arxiv: 2605.02654 · v1 · submitted 2026-05-04 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

Reductions of GL₂(mathbb Q_{p^f})-Banach spaces of slopes in (0,1)

Eknath Ghate, Shivansh Pandey

Pith reviewed 2026-05-08 18:40 UTC · model grok-4.3

classification 🧮 math.NT

keywords GL_2 representationsp-adic Banach spacessupercuspidal representationsmod p reductionintegral structuresfiltrationslocal Langlands

0 comments

The pith

Conditions ensure that irreducible quotients in mod p filtrations of GL_2 Banach spaces are supercuspidal.

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

The paper studies p-adic locally algebraic representations of GL_2 over the unramified extension Q_{p^f} of degree f. These arise from a tuple of f weights together with a parameter a_p whose p-adic valuation lies strictly between 0 and 1. The authors supply conditions guaranteeing that the natural integral structure, once reduced modulo p, carries a filtration whose successive subquotients have only supercuspidal irreducible quotients. A reader would care because such statements connect the continuous p-adic theory of automorphic forms to the discrete mod p theory that appears in the local Langlands correspondence. The authors additionally verify that the integral structure forms a lattice for small weights and small f, so the reduction is nonzero.

Core claim

Let p be an odd prime and f at least 1. We consider a p-adic locally algebraic GL_2(Q_{p^f})-representation attached to a tuple of f weights k and a p-adic integer a_p with valuation in (0,1). We give conditions under which the irreducible quotients of the subquotients in a filtration on the reduction mod p of the natural integral structure on this space are supercuspidal. We also check that for small k and f the integral structure is a lattice so that the mod p reduction is nonzero.

What carries the argument

The filtration on the reduction modulo p of the natural integral structure of the Banach space.

Load-bearing premise

A natural integral structure on the Banach space exists and admits a filtration on its mod p reduction whose subquotients have irreducible quotients that are supercuspidal precisely when the stated conditions on p, f, the weights, and a_p hold.

What would settle it

An explicit choice of weights k_i, small f, and a_p with valuation in (0,1) satisfying the paper's conditions, yet for which at least one irreducible quotient of the filtration is not supercuspidal.

read the original abstract

Let $p$ be an odd prime and $f \geq 1$. We consider a $p$-adic locally algebraic $\text{GL}_2(\mathbb Q_{p^f})$-representation attached to a tuple of $f$ weights $k=(k_i)$ for $0 \leq i \leq f-1$ and a $p$-adic integer $a_p$ with valuation in $(0,1)$. We give conditions under which the irreducible quotients of the subquotients in a filtration on the reduction mod $p$ of the natural integral structure on this space are supercuspidal. We also check that for small $k$ and $f$ the integral structure is a lattice so that the mod $p$ reduction is nonzero.

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 gives conditions for supercuspidal quotients in mod p reductions of these GL2 Banach spaces but only verifies the lattice property for small weights and f.

read the letter

The key point is that Ghate and Pandey provide conditions under which the irreducible quotients of subquotients in a filtration of the mod p reduction of a p-adic GL_2(Q_{p^f}) Banach space are supercuspidal. They also verify that the natural integral structure is a lattice for small k and f, making the reduction nonzero in those cases. What stands out is the explicit conditions tied to the tuple of weights and the valuation of a_p in (0,1). The lattice check for small parameters is concrete and likely involves direct computation or known results on integral models, which adds some value for people doing explicit work in this area. The soft spot is exactly what the stress-test note flags: the general existence and behavior of the natural integral structure and its filtration are not fully established beyond the small cases. The abstract separates the supercuspidality conditions from the lattice verification, which hints that the authors don't claim the structure always exists or is a lattice. Without a general argument or counterexamples, the main result applies only conditionally, and readers will have to check the integral structure themselves for larger k. This paper is for experts in the p-adic Langlands correspondence, particularly those studying reductions of locally algebraic representations for GL2. It won't interest a broad audience but could help specialists refine their understanding of supercuspidal phenomena mod p. I would recommend sending it to peer review; the small-case verifications are solid enough to warrant referee input on whether the conditions can be made more general or if the limitation is inherent.

Referee Report

1 major / 0 minor

Summary. The manuscript considers p-adic locally algebraic GL_2(Q_{p^f})-representations attached to a tuple of weights k=(k_i) and a_p with valuation in (0,1). It states conditions under which the irreducible quotients of the subquotients in a filtration on the mod p reduction of the natural integral structure are supercuspidal, and separately verifies that the integral structure is a lattice (hence the reduction is nonzero) only for small k and f.

Significance. If the stated conditions can be shown to guarantee both the existence of the filtration and its supercuspidal quotients in general, the result would contribute to the understanding of mod p reductions of Banach spaces in the p-adic Langlands program. The restriction of the lattice verification to small cases, however, limits the immediate applicability of the main claim.

major comments (1)

Abstract: the central claim asserts conditions under which the irreducible quotients are supercuspidal, but the existence of the natural integral structure, the filtration, and nonzero reduction for arbitrary k is only explicitly verified for small k and f; without a general construction or proof that the reduction remains nonzero and carries the stated filtration, the supercuspidality statement rests on an unverified assumption outside those cases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying an important point about the scope of our results. We agree that the abstract and introduction should more explicitly distinguish the conditional supercuspidality statement from the limited verification of the lattice property. We address the major comment below and will make the corresponding revisions.

read point-by-point responses

Referee: Abstract: the central claim asserts conditions under which the irreducible quotients are supercuspidal, but the existence of the natural integral structure, the filtration, and nonzero reduction for arbitrary k is only explicitly verified for small k and f; without a general construction or proof that the reduction remains nonzero and carries the stated filtration, the supercuspidality statement rests on an unverified assumption outside those cases.

Authors: We thank the referee for this observation. The main theorem states that, under explicit conditions on the weight tuple k and the parameter a_p (with v_p(a_p) in (0,1)), if a natural integral structure exists on the given locally algebraic Banach space and if this structure admits a filtration whose subquotients reduce mod p to nonzero representations, then the irreducible quotients of those subquotients are supercuspidal. The manuscript does not claim, and does not prove, that such an integral structure exists and yields a nonzero reduction for arbitrary k; the lattice property is verified only for small k and f, as already stated in the abstract and in the body of the paper. The supercuspidality result is therefore conditional on the existence of the filtration and the nonzero reduction. To remove any ambiguity, we will revise the abstract and the statement of the main theorem to emphasize this conditional character and to reiterate that the lattice verification is restricted to the small cases. No general construction for arbitrary k is provided, as it lies outside the scope of the present work. revision: yes

Circularity Check

0 steps flagged

No circularity: conditional claims rest on external standard constructions, not self-referential reduction.

full rationale

The paper states it considers a p-adic locally algebraic representation attached to weights k and a_p (val in (0,1)), then gives conditions for supercuspidality of irreducible quotients in the mod-p filtration of the natural integral structure, while separately verifying the lattice property only for small k and f. No derivation equations, fitted parameters renamed as predictions, or self-citation chains are exhibited that would force the central claim to equal its inputs by construction. The existence of the integral structure is taken as part of the setup (standard in p-adic representation theory), and the supercuspidality result is explicitly conditional rather than derived tautologically. This is the common case of a self-contained conditional theorem with no load-bearing circular step.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5440 in / 1066 out tokens · 32136 ms · 2026-05-08T18:40:21.179860+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost.FunctionalEquation / Foundation.AlphaCoordinateFixation (J(x)=½(x+x^{-1})−1) washburn_uniqueness_aczel unclear
Hecke operator T explicit formulas (3),(4) and Dickson-type polynomials θ_i = X_i Y_{i-1}^p − Y_i X_{i-1}^p generating V_r^*.

Reference graph

Works this paper leans on

32 extracted references · 3 canonical work pages · 1 internal anchor

[1]

Assaf, Existence of invariant norms in p -adic representations of GL_2(F) of large weights , J

E. Assaf, Existence of invariant norms in p -adic representations of GL_2(F) of large weights , J. Number Theory, 224 , 2021, 95--141

2021
[2]

Assaf, D

E. Assaf, D. Kazhdan, and E. de Shalit, Kirillov models and the Breuil–Schneider conjecture for GL_2(F) , https://arxiv.org/abs/1302.3060

work page arXiv
[3]

Bardoe and P

M. Bardoe and P. Sin, The permutation modules for GL(n, F_q) acting on P^n( F_q) and F_q^ n-1 , J. London Math. Soc., 61 , 2000, 58--80

2000
[4]

Barthel and R

L. Barthel and R. Livn\'e, Irreducible modular representations of GL _2 of a local field , Duke Math. J., 74 , 1994, 261--292

1994
[5]

Bhattacharya and E

S. Bhattacharya and E. Ghate, Reductions of Galois representations for slopes in (1,2) , Doc. Math., 20 , 2015, 943--987

2015
[6]

Bhattacharya, E

S. Bhattacharya, E. Ghate and R. Vangala, Reductions of crystalline representations of fractional slope <p , in preparation
[7]

Bhattacharya and A

S. Bhattacharya and A. Venugopal, Behaviour of certain crystalline representations modulo 2 , in preparation
[8]

Breuil, Representations of Galois and of GL_2 in characteristic p , Course at Columbia University, 2007

C. Breuil, Representations of Galois and of GL_2 in characteristic p , Course at Columbia University, 2007

2007
[9]

Breuil, Sur quelques repr\'esentations modulaires et p -adiques de GL _2( Q_p) II , J

C. Breuil, Sur quelques repr\'esentations modulaires et p -adiques de GL _2( Q_p) II , J. Inst. Math. Jussieu, 2 , no. 1, 2003, 23--58

2003
[10]

Breuil, The Emerging p -adic Langlands Programme, Proceedings of I

C. Breuil, The Emerging p -adic Langlands Programme, Proceedings of I. C. M. 2010 (Hindustan Book Agency, New Delhi; distributed by), 2010, 203--230

2010
[11]

Breuil and P

C. Breuil and P. Schneider, First steps towards p -adic Langlands functoriality, J. Reine Angew. Math., 610 , 2007, 149--180

2007
[12]

Breuil and V

C. Breuil and V. Pa s k \=u nas, Towards a modulo p Langlands correspondence for GL_2 , Mem. Amer. Math. Soc., 216 , no. 1016, 2012, vi+114 pp

2012
[13]

Breuil, F

C. Breuil, F. Herzig, Y. Hu, S. Morra and B. Schraen, Conjectures and results on modular representations of GL_n(K) for a p -adic field K , Mem. Amer. Math. Soc., 315 , no. 1598, 2025, v+163 pp

2025
[14]

Breuil, F

C. Breuil, F. Herzig, Y. Hu, S. Morra and B. Schraen, Finite length for unramified GL_2 , preprint
[15]

Buzzard and T

K. Buzzard and T. Gee, Explicit reduction modulo p of certain two-dimensional crystalline representations, Int. Math. Res. Not., 12 , no. 12, 2009, 2303--2317

2009
[16]

Buzzard and T

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

2013
[17]

Buzzard, F

K. Buzzard, F. Diamond and F. Jarvis, On Serre's conjecture for mod l Galois representations over totally real fields, Duke Math. J., 155 , no. 1, 2010, 105--161

2010
[18]

Caraiani, M

A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Pa s k \=u nas, and S. W. Shin, Patching and the p -adic local Langlands correspondence, Camb. J. Math., 4 , no. 2, 2016, 197--287

2016
[19]

On the $I(1)$-invariants: Non-abelian Hecke algebra case

A. Chitrao, A. Jana and A. Soneji, On the I(1) -invariants: Non-abelian Hecke algebra case, https://arxiv.org/abs/2604.20218

work page internal anchor Pith review Pith/arXiv arXiv
[20]

Chitrao and A

A. Chitrao and A. Soneji, Iwahori-Hecke model for the universal supersingular representation, https://arxiv.org/abs/2508.13766

work page arXiv
[21]

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

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

2010
[22]

Diamond, A correspondence between representations of local Galois groups and Lie-type groups, In L-functions and Galois representations, volume 320 of London Math

F. Diamond, A correspondence between representations of local Galois groups and Lie-type groups, In L-functions and Galois representations, volume 320 of London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 2007, 187–-206

2007
[23]

Ghate and A

E. Ghate and A. Jana, Modular representations of GL_2( F_q) using calculus , Forum Math., 37 , no. 5, 2025, 1503--1543

2025
[24]

Ghate and R

E. Ghate and R. Vangala, The monomial lattice in modular symmetric power representations, Algebr. Represent. Theory, 25 , no. 1, 2022, 121--185

2022
[25]

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

1978
[26]

Y. I. Hendel, On the universal mod p supersingular quotients for GL_2(F) over F _p for general F/ Q_p , J. Algebra, 519 , 2019, 1--38

2019
[27]

de Ieso, Existence de normes invariantes pour GL_2 , J

M. de Ieso, Existence de normes invariantes pour GL_2 , J. Number Theory, 133 , no. 8, 2013, 2729--2755

2013
[28]

Nadimpalli and M

S. Nadimpalli and M. Sheth, On the integrality of locally algebraic representations of GL_2(D) , J. Number Theory, 257 , 2024, 124--145

2024
[29]

Krattenthaler, Advanced determinant calculus, Se\'m

C. Krattenthaler, Advanced determinant calculus, Se\'m. Lothar. Combin. 42: Art. B42q, 67, 1999

1999
[30]

Schein, On the universal supersingular mod p representations of GL_2(F) , J

M. Schein, On the universal supersingular mod p representations of GL_2(F) , J. Number Theory, 141 , 2014, 242--277

2014
[31]

Schein, A family of irreducible supersingular representations of GL_2(F) for some ramified p -adic fields , Israel J

M. Schein, A family of irreducible supersingular representations of GL_2(F) for some ramified p -adic fields , Israel J. Math., 255 , no. 2, 2023, 911--930

2023
[32]

Vign\'eras, A criterion for integral structures and coefficient systems on the tree of PGL(2,F) , Pure Appl

M. Vign\'eras, A criterion for integral structures and coefficient systems on the tree of PGL(2,F) , Pure Appl. Math. Q, 4 , no. 8, 2008, Special Issue: In honor of Jean-Pierre Serre. Part 1, 1291--1316

2008