{"id":"93b8860e-db50-4649-ad58-2255800d7ab6","arxiv_id":"2505.22357","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Middle supercuspidals of GL(2N,F) are characterized by twisted gamma factors against tamely ramified quasi-characters and simple supercuspidals of GL(N,F).","lead":"The paper constructs a new family of depth 1/N supercuspidal representations of GL(2N,F), called middle supercuspidals, and proves that they are uniquely determined by their gamma factors twisted by tamely ramified characters of F^× and by simple supercuspidal representations of GL(N,F). It sharpens the local converse theorem for a natural infinite family and supports a concrete refinement of Jacquet's conjecture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The support lemmas 3.6 and 3.11 establish only the necessary direction from the containment (3.7); the 'if' direction needed to identify the integration domain is not verified, so the gamma-factor formulas may be computed over the wrong region.","rationale":"The reader's conditional verdict and the identified weakest assumption match my reading. The most load-bearing point is that the support theorem (3.7) is used as if it were an equality of supports, while it is only stated as a containment. In Lemmas 3.6 and 3.11, the printed arguments establish the necessary condition on k for an element to be in the support, but the sufficiency claim is essential for the subsequent integrals: without it, the integration domains are not controlled and the volume factors in the gamma-factor formulas are unjustified. This is not a matter of disagreement with the broader conjecture; it is an internal gap in the argument as written. The second issue, the unproved independence of P in Proposition 3.10, is also serious and is correctly flagged by the reader; had I not focused on the support lemmas, this would be the natural alternative. I would not change the verdict from CONDITIONAL: the structural strategy is sound and the computations are plausible, but the proof as written should be accepted only after the support 'iff' claims and the independence of P are independently verified. The proposed concrete test is feasible: the constructions are explicit, and for N = 2 all matrix groups and functions can be written down and checked with a small symbolic computation. No ad hominem is intended; the concern is about the completeness of the proof, not the integrity of the authors.","tokens_in":22946,"tokens_out":7899,"duration_ms":91905,"concrete_test":"Fix N = 2 and a small residue field (e.g. F_p with p odd). Choose f = X^2 + 1 and a tame additive character ψ_F, and implement the definition (2.1) of W(f,χ,ζ) together with the Bessel function J(f,χ,ζ) on the explicit matrix groups J_f, U^1, and N(2N,F). Enumerate or symbolically sample matrices α of the form (3.6) with h and x_i in the ranges claimed in Lemma 3.6: for k = -1, for k = 0, and for k = -2, compute W(α) directly. Check that W(α) is nonzero exactly when k = -1, and in particular that every α in the claimed k = -1 region is indeed in the support, not merely in N(2N,F) β_f^{-1} J_f. Repeat the same computation for matrices α of the form (3.9) in Lemma 3.11 with k = -N. If any matrix in the claimed support has W(α) = 0, the volume normalization in Proposition 3.5 or the support restriction in Proposition 3.10 is incorrect.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 rests on the explicit twisted-gamma computations of Propositions 3.5 and 3.10. Both computations use the support control (3.7), quoted as Supp(W(f,χ,ζ)) ⊂ ⊔_{k∈Z} N(2N,F) β_f^k J_f. This is a containment, not a description of the support. In Lemma 3.6, the proof shows that if α of the form (3.6) lies in the support, then one must have k = -1. That is the easy 'only if' direction: it only excludes other k. But the lemma states an 'if and only if', and the 'if' direction requires proving W(f,χ,ζ)(α) ≠ 0 for every α in the claimed k = -1 region. The containment (3.7) gives no lower bound; it cannot certify non-vanishing. The same issue occurs in Lemma 3.11 for k = -N. Without this reverse direction, the integrals in Propositions 3.5 and 3.10 are taken over a set that may be strictly larger than the actual support: the volume factors (e.g. q^{2N-2} in Proposition 3.5) and the subsequent simplifying assumption that the x_i variables do not affect W could both be wrong. A second, closely related gap is in Proposition 3.10, where the leftover rational factor P(q_F^{-s}) is asserted to be independent of χ, ζ, and the central character of the simple supercuspidal twist, but the leftover integral is never displayed or analyzed. This independence is used critically in Proposition 3.12 to vary φ over all quasi-characters of k_F^× and conclude v ∈ 1+P_F. If P actually depends on φ, equation (3.12) only gives φ(v) = P_φ(q^{-s'}), which does not force v = 1, and the distinguishing argument collapses. Since Theorem 1.1 combines the middle-vs-middle distinction (Proposition 3.12) with the middle-vs-other distinction (Proposition 4.3), a failure of either support lemma or of the independence assertion removes the proof of the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a family of depth 1/N supercuspidal representations of GL(2N,F), called middle supercuspidal representations, using Bushnell–Kutzko types attached to a simple stratum associated with a quadratic residue-field extension of inertial degree 2 and ramification index N. The central result, Theorem 1.1, asserts that if two irreducible supercuspidal representations of GL(2N,F) have the same central character and the same twisted gamma factors against all tamely ramified quasi-characters of F× and all simple supercuspidal representations of GL(N,F), then they are isomorphic. The proof proceeds by explicit Rankin–Selberg computations: Propositions 3.4 and 3.5 compute the twisted gamma factor against a tamely ramified character, and Proposition 3.10 computes it against a simple supercuspidal representation, using an explicit Whittaker function and support-containment statements from [20]. Proposition 3.12 then compares two middle supercuspidals, and Section 4 uses conductor formulas to distinguish middle supercuspidals from other depth 1/N supercuspidals.","tokens_in":23347,"tokens_out":9314,"duration_ms":97950,"significance":"If the main theorem is correct, it is a meaningful advance in the local converse problem: it shows that a new class of supercuspidal representations, beyond depth-zero and simple supercuspidals, is determined by twisting against only tamely ramified characters and simple supercuspidals of the half-size group. The explicit gamma-factor formulas are also potentially useful in themselves, and the paper makes a concrete conjecture (Conjecture 1.3) that frames the result in a broader program. The authors use established machinery (maximal simple types, explicit Whittaker functions, conductor formulas) and do not introduce fitted parameters or circular reasoning. However, the central computations hinge on several unverified support and independence claims that are load-bearing for the distinguishing argument, so the result cannot be considered fully established as written.","major_comments":[{"comment":"The proof of Lemma 3.6 establishes only the necessary direction: starting from α in the support, it derives k=-1. The claimed converse, that every α of the form (3.6) with the stated conditions on h and x_i is in the support, is not proved. The containment Supp(W(f,χ,ζ)) ⊂ ⊔_k N(2N,F) β_f^k J_f quoted from (3.7) gives no lower bound on the support and cannot certify non-vanishing. The subsequent factorization in the proof of Proposition 3.5 also asserts that the displayed matrix z lies in U^1 without a complete check: for i<N, the conditions derived imply x_i ∈ P_F^{-1}, making entries such as -x_i h^{-1} c ϖ_F^{-2} of valuation -1, which are not in P_{2N}. These points matter because the integration domain, the volume factors (e.g. q^{2N-2}), and the statement that the x_i do not affect W are used to obtain the exact gamma-factor formula in Corollary 3.7 and hence Proposition 3.8. The proof should either supply a genuine if-and-only-if statement for the support, using the equality in Theorem 2.4 if that is what is intended, or present an explicit non-vanishing argument for the integration region.","section":"3.3 (Lemma 3.6, Proposition 3.5, Eq. (3.7))"},{"comment":"In Proposition 3.10 the remaining integral is asserted to be a non-zero rational function Q(q_F^{-s}) independent of ω(u,φ,ζ'), χ, and ζ, but the integral is never displayed and no argument is given for this independence. This is not a cosmetic issue: Proposition 3.12 uses the independence to conclude, from equation (3.12), that ω(u,φ,ζ')(v) is a fixed rational function independent of φ, and then to infer v ∈ 1+P_F by varying φ. If the leftover factor, or the χ_i factors in (3.12), can depend on φ, the equation only yields φ(v)=P_φ(q^{-s'}) for some φ-dependent rational function, which does not imply v=1. The authors need to display the leftover integral and prove its independence, or else find another way to extract the desired equality from (3.12).","section":"3.4 (Proposition 3.10, Proposition 3.12, Eq. (3.12))"},{"comment":"The final step asserts that χ1=χ2 because O_{L_f}^× is generated by elements of the form 1+u'σ_f with u'∈O_F^×. This generation statement is not proved and is not immediate from the structure of unramified extensions; it is also essential, since it is the only argument identifying the characters χ after the preceding steps. A proof or a precise reference for this generation property should be supplied.","section":"3.4 (end of proof of Proposition 3.12)"}],"minor_comments":[{"comment":"There are several OCR-type artifacts in the displayed text, such as 'dept h 1/N', 'Pašk¯ unas', and the placeholder '/BD_F' in equation (2.2); these should be corrected in the final version.","section":"Abstract and Introduction"},{"comment":"The notation '/BD_F' appears to be a placeholder for the trivial quasi-character; please replace it with a standard symbol such as 1_F.","section":"§2.6, Eq. (2.2)"},{"comment":"The simplification of the integral to vol(1+P_F) is terse; a sentence explaining why the ψ_βf factor is constant on the displayed integration region would improve readability.","section":"§3.3, Proposition 3.4"},{"comment":"The determinant computations used to rule out k ≥ -N are compressed; in particular, the line 'if y_{1,1} ≡ u y_{1,N+1} mod P_F and y_{N+1,1} ≡ u y_{N+1,N+1} mod P_F, then det(x) ∈ P_F' would benefit from a brief justification that this contradicts x∈J_f.","section":"§3.4, Lemma 3.11"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: this is a good paper and the central claim is likely true. What's new: a construction of 'middle supercuspidal' representations of GL(2N,F) as depth 1/N minimax types with E/F ramification index N and residue degree 2, a parametrization by triples (f̄,χ,ζ), and explicit Rankin-Selberg gamma factors for twists by tamely ramified quasi-characters and simple supercuspidals of GL(N,F). Theorem 1.1 then gives a sharp local converse theorem for this family using only those twists. That genuinely extends the simple supercuspidal case (d=1) to d=2, and the conductor argument in Section 4 separating middle from non-middle depth 1/N representations is clever. The paper is not circular: gamma factors are computed from explicit Whittaker functions, no fitted parameters, and the literature is cited appropriately.\n\nSoft spots, in order. First, the support lemmas 3.6 and 3.11 state 'if and only if' but the proofs visibly establish only the 'only if' direction: they assume α lies in the support and derive k=-1 (or k=-N). The reverse direction—that every α of that form with k=-1 lies in the support—is essential, because Propositions 3.5 and 3.10 evaluate W on the integration region using that equivalence. The explicit factorization after Lemma 3.6 may supply the missing direction, but it is not presented as part of the lemma, and the reader should not have to assemble it. Second, Proposition 3.10 asserts without displaying the leftover integral that the rational factor P(q^{-s}) is independent of χ, ζ, and the central character of the simple supercuspidal twist. That independence is load-bearing: Proposition 3.12 varies φ over all quasi-characters of k_F^× to force v ∈ 1+P_F, and if P depended on φ that conclusion would not follow. The independence may be true, but it needs to be shown. Third, a minor point: the claim that O^×_{L_f} is generated by elements 1+u′σ_f is asserted without proof; that is probably easy to fill.\n\nNone of this kills the paper; these are referee-fixable gaps in a serious project. I would send it to a careful referee, with the explicit instruction to check the reverse directions in the support lemmas and to demand the leftover integral in 3.10 be written out. If those hold up, Theorem 1.1 is a solid new result worth publishing.","headline":"New family of depth 1/N middle supercuspidals of GL(2N,F) with a sharp local converse theorem; the main computations are credible but two load-bearing gaps need referee scrutiny.","tokens_in":23957,"tokens_out":14401,"would_cite":true,"duration_ms":148725,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50","11F70"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two families of twists uniquely determine middle supercuspidals","keywords":["local converse theorem","supercuspidal representations","twisted gamma factors","middle supercuspidal","simple supercuspidal","maximal simple types","Whittaker functions","conductor formula"],"falsifier":"Evaluate the ratio of the two sides of Proposition 3.10 for a fixed middle supercuspidal and two simple supercuspidal twists that differ only in central character; the paper predicts the ratio is the same rational function of $q_F^{-s}$. If the ratio depends on the central character, the independence claim fails and the distinguishing argument collapses.","tokens_in":22710,"feed_emoji":"","tokens_out":12045,"duration_ms":121960,"temperature":0.7,"pith_summary":"This paper constructs a new family of depth $1/N$ supercuspidal representations of $\\mathrm{GL}(2N,F)$, called middle supercuspidal representations, as a natural next step after simple supercuspidals. It proves a local converse theorem for them: two irreducible supercuspidal representations with equal central characters are isomorphic once their twisted gamma factors agree for every tamely ramified character of $F^\\times$ and every simple supercuspidal representation of $\\mathrm{GL}(N,F)$. This cuts the twist set far below what the general local converse theorem allows, where all supercuspidal twists of rank up to $N$ are permitted, and it supports a depth-based refinement of the local converse conjecture. The argument is explicit: twisted gamma factors are computed from the representation's type data, and conductor inequalities rule out every other depth $1/N$ supercuspidal.","feed_headline":"Two families of twists uniquely determine middle supercuspidals","feed_subtitle":"Gamma-factor equality on these twists forces isomorphism among middle supercuspidals.","key_machinery":"The load-bearing mechanism is the explicit Whittaker function $W_{(f,\\chi,\\zeta)}$ attached to the maximal simple type $(J_f,\\Lambda_{(f,\\chi,\\zeta)})$ that builds a middle supercuspidal. Its support is contained in the disjoint union $\\bigsqcup_{k\\in\\mathbb{Z}} \\mathrm{N}(2N,F)\\,\\beta_f^k J_f$, and the core computations show that only the $k=-1$ term contributes to tame-character twists and only the $k=-N$ term contributes to simple-supercuspidal twists. This support analysis turns each twisted gamma factor into a monomial in $q_F^{-s}$ multiplied by known characters, exposing which parameters of the type each twist sees; the conductor formula for completely distinct pairs handles all remaining depth $1/N$ supercuspidals.","core_discovery":"The paper's central claim is Theorem 1.1: if $\\pi = \\pi(\\bar f,\\chi,\\zeta)$ is a middle supercuspidal representation of $\\mathrm{GL}(2N,F)$ and $\\pi'$ is any irreducible supercuspidal representation with the same central character, then equality of $\\gamma(s,\\pi \\times \\tau,\\psi_F)$ and $\\gamma(s,\\pi' \\times \\tau,\\psi_F)$ for all tamely ramified quasi-characters $\\tau$ of $F^\\times$ and all simple supercuspidal representations $\\tau$ of $\\mathrm{GL}(N,F)$ forces $\\pi \\cong \\pi'$. The gamma factors are computed explicitly: twisting by a tame character $\\lambda$ gives $\\zeta^{-1}\\lambda(-c^{-1}\\varpi_F^2)q_F^{1-2s}$, so tame twists determine $\\zeta$ and the reduction of $c$ modulo the prime ideal; twisting by a simple supercuspidal $\\pi(u,\\varphi,\\zeta')$ gives $\\zeta^{-N}\\chi(a c u + a\\sigma_f)\\,\\omega(u,\\varphi,\\zeta')(a u\\varpi_F^2)\\,P(q_F^{-s})$, with $P$ nonzero and independent of $\\chi,\\zeta,\\zeta'$, which fixes the remaining data. A conductor comparison using complete distinctness then separates middle supercuspidals from every other depth $1/N$ supercuspidal of $\\mathrm{GL}(2N,F)$, completing the characterization.","pith_inferences":["The paper does not state this, but the explicit gamma-factor formulas suggest that the parameters $\\zeta$, $c \\bmod P_F$, and $\\chi$ can be read off directly from finitely many values of the twisted gamma factors, making the converse theorem algorithmic for this family.","If the independence of the leftover factor $P(q_F^{-s})$ survives for $d>2$, the same support analysis would likely prove a local converse theorem for all minimax depth $1/N$ supercuspidals of $\\mathrm{GL}(dN,F)$; if it fails for some $d$, the set of necessary twists would need to grow.","The conductor-based exclusion of non-middle representations suggests a more general pattern: within a fixed depth, representations with different characteristic polynomials can be separated by the conductors of their twists, independent of the detailed gamma-factor shape."],"forward_implications":["A middle supercuspidal representation is determined by far fewer twists than the general local converse theorem requires: only tame characters of $F^\\times$ and simple supercuspidals of $\\mathrm{GL}(N,F)$ are needed, not all supercuspidal twists of rank up to $N$.","Equality of twisted gamma factors against the finite set $\\Xi_{\\mathrm{middle}}$ forces equality of central characters, so the central-character hypothesis in Theorem 1.1 can be dropped at the cost of adding finitely many positive-depth character twists.","The depth-filtered converse conjecture stated as Conjecture 1.3 is verified for depth-zero, simple, and middle supercuspidal representations, giving evidence that twists of depth no larger than the representation's depth should suffice.","Every depth $1/N$ supercuspidal of $\\mathrm{GL}(2N,F)$ that is not of middle type has a simple supercuspidal twist of $\\mathrm{GL}(N,F)$ with a strictly smaller conductor, so the gamma-factor test separates middle from non-middle representations.","The same method is expected to extend to depth $1/N$ minimax supercuspidal representations of $\\mathrm{GL}(dN,F)$ for general $d$, with $d=1$ recovering simple and $d=2$ recovering middle supercuspidals."],"supporting_citations":[{"why":"Classifies supercuspidal representations through maximal simple types, the construction used to build middle supercuspidals.","marker":"[8]"},{"why":"Supplies the explicit Whittaker function and its support theorem that drive the gamma-factor computations.","marker":"[20]"},{"why":"Provides the convolution-integral definition of twisted gamma factors used throughout.","marker":"[16]"},{"why":"Gives the conductor formula for pairs of supercuspidal representations that separates middle from non-middle types.","marker":"[7]"},{"why":"Establishes the Whittaker-model properties of simple supercuspidal representations used as twists.","marker":"[2]"},{"why":"Provides the finite-twist criterion for recovering central characters that Proposition 1.2 strengthens.","marker":"[17]"},{"why":"Proves the general local converse theorem that the paper refines for middle supercuspidals.","marker":"[15]"},{"why":"Supplies the depth and central-character facts used to fix the same depth and build the finite twist set.","marker":"[21]"},{"why":"Shows that high-depth twists of a representation detect only central-character data, motivating the choice of twists.","marker":"[11]"}],"fun_headline_variants":["Two twist families uniquely identify middle supercuspidals","Gamma factors on tame and simple twists fix middle supercuspidals","Middle supercuspidals uniquely determined by two twist families","Two twist families pin down middle supercuspidals","Equality of two twist gamma factors determines middle supercuspidals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument hinges on the support formula for the explicit Whittaker function and on the assertion that the leftover factor in the twisted gamma factor is nonzero and independent of $\\chi$, $\\zeta$, and the twist's central character; if either piece gives way, the proof cannot separate representations.","fun_headline_variants_meta":{"raw":{"variants":["Two twist families uniquely identify middle supercuspidals","Gamma factors on tame and simple twists fix middle supercuspidals","Middle supercuspidals uniquely determined by two twist families","Two twist families pin down middle supercuspidals","Equality of two twist gamma factors determines middle supercuspidals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000898,"raw_usage":{"total_tokens":3882,"prompt_tokens":975,"completion_tokens":2907,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":2824}},"tokens_in":591,"tokens_out":2907,"duration_ms":24055,"temperature":1.0,"reasoning_tokens":2824,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:10:35.162724+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the ratio of the two sides of Proposition 3.10 for a fixed middle supercuspidal and two simple supercuspidal twists that differ only in central character; the paper predicts the ratio is the same rational function of $q_F^{-s}$. If the ratio depends on the central character, the independence claim fails and the distinguishing argument collapses.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies supercuspidal representations through maximal simple types, the construction used to build middle supercuspidals."},{"cited_title":"Pašk¯ unas and S","cited_arxiv_id":null,"evidence_quote":"Supplies the explicit Whittaker function and its support theorem that drive the gamma-factor computations."},{"cited_title":"Jacquet, I","cited_arxiv_id":null,"evidence_quote":"Provides the convolution-integral definition of twisted gamma factors used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the conductor formula for pairs of supercuspidal representations that separates middle from non-middle types."},{"cited_title":"Adrian and B","cited_arxiv_id":null,"evidence_quote":"Establishes the Whittaker-model properties of simple supercuspidal representations used as twists."},{"cited_title":"Jiang, C","cited_arxiv_id":null,"evidence_quote":"Provides the finite-twist criterion for recovering central characters that Proposition 1.2 strengthens."},{"cited_title":"Jacquet and B","cited_arxiv_id":null,"evidence_quote":"Proves the general local converse theorem that the paper refines for middle supercuspidals."},{"cited_title":"Deligne and G","cited_arxiv_id":null,"evidence_quote":"Shows that high-depth twists of a representation detect only central-character data, motivating the choice of twists."}],"review_version":1}