{"id":"e093d2ad-ae75-4bfd-9948-9922e54c8349","arxiv_id":"2607.04908","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For generic representations of GL_r over a p-adic field, the Jacquet-Shalika and Langlands-Shahidi exterior-square epsilon factors coincide.","lead":"The paper proves that two different definitions of exterior-square epsilon factors for generic representations of GL_r over p-adic fields agree. Completing this equality finishes a long program comparing Galois, Langlands-Shahidi, and Jacquet-Shalika constructions of these factors.","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The central claim (Thm 1.1) is established by a transparent global-to-local comparison that relies only on already-published external results (Henniart globalization for supercuspidals, Shin density, Beuzart-Plessis analytic sections and extension lemma, and the archimedean equality of JLST25). The reader's identification of Shin's theorem as the most delicate external input is accurate, but the paper's application of it is standard and does not introduce an additional soft spot. Residual verification risk is ordinary specialist checking of the cited theorems and of the odd-rank bookkeeping; nothing indicates an internal inconsistency or a missing hypothesis that would invalidate the equality for some open set of tempered representations. Consequently the ACCEPT verdict with high confidence remains appropriate.","tokens_in":20992,"tokens_out":606,"duration_ms":4984,"concrete_test":"Verify that the test functions chosen in the proof of Theorem 7.1 satisfy the support and positivity hypotheses of Shi12 Thm 4.8 (in particular that the truncated Kottwitz functions at v1,v2 and the shrinking matrix coefficients at v3 produce a non-zero Plancherel measure on the desired Bernstein components). If they do, the density statement holds and the subsequent analytic continuation covers all of Irr_gen.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest-assumption concern (possible failure of Shin's density theorem to hit every open set of Temp(GL_r) with a global cuspidal representation that is supercuspidal or unramified elsewhere) does not land as a load-bearing gap. Theorem 7.1 is obtained by a direct specialization of Shi12 Thm 4.8: choose S = S_∞ ∪ {v0}, take φ_v0 supported so that its orbital integrals are non-vanishing on the given open V ⊂ Temp(GL_r(k_v0)), take matrix coefficients of supercuspidals at three auxiliary places, and take characteristic functions of maximal compacts elsewhere. The resulting global Π has Π_v0 ∈ V and is supercuspidal or unramified at every other finite place, exactly as needed for the comparison of global functional equations (4.14) versus (4.15). Once equality holds on a dense subset of Temp, the analytic sections of Prop. 2.1, holomorphy of the local integrals (Prop. 5.2) and of the Langlands-Shahidi γ-factors (Prop. 5.1), and the extension theorem BP21 Prop. 2.8.1 push the equality first to all nearly tempered representations and then to every generic π. No hidden hypothesis is required.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","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).","tokens_in":21277,"tokens_out":911,"duration_ms":6853,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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.","section":null},{"comment":"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)).","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is a solid, carefully written contribution that fits the journal well. The reliance on Shin’s density theorem is legitimate and correctly specialized; no hidden circularity is present. I see no reason to request further external review."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","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.\n\nWhat 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.\n\nThe 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.\n\nThis 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.","headline":"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.","tokens_in":21905,"tokens_out":598,"would_cite":true,"duration_ms":5253,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F70"],"pacs":[],"model":"grok-4.5","headline":"The three definitions of exterior-square epsilon factors for GL_r agree over p-adic fields.","keywords":["exterior square L-functions","epsilon factors","Jacquet-Shalika integrals","Langlands-Shahidi method","generic representations","GL_n","p-adic fields","local functional equation"],"falsifier":"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.","tokens_in":21866,"feed_emoji":"∧","tokens_out":946,"duration_ms":6790,"temperature":0.7,"pith_summary":"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.","feed_headline":"Three exterior-square epsilon factors for GL_r now match","feed_subtitle":"Global density and analytic continuation force Jacquet-Shalika and Langlands-Shahidi factors to agree over p-adics","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Jacquet-Shalika and Langlands-Shahidi exterior-square ε-factors coincide","Three exterior-square ε-factor definitions agree for GL_r","Exterior-square ε-factors match for generic reps of GL_r over p-adics","Global methods equate Jacquet-Shalika and Langlands-Shahidi ε-factors","All three exterior-square ε-factors of GL_n agree in char zero"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Jacquet-Shalika and Langlands-Shahidi exterior-square ε-factors coincide","Three exterior-square ε-factor definitions agree for GL_r","Exterior-square ε-factors match for generic reps of GL_r over p-adics","Global methods equate Jacquet-Shalika and Langlands-Shahidi ε-factors","All three exterior-square ε-factors of GL_n agree in char zero"]},"model":"grok-4.5","effort":"low","cost_usd":0.008314,"raw_usage":{"total_tokens":1859,"prompt_tokens":662,"num_sources_used":0,"completion_tokens":113,"cost_in_usd_ticks":83140000,"prompt_tokens_details":{"text_tokens":662,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1084,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":662,"tokens_out":113,"duration_ms":7240,"temperature":1.0,"reasoning_tokens":1084,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-11T11:36:24.509952+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"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.","supporting_citations":[],"review_version":1}