{"id":"423d0321-6f06-4c0c-831d-88266a24206c","arxiv_id":"2509.01843","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For residual characteristic 2, each depth-zero supercuspidal representation of SL(2,F) restricts to a maximal compact subgroup as a direct sum of explicitly constructed representations I(1,u,ℓ), indexed by square classes, with counts that grow without bound when char(F)=2.","lead":"This paper solves the previously open wild case of branching rules: for fields of residual characteristic 2, it decomposes every depth-zero supercuspidal representation of SL(2,F) when restricted to a maximal compact subgroup, and constructs each resulting piece explicitly from nilpotent orbits. The number of pieces grows without bound for Laurent series fields of characteristic 2, and the new machinery gives a representation-theoretic version of the local character expansion","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: Theorem 7.12 is underpinned by the explicit Section 7 Mackey construction, not by the dyadic intertwining classification; recommend UNCHANGED.","rationale":"The reader's verdict identifies Corollary 3.6 as the weakest assumption, but the central branching theorem is not actually dependent on that classification. The proof of Theorem 7.12 uses Corollary 7.11, which derives the decomposition of each Mackey component σ(ℓ) via an explicit Mackey decomposition of J(ω,ℓ) and the irreducibility theorem 7.9. Neither of those arguments uses the dyadic four-way classification of Proposition 3.5/Corollary 3.6; they rely on Lemma 3.3 and square-class structure. Thus even a subtle error in the intertwining-count computations of Section 5 would not propagate to the final branching formula. This makes the paper more robust than the reader's 'weakest assumption' suggests. I independently checked the arithmetic statements: Lemma 3.3, Proposition 3.5, and Corollary 3.6 are internally consistent for the small cases I examined; the image-size computation for δ=2e is correct because the additive map x↦ιx+x^2 on the residue field has kernel {0,ι}. The proof of Theorem 7.9 contains a compressed double-coset parametrization, but the asserted representatives align with the Bruhat decomposition and with the q=2 case, so I do not regard this as a demonstrated flaw. In sum, I see no load-bearing objection to ACCEPT; the verdict should remain unchanged. A prudent verification of Corollary 3.6 would still be valuable as an independent check of the auxiliary intertwining classification.","tokens_in":33848,"tokens_out":26992,"duration_ms":265945,"concrete_test":"Verify Corollary 3.6 by brute-force enumeration for F=Q2 (e=1, q=2) and F=F2((t)) (e=∞, q=2): for δ=1..4 (with ℓ=δ+1), enumerate a,d ∈ (1+P)/(1+P^{δ+1}) satisfying a≡d mod P^δ and ad≡1 mod P^{δ+1}, compute the image of ρ(a,d)=(a−d)+P^{δ+1}, and check it matches the four-way case split, in particular |M|=q/2 for δ=2e. If any case fails, recompute the support set (5.9) and, if necessary, the branching formula.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central claim. The final branching formula Theorem 7.12 rests on Corollary 7.11, which decomposes each Mackey component σ(ℓ) explicitly as ⊕_{u∈S_⌈ℓ/2⌉} I(1,u,ℓ). That proof goes through Proposition 7.10 and Theorem 7.9 by a direct Mackey argument; it does not invoke Theorem 5.5/Corollary 3.6. A wrong case in Corollary 3.6 would invalidate the intertwining-classification Theorem 5.5 and the endomorphism-dimension Corollary 5.6, but would not by itself change the branching formula, whose index set S_m is fixed by the square-class Lemma 3.1. My spot-checks of Lemma 3.3, Proposition 3.5, Corollary 3.6, and the S_m cardinalities are consistent, including the δ=2e image of x↦ιx+x^2 of size q/2. The only mildly compressed step is the double-coset parametrization in Theorem 7.9, stated as 'following a similar strategy to Proposition 5.3'; it is consistent with the Bruhat decomposition and with the q=2 special case, and no counterexample emerged. Accordingly the paper's central claim appears sound.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper determines the complete branching rules for restriction to a maximal compact subgroup K' = SL(2,R) of every irreducible depth-zero supercuspidal representation of G' = SL(2,F) when F has residual characteristic 2. The main result (Theorem 7.12) gives an explicit multiplicity-one decomposition: either π = Ind_{K'}^{G'} σ and Res_{K'} π = σ ⊕ ⊕_{ℓ even} ⊕_{u∈S_{ℓ/2}} I(1,u,ℓ), or π has no K'_+-fixed vectors and Res_{K'} π = ⊕_{ℓ odd} ⊕_{u∈S_{(ℓ+1)/2}} I(1,u,ℓ). The irreducible components I(1,u,ℓ) are constructed by induction from characters attached to degenerate (−ℓ,−ℓ/2) cosets of nilpotent elements. The proof combines Mackey theory, a detailed dyadic arithmetic analysis of intertwining operators (Sections 3 and 5), and a new construction of irreducible K'-representations from nilpotent orbits (Section 7). Applications include representation-theoretic analogues of the local character expansion (Section 8).","tokens_in":34214,"tokens_out":30593,"duration_ms":318063,"significance":"This is a substantial advance: it completes the depth-zero supercuspidal branching problem for SL(2) in the previously inaccessible residual-characteristic-2 case. The paper reveals a genuinely new phenomenon—the number of irreducible components at fixed depth grows without bound when char(F)=2—in contrast to the constant multiplicity for odd p. It also gives explicit, geometric constructions of all components in terms of nilpotent orbits and formulates local character expansion analogues. The main proof is detailed and internally consistent; on spot checks the cardinalities, dimension bookkeeping, and the independence of the branching formula from the dyadic intertwining classification all hold. The explicit constructions and finiteness statements are important assets.","major_comments":[],"minor_comments":[{"comment":"The displayed isomorphism Res_{K_1} π ≅ (1−q)1 ⊕ Res_{K_1} τ_GL(O,1) appears to have the wrong sign. By Theorem 4.1, Res_{K_1} π = (q−1)1 ⊕ Res_{K_1} τ_GL(O,ω), and Res_{K_1} J(ω,ℓ) is independent of ω because ω is trivial on K_1. Thus the scalar should be (q−1)1, not (1−q)1. If a Grothendieck-group identity with a different formal coefficient is intended, please state this explicitly.","section":"§8.2.2"},{"comment":"The double-coset representatives for Γ(ℓ)\\K/Γ(ℓ) and Γ(ℓ)'\\K'/Γ(ℓ)' are stated without proof ('following a similar strategy to Proposition 5.3'). Since these parametrizations control irreducibility and the square-class indexing, please add a short proof or a precise lemma.","section":"Theorem 7.9"},{"comment":"The equality in Theorem 8.6 is only valid in the Grothendieck group, since nπ is negative; please include this qualification in the theorem statement and in Theorem 1.2 in the introduction.","section":"Theorem 1.2 / Theorem 8.6"},{"comment":"The notation π^{K'_ℓ} is used as a direct summand but has not been defined as a subrepresentation; earlier in §8.1 it denotes the fixed subspace. Since K'_ℓ is normal, the fixed subspace is a K′-subrepresentation; please state this explicitly.","section":"Theorem 8.10"}],"recommendation":"minor_revision","confidential_remarks":"The positive assessments of the reader and skeptic match my reading. The central branching theorem and the construction of I(ζ,u,ℓ) are sound; the issues are local and easily fixable. The paper is well suited to the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Karaganis and Nevins have done it: the complete branching rules for depth-zero supercuspidal representations of SL(2,F) when F has residual characteristic 2. This case was explicitly excluded in all earlier work, including Nevins's own, and the paper delivers not just a decomposition but an explicit construction of every irreducible component from degenerate nilpotent cosets. That is a real result, not a translation of the p-odd arguments.\n\nThe strongest part is the Section 7 construction. The representations I(1,u,ℓ) are built concretely, their irreducibility and distinctness are proved via Mackey theory, and Theorem 7.12 states the branching formula cleanly. The growth of the component count with ℓ in characteristic 2 is a genuinely new phenomenon, and the local-character-expansion analogues in Section 8 are a thoughtful payoff. The dyadic arithmetic in Section 3 is intricate; my spot checks of Lemma 3.1, Proposition 3.5, and Corollary 3.6 came out consistent, including the q/2 image in the δ=2e case.\n\nThe soft spots are all cosmetic. Theorem 1.3 says \"2qe + 1\" summands while the body consistently gives 2qe; Remark 8.7 claims e=1 in the p-odd comparison when the correct value is e=0; and the Section 6 commutator argument is a bit soft, but Section 6 is explicitly not used later. None of these touches the main theorem.\n\nI also want to note the stress-test concern about the dyadic classification. It is true that Corollary 3.6 is a delicate load-bearing piece for the intertwining results of Section 5. But the branching formula of Theorem 7.12 does not actually depend on that classification: the index set S_{⌈ℓ/2⌉} comes from the square-class lemma, and Corollary 7.11 gets the decomposition by direct Mackey construction. So even if a subcase of Proposition 3.5 needed adjustment, the central formula would survive. That is reassuring.\n\nThis paper deserves a serious referee. It is important, detailed, and the core argument is credible. The referee's job is to check the Section 7 Mackey calculations and clean up the small typos, not to reopen the main conclusion. I would accept it.","headline":"This paper closes the last open rank-one branching case for depth-zero supercuspidals on SL(2,F), and the central argument holds up despite a few cosmetic slips.","tokens_in":34822,"tokens_out":3508,"would_cite":true,"duration_ms":32933,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper gives the complete decomposition of the restriction to a maximal compact subgroup of every depth-zero supercuspidal representation of SL(2,F) when F has residual characteristic 2.","keywords":["branching rules","supercuspidal representations","residual characteristic 2","SL(2, F)","nilpotent orbits","local character expansion","Mackey theory","p-adic groups"],"falsifier":"Enumerate, for F=Q2 and δ=2, ℓ=4, all pairs (a,d)∈(1+P)² with a≡d mod P² and ad≡1 mod P⁴, and check that (a−d) mod P³ is always the same single residue; the theory predicts exactly one class. Finding two distinct residues would refute Corollary 3.6 and change the number of components at depth ℓ.","tokens_in":33649,"feed_emoji":"🧮","tokens_out":5544,"duration_ms":54566,"temperature":0.7,"pith_summary":"This paper completes the branching rules for depth-zero supercuspidal representations of SL(2,F) when F is a 2-adic field or a Laurent series field of characteristic two. It proves that restricting such a representation to a maximal compact subgroup yields a direct sum of irreducible components whose number and depth are governed by square classes modulo successive congruence subgroups. Each positive-depth component is explicitly constructed as an induced representation from a degenerate nilpotent coset, so the branching rules are controlled by the geometry of nilpotent orbits. The same machinery yields a representation-theoretic analogue of the local character expansion that remains meaningful in characteristic two, where the classical character expansion is unavailable.","feed_headline":"Branching rules solved for SL(2,F) in residual characteristic 2","feed_subtitle":"Every component is built from a nilpotent coset; counts grow without bound when char(F)=2.","key_machinery":"The Mackey components σ(ℓ)=Ind_{B'_ℓ}^{K'} g_ℓ σ, obtained from Hansen's GL(2) branching rules, are the objects to decompose. Their self-intertwining dimension is computed in Theorem 5.5 as |S_{⌈ℓ/2⌉}| using Corollary 3.6, a dyadic arithmetic classification of pairs a,d ≡ 1 mod P with a≡d mod P^δ and ad≡1 mod P^ℓ. The irreducible pieces I(ζ,u,ℓ) are built by Clifford theory from the character η_{(u,ℓ)} of the congruence subgroup K'_{⌈ℓ/2+⌉}, extended to the subgroup Γ(ℓ)', whose size sharply reflects the dichotomy at ℓ=4e.","core_discovery":"The central result (Theorem 7.12) gives the complete branching rules for any irreducible depth-zero supercuspidal representation π of SL(2,F) with F of residual characteristic 2. If π has no fixed vectors under K'_+, its restriction to the maximal compact subgroup K' is a direct sum, over odd depths ℓ, of |S_{(ℓ+1)/2}| irreducible components I(1,u,ℓ); otherwise π is induced from a cuspidal σ of the finite group SL(2,f) and the restriction is σ plus a direct sum, over even positive depths, of |S_{ℓ/2}| components. Each component I(1,u,ℓ) is constructed in (7.8) as an induction from a character attached to the degenerate (−ℓ,−ℓ/2) coset of the nilpotent element Xuϖ^{−ℓ}. The index set S_m cons","pith_inferences":["The same σ(ℓ)-breaking method may apply to positive-depth supercuspidals of SL(2,F), with the finite cuspidal σ replaced by a Bushnell–Kutzko type; the I(ζ,u,ℓ) family is expected to dominate those branching rules too.","The dyadic arithmetic lemma (Corollary 3.6) is a statement about the field's squaring map, so the same four-case dichotomy should control intertwining in other rank-one groups in residual characteristic 2.","When q=2, the Hecke-algebra computation in Section 6 suggests that End_{K'}(σ(ℓ)) is isomorphic to the group algebra of a cyclic group of order |S_{⌈ℓ/2⌉}| for ℓ ≥ 4e+1; verifying this in general would give an explicit basis of intertwiners.","The 'close cousins' phenomenon—nilpotent orbits collapsing into one degenerate coset—might offer a way to formulate orbital integrals in characteristic 2, connecting to germ-expansion analogues."],"forward_implications":["For F of residual characteristic 2, the restriction of any depth-zero supercuspidal representation to K' is now explicitly known, component by component, for every depth.","The number of irreducible components at depth ℓ is |S_{⌈ℓ/2⌉}|, which stabilizes at 2qe for ℓ ≥ 4e+1 in the 2-adic case and grows without bound when char(F)=2.","The representations I(ζ,u,ℓ) provide an explicit family of irreducible K'-representations of prescribed depth arising from nilpotent orbits, extending the p-odd picture.","The local character expansion has a representation-theoretic version (Theorem 8.6) holding on K'_{4e+1} for 2-adic fields, and a family of local expansions (Theorem 8.10) valid in all characteristics, including char F=2.","The dimension of the largest irreducible component in π^{K'_n} grows at a rate that can be computed exactly, exhibiting a half-factor at the boundary ℓ=4e."],"supporting_citations":[{"why":"supplies the GL(2,F) branching rules (Theorem 4.1) whose restriction to SL(2,F) yields the Mackey components σ(ℓ).","marker":"[Han87]"},{"why":"gives the construction of depth-zero supercuspidals from cuspidal representations, the source of π0(σ) and π1(σ).","marker":"[MP96]"},{"why":"provides the character table of GL(2,F_q) used to compute the traces χ_ℓ and intertwining numbers.","marker":"[DM91]"},{"why":"established the p-odd branching rules that this paper extends to residual characteristic 2, providing the contrast case.","marker":"[Nev13]"},{"why":"introduced the representation-theoretic local character expansion framework that Theorems 8.6 and 8.10 adapt to p=2.","marker":"[Nev24]"},{"why":"supplies the square-class representatives in 2-adic fields used to define the index sets S_m.","marker":"[Cas23]"},{"why":"provides the Mackey decomposition theorem (Proposition 2.1) used throughout Sections 4 and 5.","marker":"[Kut77]"}],"fun_headline_variants":["Branching rules for depth-zero supercuspidals in char 2","SL(2,F) branching solved, even in wild char 2","Nilpotent orbits build all branching components for SL(2)","Complete branching rules for SL(2) in residual char 2","Wild char 2: explicit branching for supercuspidal reps"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is a specific arithmetic fact about squaring in the field: for units congruent to 1, the residue of (a−d) in the quotient P^δ/P^{δ+1} follows a four-case pattern, with the delicate case δ=2e giving a half-size image because the map x↦ιx+x² has a two-element kernel.","fun_headline_variants_meta":{"raw":{"variants":["Branching rules for depth-zero supercuspidals in char 2","SL(2,F) branching solved, even in wild char 2","Nilpotent orbits build all branching components for SL(2)","Complete branching rules for SL(2) in residual char 2","Wild char 2: explicit branching for supercuspidal reps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000721,"raw_usage":{"total_tokens":3029,"prompt_tokens":659,"completion_tokens":2370,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":403,"completion_tokens_details":{"reasoning_tokens":2278}},"tokens_in":403,"tokens_out":2370,"duration_ms":19641,"temperature":1.0,"reasoning_tokens":2278,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:10:34.851712+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate, for F=Q2 and δ=2, ℓ=4, all pairs (a,d)∈(1+P)² with a≡d mod P² and ad≡1 mod P⁴, and check that (a−d) mod P³ is always the same single residue; the theory predicts exactly one class. Finding two distinct residues would refute Corollary 3.6 and change the number of components at depth ℓ.","supporting_citations":[],"review_version":1}