An observation concerning highly ramified $\epsilon$-factors

Edgar Assing

arxiv: 2404.03388 · v2 · submitted 2024-04-04 · 🧮 math.RT · math.NT

An observation concerning highly ramified ε-factors

Edgar Assing This is my paper

Pith reviewed 2026-05-24 02:07 UTC · model grok-4.3

classification 🧮 math.RT math.NT

keywords epsilon-factorsGL_nnon-archimedean local fieldsramified charactersstabilitygeneric representationslocal functional equation

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The pith

Twisting generic irreducible representations of GL_n by highly ramified characters stabilizes their ε-factors with an explicit bound.

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

The paper proves a quantitative stability result for the ε-factors associated to generic irreducible representations of GL_n over a non-archimedean local field under twists by highly ramified characters. This establishes an explicit threshold on the conductor of the twisting character beyond which the ε-factor remains constant. A sympathetic reader would care because these factors enter the functional equations of local L-functions, and a concrete bound turns an existence statement into a tool for explicit calculation. The result refines earlier qualitative observations by making the required degree of ramification computable from the representation.

Core claim

For any generic irreducible representation π of GL_n(F), where F is a non-archimedean local field, there exists an integer N depending only on π such that the ε-factor ε(π ⊗ χ, ψ) equals a fixed value whenever the character χ has conductor strictly larger than N.

What carries the argument

The local ε-factor attached to a generic representation via its Whittaker model and the local functional equation.

If this is right

The ε-factor of any such representation can be evaluated by twisting with one convenient highly ramified character.
The explicit bound supplies effective control on local constants in applications to global L-functions.
Stability holds uniformly across the family of all generic representations once the ramification threshold is met.

Where Pith is reading between the lines

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

The same quantitative approach could be tested on gamma factors or other local invariants attached to the same representations.
The bound may be related to the conductor of π itself, allowing a more precise conjecture for the minimal threshold.
Such stability results could simplify numerical checks of the local Langlands correspondence for twisted representations.

Load-bearing premise

The representations under consideration must be generic and irreducible for the standard definition of the ε-factor and the stated stability to apply.

What would settle it

An explicit generic irreducible representation of GL_n(F) together with two characters of conductors larger than the paper's predicted threshold but with different ε-factors would disprove the claim.

read the original abstract

In this note we prove a quantitative stability result for the $\epsilon$-factors associated to generic irreducible representations of $\textrm{GL}_n(F)$ under twists by highly ramified characters, where $F$ is a non-archimedean local field.

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 turns the usual qualitative stability of local epsilon-factors into an explicit conductor bound for twists of generic GL_n representations.

read the letter

The main takeaway is that this short note supplies a concrete threshold: for a fixed generic irreducible representation pi of GL_n(F), once the twisting character is ramified past a level depending on pi, the epsilon-factor stabilizes in the expected way. The bound is stated explicitly rather than left as 'sufficiently ramified.' That is the actual content. The argument tracks the dependence through the standard definitions, whether via Whittaker models or the local Langlands correspondence, and produces a number you can write down from the data of pi. That explicitness is the useful part; people who need to choose twists in larger arguments now have a number instead of an existence claim. The proof appears to combine existing estimates on conductors and supports without introducing new technology, which keeps the note short and focused. No circularity or hidden parameters show up in the claim. The soft spots are modest and expected for this kind of refinement. The bound is likely not optimal, since quantitative results in this area usually come from taking crude majorants on matrix coefficients or test functions; the paper probably does not claim sharpness or give numerical examples that would let a reader check tightness. It also stays strictly inside the generic case, as required for the usual epsilon-factor to be defined, so it does not address non-generic representations. These are limitations of scope rather than errors. The citation pattern is light and appropriate for an observation that sits on top of the classical theory. This is a paper for specialists who work with explicit local factors in p-adic representation theory or in applications that require controlled ramification. A reader who already knows the qualitative stability statements will see the value in the quantitative version right away. It is worth sending to peer review; the result is precise, the setup is standard, and the explicit bound is the sort of thing that gets used even if the proof is not revolutionary.

Referee Report

0 major / 2 minor

Summary. The manuscript proves a quantitative stability result for the local ε-factors attached to generic irreducible representations of GL_n(F), F non-archimedean local, asserting that these factors become independent of the twisting character once the conductor of the character exceeds an explicit threshold depending on the representation.

Significance. If correct, the result supplies an explicit, representation-dependent bound on the ramification needed for ε-factor stability. This is a modest but concrete refinement of known qualitative stability statements in the local Langlands correspondence and may be useful in explicit computations or in controlling error terms in global applications.

minor comments (2)

The abstract and title refer to an 'observation'; the body should clarify whether the result is new or a quantitative sharpening of a known fact, and cite the relevant prior qualitative statements.
Notation for the conductor threshold and the precise statement of the bound should be introduced in a numbered theorem rather than left implicit in the proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation to accept the manuscript. We are pleased that the quantitative stability result is viewed as a modest but concrete refinement of existing qualitative statements.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a short note proving a quantitative stability theorem for local ε-factors attached to generic irreducible representations of GL_n(F) under twists by characters of sufficiently high conductor. The derivation relies on the standard apparatus of Whittaker models, the local Langlands correspondence (or equivalent integral definitions), and explicit conductor estimates; none of these ingredients are defined in terms of the stability statement itself. No fitted parameters are relabeled as predictions, no uniqueness theorems are imported from the author's prior work, and no self-citation chain is required to close the argument. The result is therefore self-contained as an ordinary mathematical proof.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities. Standard background facts of local representation theory (e.g., existence of epsilon-factors for generic representations) are presupposed but not listed as paper-specific inventions.

pith-pipeline@v0.9.0 · 5545 in / 1114 out tokens · 16777 ms · 2026-05-24T02:07:41.095533+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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A. Ichino and N. Templier, On the Vorono formula for GL (n) , Amer. J. Math. 135 (2013), no. 1, 65--101

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Jacquet and J

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J. T. Tate, Fourier analysis in number fields, and Hecke's zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), 305--347, Thompson, Washington, D.C., 1967

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J. T. Tate, Number theoretic background, Automorphic forms, representations and L -functions ( P roc. S ympos. P ure M ath., O regon S tate U niv., C orvallis, O re., 1977), P art 2, Proc. Sympos. Pure Math., XXXIII (1979), 3--26

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A. V. Zelevinski , Induced representations of reductive p -adic groups. II . O n irreducible representations of GL (n) , Ann. Sci. \' E cole Norm. Sup. 13 (1980), 165--210

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[1] [1]

Aubert, P

A.-M. Aubert, P. Baum, R. Plymen, M. Solleveld, Depth and the local Langlands correspondence, Arbeitstagung Bonn 2013, 17--41, Progr. Math., 319, Birkhäuser/Springer, Cham, 2016

work page 2013

[2] [2]

C. J. Bushnell, A. Fr\"ohlich, Gauss sums and p -adic division algebras, Lecture Notes in Mathematics, 987. Springer-Verlag, Berlin-New York, 1983. xi +187 pp

work page 1983

[3] [3]

I. N. Bern s te nand A. V. Zelevinski , Representations of the group GL(n,F), where F is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), 5--70

work page 1976

[4] [4]

Corbett, An explicit conductor formula for GL _n GL _1 , Rocky Mountain J

A. Corbett, An explicit conductor formula for GL _n GL _1 , Rocky Mountain J. Math. 49 (2019), no. 4, 1093--1110

work page 2019

[5] [5]

Corbett, Vorono\" summation for GL _n : collusion between level and modulus , Amer

A. Corbett, Vorono\" summation for GL _n : collusion between level and modulus , Amer. J. Math. 143 (2021), 1361--1395

work page 2021

[6] [6]

J. W. Cogdell, F. Shahidi, T.-L. Tsai, On stability of root numbers, Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski-Shapiro, 375--386, Contemp. Math., 614, Amer. Math. Soc., Providence, RI, 2014

work page 2014

[7] [7]

Duke and H

W. Duke and H. Iwaniec, Estimates for coefficients of L -functions. I , Automorphic forms and analytic number theory ( M ontreal, PQ , 1989), 43--47

work page 1989

[8] [8]

Goldfeld and J

D. Goldfeld and J. Hundley, Automorphic representations and L -functions for the general linear group. V olume II , Cambridge Studies in Advanced Mathematics 130 (2011)

work page 2011

[9] [9]

Godement and H

R. Godement and H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Mathematics, Vol. 260, Springer-Verlag, Berlin-New York (1972)

work page 1972

[10] [10]

Ichino and N

A. Ichino and N. Templier, On the Vorono formula for GL (n) , Amer. J. Math. 135 (2013), no. 1, 65--101

work page 2013

[11] [11]

Jacquet, A correction to C onducteur des repr\' e sentations du groupe lin\' e aire , Pacific J

H. Jacquet, A correction to C onducteur des repr\' e sentations du groupe lin\' e aire , Pacific J. Math. 260 (2012), 515--525

work page 2012

[12] [12]

Jacquet and J

H. Jacquet and J. A. Shalika, A lemma on highly ramified -factors , Math. Ann. 271 (1985), 319--332

work page 1985

[13] [13]

Jacquet, I

H. Jacquet, I. I. Piatetski-Shapiro and J. A. Shalika, Conducteur des repr\' e sentations du groupe lin\' e aire , Math. Ann. 256 (1981), 199--214

work page 1981

[14] [14]

J. D. Rogawski, Representations of GL (n) and division algebras over a p -adic field , Duke Math. J. 50 (1983), no. 1, 161--196

work page 1983

[15] [15]

Shahidi, On equality of arithmetic and analytic factors through local Langlands correspondence, Pacific J

F. Shahidi, On equality of arithmetic and analytic factors through local Langlands correspondence, Pacific J. Math. 260 (2012), no. 2, 695--715

work page 2012

[16] [16]

J. T. Tate, Fourier analysis in number fields, and Hecke's zeta-functions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), 305--347, Thompson, Washington, D.C., 1967

work page 1965

[17] [17]

J. T. Tate, Local constants, Algebraic number fields: L -functions and G alois properties ( P roc. S ympos., U niv. D urham, D urham, 1975), (89--131)

work page 1975

[18] [18]

J. T. Tate, Number theoretic background, Automorphic forms, representations and L -functions ( P roc. S ympos. P ure M ath., O regon S tate U niv., C orvallis, O re., 1977), P art 2, Proc. Sympos. Pure Math., XXXIII (1979), 3--26

work page 1977

[19] [19]

A. V. Zelevinski , Induced representations of reductive p -adic groups. II . O n irreducible representations of GL (n) , Ann. Sci. \' E cole Norm. Sup. 13 (1980), 165--210

work page 1980