{"id":"b08ac6c8-4c46-4ca3-b0d2-a643c2779d6d","arxiv_id":"2501.18553","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For tame reductive p-adic groups with residue field of at least four elements, a Yu-type input without the second genericity condition induces a finite set of irreducible supercuspidal representations.","lead":"This paper constructs tame supercuspidal representations of p-adic reductive groups in residue characteristic two, where earlier general constructions stopped. It generalizes Yu's 2001 construction, drops a genericity condition, and gives a new treatment of Heisenberg-Weil representations over F2.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem A's q>2 claim is not proved: Lemma 4.6.7 fails at q=3, and the footnote delegation to [Yu01]+[Fin21] for the relaxed no-GE2 input is unverified; the q>3 theorem in the abstract remains plausible.","rationale":"Reading the paper in good faith, the main construction for q>3 is coherent: the Heisenberg-Weil theory for p=2 is developed in detail, the uniqueness arguments rest on R-linearizations and Lemma 4.5.5, and the supercuspidality proof follows the expected Yu-Fintzen route with Lemma 4.6.7 providing the replacement for Gérardin's Theorem 2.4(b). I could not machine-check the algebraic geometry, so the reader's MODERATE confidence is appropriate. The single place where the paper overreaches its own proof is the q=3 boundary. Lemma 4.6.7 is used at the final step to get U(k_F)⊆[P(k_F),U(k_F)]; for |k|=3 the statement is false, so q>3 is not a stylistic convention. The footnote's delegation to [Yu01] and [Fin21] is plausible but not documented; if it is meant to cover inputs that fail GE2, that needs a precise citation or argument. The inverted torsion-prime sentence in Section 4.1 and Theorem 4.6.9(c) is a genuine statement-level error: the 'for example' should involve p not being a torsion prime for the dual group, not p being one. These issues do not invalidate Theorem 4.6.9(a) for q>3, so the reader's CONDITIONAL verdict stands.","tokens_in":47063,"tokens_out":37372,"duration_ms":427278,"concrete_test":"Re-run the final contradiction in Theorem 4.6.8(a) for a concrete F_3 example: take H=SL_2 over k=F_3, P the Borel, and compute U(k)∩[P(k),U(k)]; the inclusion in Lemma 4.6.7 collapses, so the proof cannot go through for q=3. Then check the cited theorems: does [Fin21, Theorem 3.1] or [Yu01] accept the relaxed input (no GE2) when q=3? If neither does, Theorem A must be weakened to q>3 and the q=3 claim removed or proven separately.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The narrowest load-bearing point is the boundary q=3. Theorem 4.6.8(a) ends its cuspidality contradiction by applying Lemma 4.6.7, which asserts U(k) is contained in [P(k),U(k)] for |k|>3. The proof of the lemma is sharp: when H=SL_2 and k=F_3, every t in k^× satisfies t^2=1, so [S(k),U_α(k)] is trivial and the claim fails. Hence the proof of Theorem 4.6.9(a) genuinely stops at q>3, matching the abstract's q≥4 restriction. The overreach is Theorem A, which states q>2; the footnote proposes to handle q=3 by combining [Yu01] and [Fin21], but it does not show that those theorems accept the relaxed input without (GE2) or at q=3. If that delegation is not valid, the stated q>2 version is unsupported. Separately, Section 4.1 and Theorem 4.6.9(c) invert the torsion-prime implication: GE2 is automatic when p is not a torsion prime, not when it is; Example 4.1.3 itself shows a non-GE2 character at p=2 while 2 is not a torsion prime for SL_2's root system, although it is for the dual PGL_2. This is a theorem-statement error that should be corrected, but it does not enter the q>3 proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs tame supercuspidal representations of a connected reductive p-adic group G(F), attached to a 'supercuspidal G-datum' analogous to Yu's input but allowing residue characteristic 2 and omitting Yu's second genericity condition (GE2). The construction proceeds by compact induction from an open compact-mod-center subgroup, using a new Heisenberg-Weil theory for Heisenberg F_p-groups when p=2 and Clifford theory to handle the nonabelian normalizer that appears when (GE2) fails. The main theorem (Theorem 4.6.9) proves, for residue field cardinality q>3, that the resulting compactly induced representations are irreducible and supercuspidal. The abstract restricts to residue fields of size at least four, while the introduction's Theorem A claims q>2 and defers q=3 to a footnote.","tokens_in":47368,"tokens_out":11616,"duration_ms":135137,"significance":"If the q>3 theorem is correct, this is a substantial advance: it extends Yu's construction of tame supercuspidal representations to residual characteristic 2 and simultaneously removes Yu's second genericity condition. The char-2 Heisenberg-Weil linearization via real/quaternionic structures, the use of order-two character obstructions, and the Clifford-theoretic passage to a larger normalizer are genuinely new mechanisms. The paper is also commendably explicit about the q>3 boundary, and it includes useful appendices on commutator results and an explicit spin-group example. The cost is that two statement-level errors, discussed below, must be corrected before the paper can be accepted.","major_comments":[{"comment":"The q>2 claim in Theorem A is not proved by the present arguments. Theorem 4.6.9 only proves the result for q>3, and the proof of Theorem 4.6.8(a) relies on Lemma 4.6.7, whose hypothesis |k|>3 is sharp: for H=SL_2 over F_3, every t in F_3^× satisfies t^2=1, so [S(k),U_α(k)] is trivial and U(k) is not contained in [P(k),U(k)]. The footnote's delegation of q=3 to a combination of [Yu01] and [Fin21] is not a proof that those references accept the relaxed input Υ without (GE2) at q=3; [Yu01] imposes (GE2), and no specific result of [Fin21] is cited that performs the relaxation. Since the abstract already states q≥4, the simplest fix is to state Theorem A for q>3 only, or to supply a genuine q=3 argument.","section":"§1 (Theorem A), footnote 2; §4.6, Lemma 4.6.7 and Theorem 4.6.8(a)"},{"comment":"The torsion-prime implication is stated backwards in several places. The text says that generic characters satisfy (GE2) when p is a torsion prime for the Langlands dual group, and Theorem 4.6.9(c) repeats this; the correct statement is that (GE2) is automatic when p is not a torsion prime, as the paper itself states in the final sentence of Lemma 4.6.3 and as Example 4.1.3 demonstrates (p=2 is not a torsion prime for SL_2, yet the character constructed there fails (GE2)). This error does not enter the proof of Theorem 4.6.9(a), but it is a false assertion in the statement of part (c) and in the discussion in §4.3 of when the additional Clifford-theoretic choices are necessary.","section":"§4.1 (p. 20), §4.3, and Theorem 4.6.9(c)"}],"minor_comments":[{"comment":"The displayed statement reads 'Let σ∈Irr(~K,K,ρ⊗σ)', but it should read 'ρ⊗κ'.","section":"Theorem 4.6.9(a)"},{"comment":"The sentence 'For our there is no need to assume that A or π have a particular form' is missing a word; it should read 'For our purposes'.","section":"§3.2, after Lemma 3.2.2"},{"comment":"Reference [Cot25] cites a MathOverflow post; consider replacing it with a published or more permanent source if one is available.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The core q>3 construction appears sound and the paper is a serious contribution. The main issues are local and fixable: restrict Theorem A to q>3 (matching the abstract and Theorem 4.6.9) and correct the inverted torsion-prime statements. The self-citations to [Fin21] are appropriate given the direct dependency of the proof. I do not see circularity in the central construction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a real advance, not a repackaging. Fintzen and Schwein build supercuspidal representations in residue characteristic two, remove Yu's second genericity condition, and do it with a uniform input. The Heisenberg–Weil theory over F2 with R-linearizations, the Clifford theory step for disconnected normalizers, and the avoidance of Gérardin's theorem are genuinely new and non-routine. The main theorem for q>3 is supported by a long, structured proof. I could not machine-check it, but the central argument looks sound to me, and the abstract's q>=4 restriction matches Theorem 4.6.9. The new pieces—the two isomorphism classes of Heisenberg F2-groups, the nonsplit sequence for Aut_Z(H) when n>=3, and the use of real/quaternionic structure to pin down the Weil representation—are real mathematics. Credit where due: the paper honestly states its boundary and does not pretend the q=3 case is handled inside the main proof.\n\nNow the soft spots, in proportion. First, Theorem A states q>2, but the proof only gives q>3. The footnote delegates q=3 to a combination of [Yu01] and [Fin21], but does not show that those theorems accept the relaxed input without (GE2). If that delegation cannot be made rigorous, the q>2 claim in Theorem A is unsupported. The abstract is fine because it says q>=4; the discrepancy is between Theorem A and the rest of the paper. This needs to be fixed, not by hand-waving, but by proving the q=3 case or changing Theorem A.\n\nSecond, the torsion-prime implication is inverted in Section 4.1 and again in Theorem 4.6.9(c). GE2 is automatic when p is not a torsion prime for the dual group, not when it is. Example 4.1.3 itself illustrates this: an order-two character of an elliptic torus in SL2 at p=2, even though 2 is not a torsion prime for the SL2 root system (it is for PGL2). This is a genuine theorem-statement error, though it does not enter the q>3 proof. It will mislead readers and should be corrected.\n\nThe Lemma 4.6.7 boundary at |k|=3 is real and the authors know it; that is exactly why q=3 is excluded from Theorem 4.6.9. I do not see a hidden fatal flaw there—the proof genuinely stops at q>3, and the statements in the abstract match that.\n\nWho is this for? Anyone working on p-adic groups, supercuspidal constructions, or the Yu machinery. It deserves a serious referee and a prominent venue once the statement-level issues are fixed. I would send it to review and expect a revise-and-resubmit.","headline":"Genuine q>=4 construction of tame supercuspidals in residue characteristic 2; Theorem A's q>2 claim and the torsion-prime statements need correction before publication.","tokens_in":47885,"tokens_out":2243,"would_cite":true,"duration_ms":23835,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50","11F27"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs supercuspidal representations of tame reductive p-adic groups in every residue characteristic, including p=2, by compact induction from a Yu-type input that no longer needs Yu's second genericity condition; the…","keywords":["supercuspidal representations","p-adic groups","Heisenberg–Weil representations","residue characteristic two","tame ramification","compact induction","Moy–Prasad filtrations","Clifford theory"],"falsifier":"For $k=F_3$, take $H=SL_2$ with its Borel subgroup $P$ and unipotent radical $U$: then $U(F_3)\\cong F_3$ has order 3 while $[P(F_3),U(F_3)]$ is trivial, so Lemma 4.6.7 fails exactly at $q=3$ and the theorem excludes this case. To test the positive claim, carry out the construction of Example 4.4.2 for a 2-adic field with residue field $F_4$ and compute the intertwining algebra of the resulting compactly induced representation: Theorem 4.6.9(a) predicts it is one-dimensional, so any nonzero extra intertwiner would disprove the theorem.","tokens_in":46853,"feed_emoji":"🧮","tokens_out":27228,"duration_ms":254233,"temperature":0.7,"pith_summary":"The paper aims to show that supercuspidal representations—the atomic building blocks of smooth representations of p-adic groups—can be constructed uniformly in every residue characteristic, including $p=2$. It extends Yu's 2001 construction, which required odd residue characteristic and two genericity conditions on its input, by removing the second genericity condition (GE2) and by repairing the characteristic-two mechanism that had blocked earlier work. The main theorem says that for any connected reductive group over a nonarchimedean local field that splits over a tamely ramified extension, any supercuspidal datum in the paper's sense yields an irreducible supercuspidal representation of $G(F)$ by compact induction, as long as the residue field has more than three elements. If the paper is right, the supercuspidal representations built by Yu—and in the classical setting all supercuspidal representations—are special cases of a single recipe that also covers the previously inaccessible $p=2$ case.","feed_headline":"Tame supercuspidals constructed in every residue characteristic","feed_subtitle":"Works when the residue field has at least four elements and covers the previously missing p=2 case.","key_machinery":"The engine of the paper is the theory of Heisenberg $F_p$-groups and their Weil representations, rebuilt for $p=2$. A Heisenberg $F_p$-group is a finite $p$-group whose center has order $p$ and whose quotient by the center is an $F_p$-vector space; for odd $p$ these are the extraspecial $p$-groups of exponent $p$, while for $p=2$ there are two isomorphism classes of each order, built as central products of $D_8$ and $Q_8$. Each generic character $\\varphi_i$ in the input produces a Heisenberg $F_p$-group $V_i^\\natural$ inside a Moy–Prasad filtration quotient, and the Heisenberg representation $\\omega_i$ of this group is the seed of $\\kappa^-$. The new difficulty is the Weil representation: for $p=2$, the group $\\mathrm{Aut}_Z(H)$ of center-fixing automorphisms of a Heisenberg group $H$ sits in a nonsplit extension $1\\to F_2^{2n}\\to \\mathrm{Aut}_Z(H)\\to O_{2n}(F_2)\\to 1$, so the projective Weil representation cannot be linearized globally; the paper proves it linearizes over the particular subgroups arising from the p-adic group (Lemma 3.4.9) and pins down the linearization uniquely by requiring it to preserve the real or quaternionic structure of $\\omega_i$, using the absence of order-two characters in the relevant quotient when $q>2$ (Lemma 4.5.5 and Proposition 4.5.6). The final supercuspidality argument rests on two small facts: if $|k|>3$, then for a quasi-split reductive $k$-group $H$ with parabolic $P$ and unipotent radical $U$ one has $U(k)\\subseteq [P(k),U(k)]$ (Lemma 4.6.7), and a general group-theoretic observation that when $U\\subseteq [P,U]$, a representation must act trivially on $U$ (Lemma 4.6.6).","core_discovery":"The central theorem (Theorem 4.6.9(a)) asserts that, when the residue field $k_F$ has more than three elements, every supercuspidal $G$-datum $\\Upsilon$—a chain of twisted Levi subgroups $G_1\\supseteq\\cdots\\supseteq G_{n+1}$, a point $x$ in the Bruhat–Tits building, decreasing positive depths $r_i$, a depth-zero cuspidal representation $\\rho$, and characters $\\varphi_i$ that are generic of depth $r_i$ in the sense of (GE0) and (GE1) alone—yields an irreducible supercuspidal representation $\\mathrm{c\\text{-}ind}_{\\widetilde K}^{G(F)}(\\sigma)$. The subgroup $\\widetilde K$ lies between Moy–Prasad-style subgroups $K$ and $K^+$, and $\\sigma$ is built in two steps: a Heisenberg–Weil representation $\\kappa^-$ of a smaller subgroup $K^-$ is constructed from Heisenberg $F_p$-groups $V_i^\\natural$ extracted from the filtration quotients, and then Clifford theory extends $\\kappa^-$ to $\\kappa$ on $K$ and to $\\sigma$ on $\\widetilde K$. When $p$ is odd and every $\\varphi_i$ satisfies Yu's second genericity condition (GE2), the intervening groups coincide, no choices are needed, and the construction specializes to Yu's; when $p=2$, the Heisenberg–Weil machinery is rebuilt on real and quaternionic representations, since the automorphism group of a Heisenberg $F_2$-group no longer splits as a symplectic semidirect product and its projective Weil representation has no global linearization. The paper also proves that $\\widetilde K/K$ is a finite $p$-group, trivial under (GE2), and that the compact induction from $K$ decomposes as a direct sum of the new supercuspidal representations with positive multiplicities.","pith_inferences":["The Clifford-parameterized finite set attached to each datum looks like a natural bookkeeping device for the packet structure predicted by the local Langlands correspondence when the residue characteristic is small; the paper does not pursue this, but the framework is ready for it.","The real/quaternionic linearization technique should apply to other characteristic-two settings where classical Weil representations are unavailable, such as constructing types for classical groups or extending representations of unipotent radicals in small characteristic.","The natural next classification question is whether every tame supercuspidal representation arises from this construction; since it contains Yu's representations and drops (GE2), the answer may be yes in all tame settings, with the Clifford choices accounting for the finer structure.","If Lemma 4.6.7 could be sharpened to cover $|k|=3$, the construction itself would extend to residue fields of three elements, closing the gap between Theorem A's $q>2$ and the main theorem's $q>3$."],"forward_implications":["All supercuspidal representations constructed by Yu in 2001 are recovered as a special case; in particular, when $p$ does not divide the order of the absolute Weyl group of $G$, the new construction yields all supercuspidal representations of $G(F)$.","When every $\\varphi_i$ satisfies (GE2), the subgroup $\\widetilde K$ collapses to $K$ and $\\mathrm{c\\text{-}ind}_K^{G(F)}(\\rho\\otimes\\kappa)$ is itself irreducible supercuspidal, with no Clifford choices required.","The construction works in residue characteristic two, where no general supercuspidal construction existed for arbitrary tame reductive groups, and it does so without invoking the Glauberman correspondence or Gérardin's Weil-representation analysis.","A single datum $\\Upsilon$ can produce several non-isomorphic supercuspidal representations, corresponding to the Clifford-theoretic choices of $\\kappa$ and $\\sigma$; Example D.6 shows these choices can be forced and are detected by distinct formal degrees."],"supporting_citations":[{"why":"Supplies the original construction that the paper generalizes: the input datum, the subgroups $K$ and $K^+$, generic characters, and the first half of the intertwining proof; also covers the $q=3$ case together with [Fin21].","marker":"[Yu01]"},{"why":"Provides the second half of the supercuspidality proof (its Theorem 3.1), the notation for the twisted-Levi filtration subgroups, and the $q=3$ delegation.","marker":"[Fin21]"},{"why":"Establishes the Heisenberg–Weil representation theory for odd $p$ (Lemma 1.2, Lemma 1.5, Theorem 2.4(a)) that the paper extends to $p=2$ and avoids in the supercuspidality argument.","marker":"[Gér77]"},{"why":"Gives the torsion-prime theory for reductive groups, explaining exactly when (GE2) can fail and providing the centralizer results used in Lemma 4.6.2 and Appendix D.","marker":"[Ste75]"},{"why":"Supplies the enlarged Bruhat–Tits building, Moy–Prasad subgroups, admissible embeddings, and structural facts about $\\pi_1$ and finite abelian quotients used throughout.","marker":"[KP23]"},{"why":"Classifies extraspecial $p$-groups and their automorphism groups, yielding the short exact sequences for $\\mathrm{Aut}_Z(H)$ in Fact 3.4.1.","marker":"[Win72]"},{"why":"Proves that simply-connected quasi-split groups over fields with at least four elements equal their own commutator subgroup, underpinning Lemma C.4 and the absence of order-two characters in Lemma 4.5.5.","marker":"[Tit64]"},{"why":"Supplies Lemma A.4.3 on the transitivity of the character-group action over the set of Clifford extensions, used in Lemma 4.3.1(a).","marker":"[Kal]"}],"fun_headline_variants":["Supercuspidals in every residue characteristic, including p=2","Tame supercuspidals without Yu's second genericity","Construction completes Yu's supercuspidals at p=2","Heisenberg-Weil supercuspidals for all residue fields"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the residue field $k_F$ has more than three elements: the final step of the supercuspidality proof needs the commutator fact that for a quasi-split reductive group over a field $k$ with $|k|>3$, the $k$-points of the unipotent radical of a parabolic subgroup lie in $[P(k),U(k)]$, and this fails for $k=F_3$.","fun_headline_variants_meta":{"raw":{"variants":["Supercuspidals in every residue characteristic, including p=2","Tame supercuspidals without Yu's second genericity","Construction completes Yu's supercuspidals at p=2","Heisenberg-Weil supercuspidals for all residue fields"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000528,"raw_usage":{"total_tokens":2601,"prompt_tokens":1056,"completion_tokens":1545,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":672,"completion_tokens_details":{"reasoning_tokens":1473}},"tokens_in":672,"tokens_out":1545,"duration_ms":13457,"temperature":1.0,"reasoning_tokens":1473,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T23:00:23.752134+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $k=F_3$, take $H=SL_2$ with its Borel subgroup $P$ and unipotent radical $U$: then $U(F_3)\\cong F_3$ has order 3 while $[P(F_3),U(F_3)]$ is trivial, so Lemma 4.6.7 fails exactly at $q=3$ and the theorem excludes this case. To test the positive claim, carry out the construction of Example 4.4.2 for a 2-adic field with residue field $F_4$ and compute the intertwining algebra of the resulting compactly induced representation: Theorem 4.6.9(a) predicts it is one-dimensional, so any nonzero extra intertwiner would disprove the theorem.","supporting_citations":[],"review_version":1}