On the exterior square $\varepsilon$-factors of $GL_n$

Ravi Raghunathan

On the exterior square $\varepsilon$-factors of $GL_n$

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Pith's one-line read The three definitions of exterior-square epsilon factors for GL_r agree over p-adic fields.

desk verdict Solid completion of the three-way equality of exterior-square ε-factors for p-adic GL_r, with a clean global proof of the local functional equation. read the letter →

arxiv 2607.04908 v1 pith:LKLOECJO submitted 2026-07-06 math.NT

Ravi Raghunathan This is my paper · ORCID

classification math.NT MSC 11F70

keywords exteriorsquareL-functionsepsilonfactorsJacquet-ShalikaintegralsLanglands-ShahidimethodgenericrepresentationsGL_np-adicfieldslocalfunctionalequation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Three independent recipes attach exterior-square epsilon factors to a generic representation of GL_r over a local field: one from Galois theory via the local Langlands correspondence, one from the Langlands-Shahidi method, and one from Jacquet-Shalika integral representations. The paper proves that, when the field is p-adic, the Jacquet-Shalika and Langlands-Shahidi factors are identical. Combined with earlier equalities already known for the other pairs, this forces all three factors to coincide for every local field of characteristic zero. The argument proceeds by globalising dense sets of tempered representations, comparing global functional equations, and then extending by analytic continuation along Bernstein components. Along the way a new global proof of the local functional equation for the Jacquet-Shalika integrals is obtained.

What carries the argument

Global comparison of functional equations after Shin-type density globalisation of tempered representations, followed by analytic continuation of the resulting equality along connected components of the Bernstein variety using holomorphic sections of Whittaker models.

What would settle it

Exhibit a single generic p-adic representation for which an explicit numerical evaluation of the Jacquet-Shalika epsilon factor differs from the Langlands-Shahidi (or Galois) value at one fixed s and ψ, or show that some open set in the tempered dual cannot be globalised as required by the density theorem used here.

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Extended reading notes

Core claim

For every generic representation π of GL_r(F) with F p-adic, the exterior-square epsilon factor defined by Jacquet-Shalika integrals equals the exterior-square epsilon factor defined by the Langlands-Shahidi method. Consequently all three standard definitions (Jacquet-Shalika, Langlands-Shahidi, Galois) of the exterior-square epsilon factor agree over every local field of characteristic zero.

Load-bearing premise

Every open set of tempered representations at a fixed finite place must contain a local component of a global cuspidal representation that is supercuspidal or unramified at every other finite place; if that density fails for some open sets, the equality cannot be extended from a dense subset to all tempered (and then all generic) representations.

Editorial extensions

If this is right

All three exterior-square L- and ε-factors may be used interchangeably for any generic representation of GL_r over a local field of characteristic zero.
The Jacquet-Shalika epsilon factor now inherits the known multiplicativity formula in terms of the inducing data that was previously available only for the Langlands-Shahidi and Galois factors.
The local functional equation for Jacquet-Shalika integrals holds for every generic p-adic representation, not merely for discrete series or globalisable ones.
Global exterior-square L-functions defined by any of the three methods satisfy identical functional equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same density-plus-analytic-continuation strategy should apply verbatim to other local factors (symmetric-square, Asai, Rankin-Selberg of higher rank) once a suitable local integral representation is known.
Once multiplicativity is available for the Jacquet-Shalika factors, inductive arguments that reduce statements about arbitrary generic representations to supercuspidals become available for the integral side as well.
The result removes one of the last remaining mismatches among the three standard constructions of exterior-square factors, tightening the dictionary between integral representations and the Langlands correspondence.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper finishes the comparison of the three exterior-square ε-factors for generic representations of GL_r over p-adic fields: Jacquet-Shalika equals Langlands-Shahidi (Theorem 1.1), and therefore all three coincide once you fold in the earlier Galois equalities (Theorem 1.2). That was the remaining gap after JLST25 settled the archimedean case and CST17 handled Galois vs Shahidi.

What is new is the global-to-local route that also re-proves the local functional equation for the Jacquet-Shalika integrals (Theorems 4.6-4.7). The chain is transparent: Henniart globalization gives supercuspidals (Prop. 6.2), Shin's Plancherel density (specialized as Thm 7.1) gives a dense set of tempered representations (Prop. 7.4), analytic sections of Whittaker models (BP21 Prop. 2.1) plus holomorphy of the integrals and γ-factors push to nearly tempered (Prop. 7.5-7.6), and BP21 Prop. 2.8.1 extends across the Bernstein components to all generic π. The odd-rank bookkeeping is handled cleanly by the same argument. Citations are accurate and the logical order is careful; nothing is defined circularly.

The soft spots are ordinary specialist ones. The density step rests on a direct specialization of Shi12 Thm 4.8 (matrix coefficients of supercuspidals at three auxiliary places, characteristic functions of max compacts elsewhere); the stress-test note is right that this hits every open set of Temp(GL_r) as claimed, so the residual risk is just verification of the orbital-integral non-vanishing, not a structural hole. The inductive formula for ε_JS was previously missing, which is why the reduction could not be done earlier; the paper supplies it as a byproduct rather than assuming it.

This is for people who work with exterior-square L-functions, local Langlands for GL_n, or integral representations. It removes a technical obstruction that papers have had to work around. The math is pure and the argument is standard in the subject, so it deserves a serious referee. I would accept it for peer review and would cite the equality when I need the three factors to be interchangeable.

Referee Report

0 major / 4 minor

Summary. The paper proves that for a p-adic field F and any generic irreducible representation π of GL_r(F), the exterior-square ε-factors defined by Jacquet–Shalika integrals and by the Langlands–Shahidi method coincide (Theorem 1.1). Combined with earlier equalities of Langlands–Shahidi and Galois factors (CST17, Shahidi) and the archimedean equality of Jacquet–Shalika and Galois factors (JLST25), this yields the coincidence of all three families of ε-factors over any local field of characteristic zero (Theorem 1.2). The argument proceeds by global comparison of functional equations: equality is first obtained for supercuspidals via Henniart globalization (Prop. 6.2), then for a dense subset of tempered representations via a specialization of Shin’s Plancherel density theorem (Thm 7.1 / Prop. 7.4), then extended to nearly tempered representations by analytic sections of Whittaker models (Prop. 2.1) and holomorphy of the integrals and γ-factors (Props. 5.1–5.2, 7.5–7.6), and finally to all generic representations by the extension theorem of Beuzart-Plessis (Prop. 2.8.1 of BP21). Along the way a new global proof of the local functional equation for the Jacquet–Shalika integrals is obtained (Thms 4.6–4.7).

Significance. The result closes a long-standing gap in the comparison of the three standard constructions of exterior-square ε-factors for GL_n. The equality is load-bearing for the local Langlands correspondence and for the theory of automorphic L-functions, where consistency of the various definitions is required. The paper supplies a clean global proof of the local functional equation that avoids the purely local arguments of Matringe and Cogdell–Matringe, and it carefully adapts the analytic-continuation machinery of Beuzart-Plessis from the Asai setting. The logical chain is fully explicit, cites the necessary external theorems at each step, and does not introduce free parameters or ad-hoc entities.

minor comments (4)

Page 2, line 3 of the introduction: the phrase “the odd case of Theorem 1.1 from the even case” is slightly misleading; the reduction is only sketched later (via the definitions of the integrals) and could be flagged more clearly in §3.
Notation for the two families of Jacquet–Shalika integrals (even/odd) is introduced in §4.1 but the subsequent global comparison (4.12)–(4.15) switches between them without always restating which case is under discussion; a short clarifying sentence would help the reader.
In the statement of Theorem 7.1 the three auxiliary supercuspidal places are chosen for technical convenience; a parenthetical remark that two would probably suffice (as the author suspects in Remark 7.2) would make the dependence on Shin’s hypotheses more transparent.
Several typographical slips: “ap-adic” for “a p-adic”, “continous” for “continuous”, and occasional missing spaces after commas in displayed equations (e.g., (2.1), (4.4)).

Circularity Check

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No significant circularity: equality of independently defined ε-factors is transferred from known places (archimedean, supercuspidal, unramified) via globalization/density and analytic continuation.

full rationale

The derivation of Theorem 1.1 proceeds by comparing two independently defined global functional equations (the Jacquet-Shalika integral equation of Theorem 4.10 from JS90 versus the Langlands-Shahidi equation of Theorem 3.7). Equality of local factors is already known at archimedean places (JLST25), unramified places (explicit computation via Proposition 4.9), and supercuspidals (Proposition 6.2 via Henniart globalization). Shin density (Theorem 7.1, specializing Shi12 Thm 4.8) produces a dense set of tempered representations that arise as local components of global cuspidals that are supercuspidal or unramified elsewhere; the global comparison therefore yields equality on that dense set (Proposition 7.4). Analytic sections of Whittaker models (Proposition 2.1 from BP21), holomorphy/continuity of the local integrals (Proposition 5.2) and of the Langlands-Shahidi γ-factors (Proposition 5.1), and the extension theorem BP21 Prop. 2.8.1 then push the equality first to all nearly tempered representations and finally to every generic π. The local functional equation itself is likewise extended from the globalizable case (already available in the author's prior KR12) by the same continuity. No quantity is defined in terms of the target equality, no parameter is fitted to data and then re-predicted, and the self-citations (KR12 for the globalizable base case, BP21 for analytic techniques) are ordinary dependence on prior literature rather than a load-bearing unverified loop. The argument is therefore self-contained against external benchmarks and exhibits no circular reduction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

Pure local-global number theory paper. No free parameters or invented physical entities. The load-bearing background consists of standard theorems of the Langlands program (LLC for GL_n, Langlands-Shahidi multiplicativity, Jacquet-Shalika integral theory) plus three external globalization/analytic results (Henniart, Shin, Beuzart-Plessis) and the recent archimedean equality (JLST25).

assumptions (8)

domain assumption Local Langlands correspondence for GL_n identifies irreducible admissible representations with Weil-Deligne representations and preserves Rankin-Selberg and exterior-square L- and ε-factors (HT01, Hen00, CST17).
Used to define Galois factors and to equate them with Langlands-Shahidi factors (Thm 3.1, Prop 3.4).
domain assumption Langlands-Shahidi global functional equation and holomorphy for exterior-square L-functions of unitary cuspidal automorphic representations (Sha90b, Kim99).
Compared with the Jacquet-Shalika global functional equation in (4.14)-(4.15) to transfer local equalities.
domain assumption Jacquet-Shalika local integrals admit meromorphic continuation and generate the exterior-square L-factor (JS90, Bel11, Jo20, KR12).
Defines L_JS and the normalized integrals Ξ used in the local functional equation (Thms 4.4, 4.6).
domain assumption Henniart globalization: every supercuspidal of GL_r(F) appears as the local component at one place of a global cuspidal representation unramified at all other finite places (Hen84, PSP08).
Prop 6.2 reduces the supercuspidal case of Thm 1.1 to a comparison of global functional equations.
domain assumption Shin's automorphic Plancherel density theorem yields, for any open set of tempered representations at a fixed finite place, a global cuspidal representation whose local component lies in that set and is supercuspidal or unramified elsewhere (Shi12 Thm 4.8).
Thm 7.1 / Prop 7.4 produce a dense set of tempered representations where the epsilon equality holds.
domain assumption Existence of analytic sections of Whittaker models in the induction parameter and extension of holomorphic functions of finite order from half-planes times open sets in hyperplanes (BP21 Cor 2.7.1, Prop 2.8.1).
Prop 2.1 and the final extension step that passes from nearly tempered to all generic representations.
domain assumption Archimedean equality ε_JS = ε_G = ε_Sh (JLST25 Thm 2.1, Sha85).
Used at infinite places when comparing global functional equations (4.14)-(4.15).
standard math Standard facts of smooth admissible representations of p-adic GL_r, Whittaker models, and the Fell topology on tempered representations.
Background throughout Sections 2-5.

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Pith. "Pith review of On the exterior square $\varepsilon$-factors of $GL_n$." pith.science (2026). https://pith.science/paper/LKLOECJO

@misc{pith2026260704908,
  author       = {Pith},
  title        = {Pith review of: On the exterior square $\varepsilon$-factors of $GL_n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LKLOECJO}},
  note         = {Machine review of arXiv:2607.04908}
}

abstract

Let $\pi$ be a generic representation of ${\mathrm{GL}}_r(F)$, where $F$ is a $p$-adic field. We show using global methods that the $\varepsilon$-factors associated to the exterior square of $\pi$ via the Jacquet-Shalika integrals and Langlands-Shahidi methods coincide. Along the way we also give a new proof of the local functional equation in the $p$-adic case following the techniques in [Beuzart-Plessis, Relative trace formulas, Simons Symp.(2021), pages 1-50].

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.

[1] [1]

On the H olomorphy of E xterior- S quare L -functions for GL (n)

Dustin Belt. On the H olomorphy of E xterior- S quare L -functions for GL (n) . preprint, arXiv:1108.2200v4 [math.NT] , 2011

arXiv 2011

[2] [2]

Archimedean theory and -factors for the A sai R ankin- S elberg integrals

Rapha\" e l Beuzart-Plessis. Archimedean theory and -factors for the A sai R ankin- S elberg integrals. In Relative trace formulas , Simons Symp., pages 1--50. Springer, Cham, [2021] 2021

2021

[3] [3]

On limit multiplicities of discrete series representations in spaces of automorphic forms

Laurent Clozel. On limit multiplicities of discrete series representations in spaces of automorphic forms. Invent. Math. , 83(2):265--284, 1986

1986

[4] [4]

Cogdell and Nadir Matringe

James W. Cogdell and Nadir Matringe. The functional equation of the J acquet- S halika integral representation of the local exterior-square L -function. Math. Res. Lett. , 22(3):697--717, 2015

2015

[5] [5]

Casselman and J

W. Casselman and J. Shalika. The unramified principal series of p -adic groups. II . T he W hittaker function. Compositio Math. , 41(2):207--231, 1980

1980

[6] [6]

J. W. Cogdell, F. Shahidi, and T.-L. Tsai. Local L anglands correspondence for GL _n and the exterior and symmetric square -factors. Duke Math. J. , 166(11):2053--2132, 2017

[7] [7]

La conjecture de L anglands locale pour GL (3)

Guy Henniart. La conjecture de L anglands locale pour GL (3) . M\' e m. Soc. Math. France (N.S.) , (11-12):186, 1984

1984

[8] [8]

Une preuve simple des conjectures de L anglands pour GL (n) sur un corps p -adique

Guy Henniart. Une preuve simple des conjectures de L anglands pour GL (n) sur un corps p -adique. Invent. Math. , 139(2):439--455, 2000

2000