{"id":"5479eea6-0da6-4a02-9ed9-ea9f8e4dd238","arxiv_id":"2509.22390","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The optimal standard local converse theorem for Sp_{2N}, SO_{2N}, SO_{2N+1}, and G2 requires twisting up to roughly half the dual group's standard representation dimension, except for an improved SO_{2N} bound for odd N.","lead":"This paper proves that local converse theorems for symplectic groups, even and odd special orthogonal groups, and the exceptional group G2 are sharp: twisting by GL_r gamma factors with r about half the dimension of the standard representation of the dual group is necessary. It also gives a counterexample to a related conjecture for GL_4, showing that character twists alone cannot distinguish all supercuspidal representations even with all exterior-power gamma factors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.7 overstates a Galois-side counterexample as refuting the automorphic Ramakrishnan conjecture; the GL4 exterior-power gamma-factor compatibility is not established.","rationale":"The reader's verdict cites Theorem 1.7 scope as the reason for CONDITIONAL, and my stress-test agrees that this is the most load-bearing concern. The reader's weakest_assumption, however, focused on Proposition 2.10 and the matching conditions; those constructions appear internally consistent after correcting the evident degree typo in Theorem 3.8 (the self-dual pairs must have degree M, not 2M, for the dimension count in the displayed parameter to match N* = 2N or 2N+1). The main sharpness results for classical groups and G2 are proven at the level of Langlands parameters and then transferred to generic supercuspidal representations via known LLC results; that transfer is standard and I found no countervailing gap. The genuine soft spot is the framing of Theorem 1.7: the equality of Galois-side gamma factors does not automatically refute an automorphic conjecture unless the compatibility of those gamma factors with automorphic ones is established. The paper partially discloses this in Section 5 but the abstract and Theorem 1.7 state the conclusion unconditionally. My recommended verdict is therefore UNCHANGED: CONDITIONAL is appropriate pending this clarification.","tokens_in":27385,"tokens_out":51908,"duration_ms":440431,"concrete_test":"Check whether the automorphic exterior-square gamma factor for the GL4(F) supercuspidal representation associated to the admissible pair (E/F, chi) in Section 5 equals the Galois-side gamma(s,(V^2 o rho_chi) otimes eta, psi) used in Theorem 1.7. A citation of such a compatibility result, e.g. via Kim's exterior-square functoriality for GL4, would settle whether the Ramakrishnan conjecture is actually disproved. If no such compatibility exists, Theorem 1.7 should be restated as a Galois-side counterexample only.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The sharpness theorems for classical groups and G2 rest on Proposition 2.10 and the LLC framework, and I do not find a fatal gap there. The load-bearing weakness is in Section 5: Theorem 1.7 is presented as a disproof 'in general' of the Ramakrishnan/YZ22 conjecture, but the equality in (5.1) is for exterior-power gamma factors defined on the Galois side, via gamma(s,(V^i o phi) otimes eta, psi). The conjecture it targets concerns automorphic exterior-power gamma factors for representations of GL4(F). The paper notes in Section 5 that for N>=6 it is interpreting factors on the Galois side, but the theorem and abstract do not carry that caveat. Unless the local Langlands correspondence for GL4 is known to preserve exterior-square gamma factors, or these factors are shown to match Langlands-Shahidi factors, the counterexample does not logically touch the automorphic conjecture. This does not affect Corollaries 3.6/3.9 or Theorem 4.6, which are parameter-level and LLC-based, but it is a real scope overstatement in a headline result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies sharpness in local converse theorems for split classical groups and G2 over non-archimedean local fields of characteristic 0. The main technical engine is a pair of propositions (Propositions 2.7 and 2.10), proved from Moy's formulas, giving criteria for two direct sums of tame supercuspidal Weil-group representations to have equal twisted gamma factors against all representations of dimension below a given bound. Using these criteria, the authors construct generic tempered non-cuspidal representations of Sp_{2N}, SO_{2N}, SO_{2N+1} that are gamma-equivalent to level N-1 but not outer conjugate, and supercuspidal examples at level M-1. For odd N they prove an unconditional improved bound for SO_{2N} and reduce the symplectic and odd orthogonal cases to Conjecture 3.10. For G2 they prove a local converse theorem at level 3 and show sharpness at level 2 even when both fundamental representations of the dual group are used. The final section gives Galois-side parameters for GL_4 whose exterior-power gamma factors agree under all character twists, and the authors interpret this as disproving a conjecture of Ramakrishnan and Ye--Zelingher.","tokens_in":27651,"tokens_out":12817,"duration_ms":113393,"significance":"If the main results stand, the paper gives a substantial and essentially complete answer to the sharpness question for standard local converse theorems in the tame setting, including the first such results for G2. The strengths of the paper are the self-contained proof of the key technical propositions from Moy's Gauss-sum formulas, the explicit and checkable constructions, and the honest separation of proved results from Conjecture 3.10. The authors also correctly note the conditional nature of the improved bound for symplectic and odd orthogonal groups. However, the headline claim in Section 5 goes beyond what is logically established, and the proof of Corollary 3.18 has a gap concerning local conjugacy. These issues do not appear to affect the central classical-group and G2 sharpness theorems, but they do require correction before the paper can be accepted.","major_comments":[{"comment":"Theorem 1.7 is presented as a disproof of the Ramakrishnan/Ye–Zelingher conjecture, but the equality proved in (5.1) is for Galois-side exterior-power gamma factors γ(s,(∧^i φ_j)⊗η,ψ). The conjecture concerns automorphic exterior-power gamma factors for representations of GL_N(F). The paper does not cite or prove a compatibility theorem asserting that the local Langlands correspondence for GL_4 preserves these exterior-power gamma factors for i≥2; hence the constructed parameters need not contradict the automorphic conjecture as stated. The caveat in §5 that the factors are interpreted on the Galois side is not carried into Theorem 1.7 or the abstract. Please restrict the theorem to the Galois-side/parameter-level statement, or supply the missing compatibility argument.","section":"§5, Theorem 1.7 and Eq. (5.1)"},{"comment":"The proof of Corollary 3.18 shows that the inflated parameters φ'_1 and φ'_2 are outer equivalent and not conjugate, but it does not verify the defining condition of local conjugacy, namely equality after composing with every algebraic representation of SO_{2N}(C). In particular, the half-spin representations V_N^± are not discussed. Example 3.17 supplies the needed half-spin equality only for SO_6, and the inflation step from SO_6 to SO_{2N} is not shown to preserve it. Please add the missing argument, or cite a lemma from [Yu22] that establishes local conjugacy directly.","section":"§3.4, Corollary 3.18"}],"minor_comments":[{"comment":"The phrase 'all irreducible irreducibler-dimensional representations' should read 'all irreducible r-dimensional representations'.","section":"§4.1, Definition 4.1"},{"comment":"The sentence 'for N≥6 the requisite γ-functions have not yet been defined on the automorphic side' is confusing immediately before a theorem about N=4; please clarify that the Galois-side interpretation is needed for the exterior powers under consideration even when N=4.","section":"§5, first paragraph"},{"comment":"The reference [KT] is listed as a URL without a year; please add publication details if available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is a strong contribution to the local converse problem, and the classical-group and G2 sharpness results appear sound and well-motivated. The Section 5 overstatement is fixable by rephrasing, and the Corollary 3.18 issue is a genuine but localized gap that the authors should be asked to repair before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know about arXiv:2509.22390 is that the sharpness results are real and the main machinery is proven in the paper. Adrian and Stevens show that for split Sp and SO groups and for G2, under a largeness condition on p, the standard local converse theorem cannot be improved: you need GL_r twists up to roughly half the dimension of the standard representation. For supercuspidals of SO_{2N} with N odd they get the better bound N−1 and prove it is optimal, and for G2 they show even adding the 14-dimensional adjoint representation at level 2 does not distinguish their examples. These are substantial results, carrying the earlier GL_N sharpness methods over to all split classical groups and an exceptional group. The technical engine, Proposition 2.10, is proved from Moy's Gauss-sum formulas rather than quoted. That gives me confidence in the unconditional results.\n\nThe soft spot is Section 5. As stated in the abstract and Theorem 1.7, the paper claims to disprove the Ramakrishnan/YZ22 conjecture for GL4. What is actually proved is equality of exterior-power gamma factors defined on the Galois side, using the LLC for GL4. The paper itself notes in Section 5 that for N≥6 it is interpreting the factors on the Galois side, but the headline and abstract do not carry that caveat for the GL4 case. Unless those Galois-side exterior-power factors are known to match the automorphic Langlands–Shahidi factors, the example is a counterexample to a Galois-side analogue, not to the automorphic conjecture. This is a scope overstatement in a headline result, not a gap in the main body. The classical-group and G2 sections do not depend on it.\n\nI also want to give credit where it is earned: the paper is honest elsewhere. Conjecture 3.10 is labeled a conjecture, with no evidence claimed for the scope conditions, and the conditional improvement for Sp and SO_{2N+1} is clearly flagged. The removal of the central-character assumption in Theorem 3.2 is a useful stand-alone contribution.\n\nWho gets value from this paper: anyone working on local converse problems, sharpness, or gamma-factor characterizations of L-packets. It deserves a serious referee, and the referee should push for a rewritten Section 5 that explicitly separates the Galois-side statement from the automorphic conjecture. My recommendation: send it to peer review, expecting that revision rather than any reworking of the main results.","headline":"A strong sharpness paper for local converse theorems in classical groups and G2, with one overreach in the GL4/Ramakrishnan section that should be reframed.","tokens_in":28144,"tokens_out":3721,"would_cite":true,"duration_ms":33459,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11S70","22E50","11F85","22E55"],"pacs":[],"model":"deepseek-v4-flash","headline":"For symplectic, orthogonal, and $G_2$ groups with large residual characteristic, the optimal local converse theorem requires twisting through half the dimension of the standard representation of the dual group.","keywords":["local converse problem","twisted gamma factors","admissible pairs","supercuspidal representations","classical groups","G2","sharpness","Langlands parameters"],"falsifier":"Check the explicit pairs of Theorem 3.8 at level $M-1$: if any irreducible twist $\\tau$ of dimension $M-1$ produced unequal gamma-factors, the criterion of Proposition 2.10 would be violated and the sharpness claim would collapse. For the conjectural odd-$N$ improvement, find self-dual supercuspidal representations of $GL_{2N}(F)$ of the same parity that are $\\gamma$-equivalent to level $N-1$ but not isomorphic; such a pair would refute Conjecture 3.10 and the conjectured converse for symplectic and odd orthogonal groups.","tokens_in":27167,"feed_emoji":"🎯","tokens_out":12520,"duration_ms":98767,"temperature":0.7,"pith_summary":"The paper determines, for large residual characteristic, the exact number of $GL_r$-twists needed to distinguish generic representations of split classical groups and of the exceptional group $G_2$ by their twisted $\\gamma$-factors. It shows that for $G_N=Sp_{2N},SO_{2N},SO_{2N+1}$ with $p>N$, the standard local converse theorem is optimal: twisting by $GL_r$ through $r=N$ (half the dimension of the standard representation of the dual group) is necessary, already among generic tempered non-cuspidal representations. For generic supercuspidal representations the required level drops by one when $G=SO_{2N}$ with $N$ odd, and the paper conjectures the same drop for symplectic and odd orthogonal groups. For $G_2$ with $p>3$, level 3 is necessary and sufficient, and level 2 fails even if both fundamental representations of the dual group are used. A separate construction for $GL_4$ disproves the idea that character twists together with all exterior-power $\\gamma$-factors can separate supercuspidal representations.","feed_headline":"Converse theorems for classical groups proven sharp at level N","feed_subtitle":"Symplectic and orthogonal groups need GL_r twists through r=N; G2 needs level 3 even with both fundamental representations.","key_machinery":"The engine is Proposition 2.10, a $\\gamma$-factor comparison criterion for direct sums $\\bigoplus_i \\rho_{\\chi_i}$ of Weil-group representations induced from totally ramified admissible pairs. Admissible pairs $(E/F,\\chi)$ are tame objects: a tamely ramified extension $E/F$ together with a quasi-character $\\chi$ of $E^\\times$ not inflated from a proper subfield, and their induced representations exhaust the tame irreducible Weil-group representations (Theorem 2.2). Proposition 2.10 says that equality of all twisted $\\gamma$-factors against representations of dimension less than $r$ follows from three conditions: the characters $\\chi_i,\\chi_i'$ agree on the filtration subgroup $U^{t_i}_{E_i}$; the products $\\prod_i \\chi_i(\\beta_i)$ and $\\prod_i \\chi_i'(\\beta_i)$ agree; and the products $\\prod_i \\chi_i$ and $\\prod_i \\chi_i'$ agree on $F^\\times$. The proof reduces to a product formula for $\\gamma$-factors of tensor products (Proposition 2.7) and a Gauss-sum computation; the sharpness examples are precisely choices of admissible pairs that satisfy these conditions while remaining inequivalent. A second mechanism is Lemma 2.12, which produces at least $p$ self-dual admissible characters of either parity, needed for the supercuspidal counterexamples.","core_discovery":"The central discovery is a family of sharpness counterexamples built from minimal totally ramified admissible pairs $(E/F,\\chi)$ and $(E/F,\\chi')$ that agree on $U_E$ but differ by a sign on a uniformizer. Under the condition $p>N$, Proposition 2.10 gives equality of all standard twisted $\\gamma$-factors against every irreducible twist of dimension below $N$, while the two parameters are not outer conjugate. These pairs produce generic tempered counterexamples at level $N-1$ and, after passing through self-dual pairs constructed by a fixed-point correspondence for pro-$p$ characters, supercuspidal counterexamples at level $2\\lfloor N/2\\rfloor-1$ (Corollaries 3.6 and 3.9). For $SO_{2N}$ with odd $N$, a determinant obstruction rules out the last potential irreducible $2N$-dimensional orthogonal component, yielding the unconditional level $N-1$ converse for supercuspidals (Theorem 3.16). For $G_2$, cubic admissible pairs pulled back through $SL_3\\subset G_2$ give inequivalent discrete parameters that agree at levels 1 and 2 against both the 7-dimensional standard and 14-dimensional adjoint representations, while Theorem 4.2 makes level 3 sufficient. The $GL_4$ counterexample takes a totally ramified quartic admissible character and its twist by the unramified quadratic character; these are inequivalent, yet every exterior-power twist by every character agrees.","pith_inferences":["One could test whether the same unramified-quadratic-twist construction produces $GL_{2m}$ counterexamples for every even $2m$, not just $GL_4$; if so, character-twist-only converse theorems fail in all even ranks.","The determinant obstruction behind Theorem 3.16 suggests a general principle: whenever the relevant self-dual Weil-group representations are forced to have nontrivial determinant, the odd-$N$ improvement should hold; this may transfer to other groups whose duals have similar parity restrictions.","The sharpness examples are constructed at the level of Langlands parameters; with currently available local Langlands correspondences for other groups, the same admissible-pair technique could plausibly produce analogous counterexamples for classical groups in positive characteristic."],"forward_implications":["For $p>N$, the standard local converse theorem for $Sp_{2N}$, $SO_{2N}$, and $SO_{2N+1}$ cannot be improved: the bound $r\\le N$ is necessary, so the existing theorems are optimal in the setting of generic tempered representations.","For generic supercuspidal representations, the $SO_{2N}$ case with odd $N$ is settled at level $N-1$; if Conjecture 3.10 holds, all classical groups attain the same improved bound and Corollary 3.9 becomes optimal.","For $G_2$ with $p>3$, a local converse theorem must twist by $GL_3$; the counterexample at level 2 works with both fundamental representations, so including the adjoint representation does not rescue a level-2 theorem.","The $SO_6$ construction makes $SO_{2N}(C)$ generically $WD_F$-unacceptable for $q\\equiv 3\\pmod 4$: some outer-equivalent parameters have identical twisted $\\gamma$-factors for every algebraic representation of the dual group, so for even orthogonal groups no local converse theorem based on all dual-group representations can distinguish such parameters.","The $GL_4$ example rules out the proposed shortcut of using only character twists with all exterior-power $\\gamma$-factors to separate supercuspidal representations, even before the same question is considered for $GL_N$ with $N\\ge 5$."],"supporting_citations":[{"why":"Supplies the tame admissible-pair parametrization of irreducible Weil-group representations used by Proposition 2.10.","marker":"[Moy86]"},{"why":"Gives the sharpness strategy for the $GL_N$ local converse theorem in the prime case and the tensor-product gamma-factor lemmas (Lemmas 2.4 and 2.5) used in the proof.","marker":"[ALST18]"},{"why":"Extends $GL_N$ sharpness to general $N$ and provides the gamma-factor comparison underlying Proposition 2.7.","marker":"[Adr23]"},{"why":"Establishes the generator-based Local Converse Theorem and the $WD_F$-acceptability framework that the paper sharpens and extends.","marker":"[Mat24]"},{"why":"Provides the $G_2$ local Langlands correspondence that turns the constructed parameters into supercuspidal representations.","marker":"[GS23a]"},{"why":"Defines $G_2\\times GL_r$ gamma-factors via Langlands parameters, making the $G_2$ converse theorem and its sharpness statement meaningful.","marker":"[GS23b]"},{"why":"Gives the Arthur classification for classical groups used in Section 3 to identify representations with Langlands parameters.","marker":"[Art13]"},{"why":"Ensures uniqueness of the generic representation in an outer conjugacy class, converting gamma-factor equality into outer conjugacy in Lemma 3.1.","marker":"[Var17]"},{"why":"Prove the standard local converse theorems for odd orthogonal, symplectic, and even orthogonal groups respectively; these are the baselines whose sharpness is established.","marker":"[JS03, Zha18, HL25]"}],"fun_headline_variants":["Optimal standard twists: GL_r up to half-dim for classical groups","G2 needs level 3 twists; level 2 fails to distinguish","Generic supercuspidals improve SO_2N converse to level N-1","Counterexamples: no non-standard twists for GL, G2, SO","Sharp local converse: half-dim GL twists for Sp, SO, G2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on having residual characteristic $p$ larger than the twisting level ($p>N$, or $p>3$ for $G_2$), so that every irreducible representation of dimension below $r$ is induced from an admissible pair; if that tameness hypothesis fails, the constructed counterexamples are not known to exist.","fun_headline_variants_meta":{"raw":{"variants":["Optimal standard twists: GL_r up to half-dim for classical groups","G2 needs level 3 twists; level 2 fails to distinguish","Generic supercuspidals improve SO_2N converse to level N-1","Counterexamples: no non-standard twists for GL, G2, SO","Sharp local converse: half-dim GL twists for Sp, SO, G2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001167,"raw_usage":{"total_tokens":4881,"prompt_tokens":1052,"completion_tokens":3829,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":668,"completion_tokens_details":{"reasoning_tokens":3730}},"tokens_in":668,"tokens_out":3829,"duration_ms":26128,"temperature":1.0,"reasoning_tokens":3730,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:44:40.242241+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the explicit pairs of Theorem 3.8 at level $M-1$: if any irreducible twist $\\tau$ of dimension $M-1$ produced unequal gamma-factors, the criterion of Proposition 2.10 would be violated and the sharpness claim would collapse. For the conjectural odd-$N$ improvement, find self-dual supercuspidal representations of $GL_{2N}(F)$ of the same parity that are $\\gamma$-equivalent to level $N-1$ but not isomorphic; such a pair would refute Conjecture 3.10 and the conjectured converse for symplectic and odd orthogonal groups.","supporting_citations":[],"review_version":2}