, the sub-quotient that survives in Θ¯ k,ap is marked by ✓. In the above picture, we notice that ✓ always appears along the diagonal (t = T + 1) [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Contribution of JH factors when a < 2i − 2n − 1 and r ≡ a − i + n mod p. The two pictures above explain why Theorem 5.5 was in the run up to our proof of it called the “Diagonal Conjecture”. Remark 5.6. We give some numerical examples. • When p = 7, r = 65 and i = 4, so that a = 5, n = 1, s = 23 and t = 1, then V (4) r /V (5) r contributes, corroborating part (i) (a) of Theorem 5.5. • When p = 5, r = 116… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Here 2i − 2n − 1 < a < 2i and a is odd, and k = 2i − a − n [PITH_FULL_IMAGE:figures/full_fig_p112_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Here 2i − 2n − 1 < a < 2i and a is even, and k = 2i − a − n. We next consider the case 2i − n ≤ a < 2i. Theorem 5.35 (Hybrid Conjecture). Let r ≡ a mod (p − 1) with 1 ≤ a ≤ p − 1 and r ≥ i(p + 1) + p with v(ap) ∈ (i, i+ 1). Let s = a−i+n+ (i−n)p and v(r −s) = t with 1 ≤ n ≤ i−1 and t ≥ 1. Assume that i < a and 2i − n ≤ a < 2i. Then (i) indG KZ(V (a+t−i−1) r /V (a+t−i) r ) ↠ Θ¯ k,ap if 0 < t ≤ i − ⌊ a 2 ⌋… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Contribution of JH factors when a ≥ 2i and r ≡ a − i + n mod p. In the above figure, × inside a grid at position (t, T) indicates that, for a given value of t the image of indG KZ(V (i−T) r /V (i−T +1) r ) vanishes in Θ¯ k,ap . Thus for a given value of t, the sub-quotient that possibly survives in Θ¯ k,ap is marked by ✓. In the above picture, we notice that ✓ always appears along the diagonal (t = T + 1… view at source ↗
read the original abstract

Let $p$ be an odd prime and let $V_{k,a_p}$ be the two-dimensional crystalline representation of the Galois group of ${\mathbb Q}_p$ of weight $k \geq 2$ and parameter $a_p \in \bar{\mathbb{Q}}_p$. We study the semi-simplification $\bar{V}_{k,a_p}$ of the mod $p$ reduction of $V_{k,a_p}$ when the slope (valuation of $a_p$) is a positive fraction $< p-1$ using the mod $p$ local Langlands correspondence. We describe the $\textit{exact shape}$ of $\bar{V}_{k,a_p}$ for all such slopes and all (sufficiently large, depending on the slope) weights $k$, as long as certain Jordan-H\"older factors of dimension $p-1$ do not intervene in the computation (when $k$ is odd), though we also provide some criteria which further determine the shape of $\bar{V}_{k,a_p}$ in some of these exceptional cases. To keep this paper a reasonable length, we assume that for certain bad congruence classes of $k$ mod $p$, the slope is less than the representative - taken in the range $[1,p-1]$ - of the congruence class of $k-2$ mod $(p-1)$, which is generically the case if the slope is small. Finally, a folklore conjecture predicts that the reduction $\bar{V}_{k,a_p}$ is $\textit{irreducible}$ for fractional slopes if $k$ is even. We deduce this conjecture for all fractional slopes $< p-2$ and all (sufficiently large, even) weights $k$ under the aforementioned slope assumption.

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

2 major / 0 minor

Summary. The paper claims to describe the exact shape of the semi-simplification ar{V}_{k,a_p} of the mod p reduction of two-dimensional crystalline Galois representations V_{k,a_p} (weight k ≥ 2, parameter a_p with positive fractional slope < p-1) for all sufficiently large k (depending on the slope), using the mod p local Langlands correspondence. This holds provided certain Jordan-Hölder factors of dimension p-1 do not intervene (with partial criteria given for some exceptional cases when k is odd). To keep the paper a reasonable length, the authors impose the assumption that for certain bad congruence classes of k mod p, the slope is less than the representative (in [1,p-1]) of the congruence class of k-2 mod (p-1). Under the same assumption, the paper deduces the folklore irreducibility conjecture for all fractional slopes < p-2 and all sufficiently large even weights k.

Significance. If the results hold, the explicit shape descriptions advance the understanding of reductions of crystalline representations in the fractional slope range, providing concrete output from the mod p local Langlands correspondence without parameter fitting. The partial deduction of the irreducibility conjecture for even k strengthens the evidence for this folklore statement in a range of cases. The approach relies on the established correspondence rather than ad-hoc fitting, which is a strength.

major comments (2)
  1. [Abstract] Abstract: the slope assumption for bad congruence classes of k mod p (that the slope is less than the representative in [1,p-1] of the class of k-2 mod (p-1)) is load-bearing for the central claim, as the mod p local Langlands analysis of the reduction and the shape computation are performed under this restriction; without it the Jordan-Hölder factors of dimension p-1 could intervene differently for the same slope, changing the described shape.
  2. [Abstract] Abstract: the deduction of the folklore irreducibility conjecture for slopes < p-2 and even k is explicitly conditioned on the same slope assumption, so the result does not apply in the complementary cases where the assumption fails.

Simulated Author's Rebuttal

2 responses · 0 unresolved

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

read point-by-point responses

  1. Referee: [Abstract] Abstract: the slope assumption for bad congruence classes of k mod p (that the slope is less than the representative in [1,p-1] of the class of k-2 mod (p-1)) is load-bearing for the central claim, as the mod p local Langlands analysis of the reduction and the shape computation are performed under this restriction; without it the Jordan-Hölder factors of dimension p-1 could intervene differently for the same slope, changing the described shape.

    Authors: We agree that the slope assumption is load-bearing for the central claims. The manuscript already states explicitly in the abstract that the assumption is imposed to keep the paper at a reasonable length, precisely because the mod p local Langlands analysis and resulting shape descriptions are performed under this restriction; without it, additional cases would arise in which Jordan-Hölder factors of dimension p-1 could intervene differently. revision: no

  2. Referee: [Abstract] Abstract: the deduction of the folklore irreducibility conjecture for slopes < p-2 and even k is explicitly conditioned on the same slope assumption, so the result does not apply in the complementary cases where the assumption fails.

    Authors: We agree that the deduction of the irreducibility conjecture is conditioned on the slope assumption, as already stated in the abstract. The result therefore applies only under this assumption (for sufficiently large even weights k and fractional slopes less than p-2); it does not claim to cover the complementary cases. revision: no

Circularity Check

0 steps flagged

No significant circularity detected; relies on external mod p local Langlands correspondence

full rationale

The paper's derivation of the exact shape of bar V_{k,a_p} proceeds via the established mod p local Langlands correspondence (external to this work) together with explicit case analysis under a stated slope restriction for bad congruence classes. No equations or claims reduce by construction to fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations whose justification collapses into the present paper. The slope assumption is presented as an explicit scope limitation rather than a derived result, and the folklore conjecture deduction remains conditional on that assumption plus the external correspondence. The work is therefore self-contained against external benchmarks with no circular steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract: work rests on the mod p local Langlands correspondence (standard in the field) and basic properties of crystalline representations and their reductions; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • domain assumption The mod p local Langlands correspondence correctly associates Galois representations to representations of GL(2) over finite fields.
    Invoked to compute the semi-simplification of the reduction.
  • standard math Crystalline representations of weight k and parameter a_p exist and have well-defined slopes given by the valuation of a_p.
    Background fact used throughout the description of V_{k,a_p}.

pith-pipeline@v0.9.1-grok · 5861 in / 1541 out tokens · 19493 ms · 2026-07-02T06:57:57.991417+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

32 extracted references · 1 canonical work pages

  1. [1]

    Arsovski, On the reductions of certain two-dimensional crystalline representations, Doc

    B. Arsovski, On the reductions of certain two-dimensional crystalline representations, Doc. Math. 26 (2021), 1929--1979

    2021

  2. [2]

    Barthel and R

    L. Barthel and R. Livn´e, Irreducible modular representations of GL_2 of a local field Duke J. Math. (1994) 75, no. 2, 261--292

    1994

  3. [3]

    Bergdall and B

    J. Bergdall and B. Levin, Reductions of some two-dimensional crystalline representations via Kisin modules, Int. Math. Res. Not. IMRN 2022 (2022), no.4, 3170--3197

    2022

  4. [4]

    Berger, Errata for my articles, https://perso.ens-lyon.fr/laurent.berger/articles.php

    L. Berger, Errata for my articles, https://perso.ens-lyon.fr/laurent.berger/articles.php

  5. [5]

    Berger, Repr\'esentations modulaires de GL_2( _p) et repr\'esentations galoisiennes de dimension 2

    L. Berger, Repr\'esentations modulaires de GL_2( _p) et repr\'esentations galoisiennes de dimension 2 . Ast\'erisque (2010), no. 330, 263--279

    2010

  6. [6]

    Berger, La correspondance de Langlands locale p-adique pour _ 2 ( _p) , Ast\'erisque (2011), no

    L. Berger, La correspondance de Langlands locale p-adique pour _ 2 ( _p) , Ast\'erisque (2011), no. 339, 157--180

    2011

  7. [7]

    Berger, Local constancy for the reduction mod p of 2 -dimensional crystalline representations, Bull

    L. Berger, Local constancy for the reduction mod p of 2 -dimensional crystalline representations, Bull. Lond. Math. Soc. 44 (2012), no.3, 451--459

    2012

  8. [8]

    Berger, H

    L. Berger, H. Li and H. J. Zhu, Construction of some families of 2 -dimensional crystalline representations, Math. Ann. 329 (2004), no.2, 365--377

    2004

  9. [9]

    Bhattacharya, Reductions of certain crystalline representations and local constancy in the weight space, J

    S. Bhattacharya, Reductions of certain crystalline representations and local constancy in the weight space, J. Th\'eor. Nombres Bordeaux 32 (2020), no. 1, 25--47

    2020

  10. [10]

    Bhattacharya and E

    S. Bhattacharya and E. Ghate, Reductions of Galois representations for slopes in (1,2) , Doc. Math. 209 (2015), 43--987

    2015

  11. [11]

    Bhattacharya, E

    S. Bhattacharya, E. Ghate and S. Rozensztajn, Reductions of Galois representations of slope 1 , J. Algebra, 508 (2018), 98--156

    2018

  12. [12]

    Buzzard and T

    K. Buzzard and T. Gee, Explicit reduction modulo p of certain two-dimensional crystalline representations, Int. Math. Res. Not. IMRN 12 (2009), no. 2, 303--2317

    2009

  13. [13]

    Buzzard and T

    K. Buzzard and T. Gee, Explicit reduction modulo p of certain 2 -dimensional crystalline representations, II , Bull. Lond. Math. Soc. 45 (2013), no. 4, 779--788

    2013

  14. [14]

    Breuil, Sur quelques repr\'esentations modulaires et p -adiques de _ 2 ( _ p ) I , Compositio Math

    C. Breuil, Sur quelques repr\'esentations modulaires et p -adiques de _ 2 ( _ p ) I , Compositio Math. 138 (2003), no. 2, 165--188

    2003

  15. [15]

    Breuil, Sur quelques repr\'esentations modulaires et p -adiques de _ 2 ( _ p ) II , J

    C. Breuil, Sur quelques repr\'esentations modulaires et p -adiques de _ 2 ( _ p ) II , J. Inst. Math. Jussieu 2 (2003), no. 1, 23--58

    2003

  16. [16]

    Chitrao, An Iwahori theoretic mod p Local Langlands Correspondence, Canad

    A. Chitrao, An Iwahori theoretic mod p Local Langlands Correspondence, Canad. Math. Bull. 68 (2025), no. 3, 805--817

    2025

  17. [17]

    Colmez, Repr\'esentations de GL_2( _p) et ( , ) -modules , Ast\'erisque (2010), no

    P. Colmez, Repr\'esentations de GL_2( _p) et ( , ) -modules , Ast\'erisque (2010), no. 330, 281--509

    2010

  18. [18]

    Edixhoven, The weight in Serre’s conjectures on modular forms, Invent

    B. Edixhoven, The weight in Serre’s conjectures on modular forms, Invent. Math. 109 (1992), no. 3, 563–594

    1992

  19. [19]

    Gessel and G

    I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae, Adv. in Math. 58 (1985), no. 3, 300--321

    1985

  20. [20]

    Ganguli and E

    A. Ganguli and E. Ghate, Reductions of Galois representations via the mod p Local Langlands Correspondence, J. Number Theory 147 (2015), 250--286

    2015

  21. [21]

    Ganguli and S

    A. Ganguli and S. Kumar, On the local constancy of certain mod p Galois representations, Res. Number Theory 10 (2024), no. 2, Paper No. 52, 43 pp

    2024

  22. [22]

    Ghate, A zig-zag conjecture and local constancy for Galois representations, in Algebraic Number Theory and Related Topics 2018, RIMS K\^ oky\^ uroku Bessatsu B86 Res

    E. Ghate, A zig-zag conjecture and local constancy for Galois representations, in Algebraic Number Theory and Related Topics 2018, RIMS K\^ oky\^ uroku Bessatsu B86 Res. Inst. Math. Sci., Kyoto, (2021), 249--268

    2018

  23. [23]

    Ghate, Zig-zag holds on inertia for large weights, Preprint, https://arxiv.org/abs/2211.12114

    E. Ghate, Zig-zag holds on inertia for large weights, Preprint, https://arxiv.org/abs/2211.12114

    work page arXiv

  24. [24]

    Ghate and S

    E. Ghate and S. Majumder, Reductions of Galois representations of slope 2 , in preparation

  25. [25]

    Ghate and V

    E. Ghate and V. Rai, Reductions of Galois representations for slope 3/2 , Kyoto Journal of Mathematics 65 (2025), no. 3, 1--42

    2025

  26. [26]

    Ghate and V

    E. Ghate and V. Ravitheja, The Monomial Lattice in Modular Symmetric Power Representations, Algebra and Representation Theory 25 (2022), no. 1, 121–185

    2022

  27. [27]

    D. J. Glover, A study of certain modular representations, J. Algebra 51 (1978), no. 2, 425--475

    1978

  28. [28]

    Krattenthaler, Advanced determinant calculus, S\'em

    C. Krattenthaler, Advanced determinant calculus, S\'em. Lothar. Combin. 42 (1999), Art. B42q, 67 pp

    1999

  29. [29]

    R. Liu, N. Truong, L. Xiao and B. Zhao, Slopes of modular forms and geometry of eigencurves, Annals of Math. (2026), to appear

    2026

  30. [30]

    Nagel and A

    E. Nagel and A. Pande, Reductions of Galois representations for slopes in (2,3) , Ramaujan J. Math. 67 (2025), no. 3, Paper No. 70, 62 pp

    2025

  31. [31]

    Rozensztajn, A script to compute the reduction modulo p of some crystalline Galois representations, https://perso.ens-lyon.fr/sandra.rozensztajn/software.html

    S. Rozensztajn, A script to compute the reduction modulo p of some crystalline Galois representations, https://perso.ens-lyon.fr/sandra.rozensztajn/software.html

  32. [32]

    Rozensztajn, An algorithm for computing the reduction of 2 -dimensional crystalline representations of Gal( _p/ _p) , Int

    S. Rozensztajn, An algorithm for computing the reduction of 2 -dimensional crystalline representations of Gal( _p/ _p) , Int. J. Number Theory 14 (2018), no. 7, 1857--1894

    2018