{"id":"248838ba-a9e6-40d5-a096-b513a3ed7890","arxiv_id":"2502.01505","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every reductive group over a p-adic field, a character has depth zero if and only if it is trivial on the pro-unipotent radical of every parahoric subgroup, and this agrees with the torus-based definition.","lead":"This mathematics paper proves that several natural ways to define depth zero for characters of p-adic groups agree, even for groups with wild ramification. It also proves structural properties of the Langlands correspondence for tori, giving a reliable base for classifying depth-zero representations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lie-algebra identification in Proposition 3.3 fails in positive characteristic, leaving Theorem 1.4 unsupported for non-archimedean local fields of characteristic p.","rationale":"I agree with the reader that Proposition 3.3 is the load-bearing step, but the reader's formulation locates the risk in the Kaletha–Prasad isomorphism. The sharper problem is in the sentence before (3.9): the Lie algebra of Gsc is identified with gder. For local fields of characteristic p this is false whenever the isogeny from the simply connected cover is inseparable, e.g. SL_p → PGL_p. The diagram (3.10) then compares the wrong associated graded spaces; the surjectivity of the lower-right map with gder overestimates the tangent space of the group-level source. The rest of the depth-zero equivalence (Theorem 3.4, Lemma 3.5) depends exactly on Proposition 3.3, so the central claim is not fully proved at the stated level of generality. The concern is internal, not a disagreement with standard consensus, and it can be tested by the PGL_p computation above. I would therefore move the verdict to CONDITIONAL: accept if the authors add a characteristic-zero hypothesis or supply a corrected proof for characteristic p.","tokens_in":11021,"tokens_out":40250,"duration_ms":369567,"concrete_test":"Set G = PGL_p over F = F_p((t)), let T′ be the diagonal split torus, and choose a vertex f in the standard apartment. Compute the associated graded map d: t′(Fnr)_r ⊕ sl_p(Fnr)_f,r → pgl_p(Fnr)_f,r induced by the product map. If d is surjective for all r, Proposition 3.3 may be repairable by replacing gder with the true image of Lie(Gsc); if not, compute the group-level product T′_{0+} · Im(SL_p(F) → PGL_p(F)) and check whether it contains G_{f,0+}. This directly settles whether the stated proof's Lie-algebra step, and Proposition 3.3 itself, holds in characteristic p.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Proposition 3.3 is the load-bearing technical step for Theorems 3.4 and Lemma 3.5. Its proof states: 'The Lie algebra of T′ × Gsc is t′ ⊕ gder' (before (3.9)) and then uses the surjectivity of (3.11) with gder in the lower row of diagram (3.10). This identification is valid only when the central isogeny Gsc → Gder is separable. Over a non-archimedean local field of characteristic p, it fails for groups with p-torsion in the fundamental group. For example, for G = PGL_p over F_p((t)), the differential Lie(SL_p) → Lie(PGL_p) has image the derived subalgebra, of codimension 1, so Lie(Gsc) is not Lie(Gder). Therefore the associated graded space in the lower row of (3.10) is not the tangent space of the domain (T′_r × Gsc,f,r)/(T′_s × Gsc,f,s), and the surjectivity proved for t′ ⊕ gder does not imply the needed surjectivity for t′ ⊕ dq(Lie(Gsc)). Since the paper claims Theorem 1.4 for every non-archimedean local field without a characteristic-zero assumption, the depth-zero equivalence is not established in positive characteristic. The theorem may still be true, but this step is a gap as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the local Langlands correspondence (LLC) for arbitrary tori over local fields and gives a detailed analysis of depth-zero characters of reductive p-adic groups, allowing wildly ramified groups. The main results are: (i) a norm-compatibility property for the LLC for tori (Proposition 1.2/2.1); (ii) a depth-zero restriction of the LLC for arbitrary tori (Proposition 1.3/2.2); (iii) a proof of bijectivity of Langlands' map H^1(W_F, Z(G^∨)) → Hom(G/G_sc, C^×) (Theorem 3.1); and (iv) several equivalent characterizations of depth-zero characters of G, defined either by restriction to maximal tori or by triviality on pro-unipotent radicals of parahoric subgroups (Theorems 3.4 and Lemma 3.5, summarized as Theorem 1.4). The paper is written for non-archimedean local fields of arbitrary characteristic, with the wildly ramified case highlighted as the main difficulty.","tokens_in":11287,"tokens_out":9963,"duration_ms":92411,"significance":"If the results are correct, the paper provides a clean and uniform description of depth-zero characters of arbitrary reductive p-adic groups, removing tameness hypotheses that appear in earlier work. Proposition 2.2 is a genuinely new statement for wildly ramified tori, and Theorem 3.1 fills a gap in the literature by proving bijectivity of Langlands' homomorphism. The equivalence of the torus and parahoric notions of depth zero is an important structural result that is likely to be useful in the authors' companion work on the LLC for depth-zero representations. The proofs are explicit, use commutative diagrams, and carefully distinguish new content from cited results. The main weakness is a gap in Proposition 3.3 for positive characteristic, which currently leaves Theorem 1.4 unsupported in that setting.","major_comments":[{"comment":"The proof states 'The Lie algebra of T′ × Gsc is t′ ⊕ gder' immediately before (3.9). This identification is valid only when the differential of the central isogeny q: Gsc → Gder is surjective, which fails for inseparable isogenies in positive characteristic. For example, for G = PGL_p over F_p((t)), the image of dq: Lie(SL_p) → Lie(PGL_p) is the derived subalgebra of Lie(PGL_p), a proper subspace of codimension 1. Consequently, the lower row of diagram (3.10) does not represent the tangent space of the domain (T′_r × Gsc,f,r)/(T′_s × Gsc,f,s), and the surjectivity of the addition map (3.11) for t′ ⊕ gder does not, as written, imply the surjectivity of the left vertical map (3.12). Since Proposition 3.3 is the load-bearing step for Theorem 3.4 and Lemma 3.5, and since Theorem 1.4 is claimed for every non-archimedean local field without a characteristic-zero assumption, the proof is incomplete in positive characteristic. The authors should either add a characteristic-zero hypothesis to Theorem 1.4 or supply a separate argument for inseparable isogenies verifying that t′(Fnr)_r + dq(Lie(Gsc))(Fnr)_{f,r} = g(Fnr)_{f,r} for the appropriate graded pieces.","section":"§3, Proposition 3.3, diagram (3.10)–(3.12)"}],"minor_comments":[{"comment":"The proof invokes the existence of a z-extension (3.1) with H^1(F,D) = 1 without giving a proof or a precise reference. This is a standard construction (e.g., via induced tori R_{E/F} G_m and Hilbert 90), but a citation would help the reader, especially since the argument relies on it.","section":"§3, Theorem 3.1, equation (3.1)"},{"comment":"There is a typographical error in the abstract: 'a rbi-trary' should be 'arbitrary'.","section":"Abstract"},{"comment":"In the proof of Proposition 2.1, representatives ̅γ for cosets in W_F/W_K are used in formulas (2.2) and (2.3) without an explicit statement that such representatives are chosen once and for all; a short clarifying sentence would improve readability.","section":"§2, Proposition 2.1, proof of (2.2)–(2.3)"},{"comment":"The phrase 'for one chamber C' could be misread as 'for a single, unspecified chamber'; the intended meaning is 'for some chamber C' or 'for any chamber C', since the proof shows all chambers are equivalent by transitivity of G on chambers.","section":"§3, Theorem 3.4(b)"}],"recommendation":"major_revision","confidential_remarks":"The decisive issue is the positive-characteristic gap in Proposition 3.3. The tori results (Propositions 2.1 and 2.2) appear sound, and Theorem 3.1 is a useful contribution, but Theorem 1.4 as stated over all non-archimedean local fields cannot stand without a fix. I would recommend that the authors either prove the required surjectivity for inseparable isogenies or explicitly restrict the main theorem to characteristic zero. The latter would still be a valuable paper, but the stated scope would be weaker."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth reading for two results that look solid: Proposition 2.2 gives the depth-zero piece of the LLC for tori without tameness assumptions, and Theorem 3.1 finally writes down a proof of the bijectivity of Langlands' map from H^1(W_F, Z(G∨)) to Hom(G/G_sc, C×). Both are clean and the exposition is honest about what is new.\n\nThe soft spot is real and specific. In Proposition 3.3, the proof states that the Lie algebra of T' × G_sc is t' ⊕ g_der. That is only true when the central isogeny G_sc → G_der is separable. For local fields of characteristic p, e.g. G = PGL_p over F_p((t)), the differential of SL_p → PGL_p is not surjective; its image is the derived subalgebra of Lie(PGL_p), one codimension short. So diagram (3.10) has the wrong object in the lower row. The surjectivity proven for t' ⊕ g_der → g does not imply surjectivity of t' ⊕ Lie(G_sc) → g, and the p-adic limit step that transfers the surjectivity back to the group does not go through. This gap is load-bearing: Theorem 3.4, Lemma 3.5, and ultimately Theorem 1.4 rely on Proposition 3.3. For characteristic zero or tamely ramified groups the proof is fine; for wildly ramified groups in positive characteristic, the equivalence of the depth-zero definitions is not established as written. The theorem may be true, but the proof needs repair.\n\nThe rest of the paper looks careful. The citation to Kaletha-Prasad is weighty but appropriate; the companion papers are used only as motivation, not as premises. No fitted constants, no circularity.\n\nMy verdict: send to a serious referee. The torus results alone are publishable, and the main theorem is probably fixable. But a referee should insist on a corrected proof of Proposition 3.3 for the inseparable case, or an explicit restriction on F.","headline":"Solid torus results and a needed proof of Langlands' bijection, but the main theorem's proof has a characteristic-p gap in Proposition 3.3.","tokens_in":11831,"tokens_out":6596,"would_cite":true,"duration_ms":62531,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11S37","22E50","20G25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Depth-zero characters of reductive p-adic groups admit one uniform description, even when the group is wildly ramified.","keywords":["local Langlands correspondence","depth-zero characters","p-adic groups","wildly ramified groups","Moy–Prasad filtrations","Bruhat–Tits building","tori","parahoric subgroups"],"falsifier":"The claim would be refuted by exhibiting a wildly ramified torus $T$ and a smooth character $\\chi$ trivial on $T_{0+}$ whose Langlands parameter is nontrivial on the wild inertia group $P_F$, or by a reductive group $G$ with a character $\\chi$ trivial on $G_{sc}$ that satisfies one of the four depth-zero conditions and violates another.","tokens_in":10806,"feed_emoji":"🧪","tokens_out":9295,"duration_ms":72904,"temperature":0.7,"pith_summary":"The paper establishes that, for any connected reductive group over a non-archimedean local field, the several natural definitions of a depth-zero character all agree, with no tameness assumption on the group. A character trivial on the simply connected cover of the derived group has depth zero on every maximal torus exactly when it kills the pro-unipotent radical of every parahoric subgroup, and checking one suitably chosen torus or one Iwahori subgroup is enough. Along the way, the authors prove that the local Langlands correspondence for arbitrary tori restricts to an isomorphism between depth-zero characters and depth-zero Langlands parameters, and that Langlands' natural map from $H^1(W_F, Z(G^\\vee))$ to characters of $G/G_{sc}$ is bijective. These results make depth-zero characters a uniformly accessible class, which the authors use as input for local Langlands correspondences for depth-zero representations.","feed_headline":"Depth-zero characters agree for wildly ramified p-adic groups","feed_subtitle":"A single torus check or single parahoric check decides depth zero, opening depth-zero local Langlands to wild cases.","key_machinery":"The central mechanism is the surjectivity of the product map $T'_r \\times G_{sc,f,r} \\to G_{f,r}$ at level $r = 0+$ (Proposition 3.3), proved for every facet $f$ of the Bruhat–Tits building. It is obtained by passing to the associated graded Moy–Prasad isomorphisms $G_{f,r}/G_{f,s} \\cong \\mathfrak{g}_{f,r}/\\mathfrak{g}_{f,s}$ and checking surjectivity of the Lie-algebra addition map $\\mathfrak{t}'(\\mathbb{F}^{nr})_r \\oplus \\mathfrak{g}_{der}(\\mathbb{F}^{nr})_{f,r} \\to \\mathfrak{g}(\\mathbb{F}^{nr})_{f,r}$ on the unramified splitting, then transferring the surjectivity back to the group level via $p$-adic completeness. For the torus results, the load-bearing identity is Proposition 2.1: the LLC for tori sends the norm map $N_{E/F}$ to restriction $\\operatorname{Res}^{W_F}_{W_E}$, which makes the depth-zero restriction follow from Pontryagin duality and the structure of $I_F/P_F$.","core_discovery":"For an arbitrary torus $T$ over a non-archimedean local field $F$, the local Langlands correspondence restricts to an isomorphism\n$$\\operatorname{Hom}(T/T_{0+},\\mathbb{C}^\\times) \\cong $H^{1}$(W_F/P_F, T^\\vee{}^{P_F}),$$\nidentifying characters trivial on the pro-$p$ radical $T_{0+}$ of the parahoric subgroup $T_0$ with cocycles trivial on wild inertia $P_F$. For any connected reductive group $G$, a character $\\chi$ trivial on the image of the simply connected cover $G_{sc}$ has depth zero on every maximal torus if and only if its kernel contains the pro-unipotent radical $G_{f,0+}$ of every parahoric subgroup; one Iwahori subgroup or one maximal torus containing a maximal unramified torus already suffices. The proof also establishes that Langlands' map $H^1(W_F, Z(G^\\vee)) \\to \\operatorname{Hom}(G/G_{sc},\\mathbb{C}^\\times)$ is a natural topological isomorphism.","pith_inferences":["Beyond the paper, the same equivalence of depth-zero conditions might extend to characters not assumed trivial on $G_{sc}$, provided the anisotropic part of the derived group is handled separately; the authors do not claim this.","Beyond the paper, Theorem 3.1 implies that the full character group of $G/G_{sc}$ is governed by $H^1(W_F, Z(G^\\vee))$, and Lemma 3.2 identifies the depth-zero part in the tame case; whether the equality $X_0(G^\\vee) = H^1(W_F/P_F, Z(G^\\vee)^{P_F})$ holds for wildly ramified groups is not settled here.","Beyond the paper, Proposition 2.1 invites a base-change statement for depth-zero LLC: restriction of a parameter to $W_E$ should correspond to the norm of the character, which could give a purely local route to cyclic base change for depth-zero representations."],"forward_implications":["For every connected reductive group over a non-archimedean local field, tensoring a depth-zero representation by a character in $X_0(G)$ preserves depth zero, so the depth-zero category is stable under such twists.","The depth-zero part of the LLC for tori is now available for arbitrary tori: $\\operatorname{Hom}(T/T_{0+},\\mathbb{C}^\\times) \\cong H^1(W_F/P_F, T^\\vee{}^{P_F})$ even when $T$ splits only over a wildly ramified extension.","Langlands' map $H^1(W_F, Z(G^\\vee)) \\to \\operatorname{Hom}(G/G_{sc},\\mathbb{C}^\\times)$ is a natural isomorphism of topological groups for all connected reductive groups over local fields.","The equivalence of the four depth-zero conditions reduces checking depth zero to a single maximal torus containing a maximal unramified torus, which is the practical criterion for applications.","For separable extensions, the character associated to the restriction of a parameter is the norm-twisted character, a naturality property not previously recorded."],"supporting_citations":[{"why":"Establishes the local Langlands correspondence for tori, the object that Proposition 2.2 refines at depth zero.","marker":"[Lan2]"},{"why":"Supplies the LLC for tori with depth preservation in the tame case and the construction used for the norm-compatibility proof.","marker":"[Yu]"},{"why":"Provides the LLC for tori and Langlands' map from $H^1(W_F, Z(G^\\vee))$ to characters of $G/G_{sc}$ that Theorem 3.1 proves bijective.","marker":"[Bor]"},{"why":"Gives the weakly unramified isomorphism $\\operatorname{Hom}(T/T_0,\\mathbb{C}^\\times) \\cong H^1(W_F/I_F, T^\\vee{}^{I_F})$ used as a base step in Proposition 2.2.","marker":"[Hai]"},{"why":"Provides the corestriction formula and the structure of $I_F/P_F$ as having no nontrivial pro-$p$ quotients, used in Proposition 2.2.","marker":"[NSW]"},{"why":"Supplies the Bruhat–Tits theory and the Moy–Prasad isomorphism theorem that Proposition 3.3 relies on.","marker":"[KaPr]"},{"why":"Defines the Moy–Prasad filtration subgroups $G_{f,r}$ that underlie all parahoric depth-zero conditions.","marker":"[MoPr]"},{"why":"Provides the Kottwitz isomorphism used in Theorem 3.1 to show $(D^\\vee)^{W_F}$ is connected and $H^1(F,G_{sc})$ is trivial.","marker":"[Kot]"}],"fun_headline_variants":["Depth-zero: one check suffices for wildly ramified p-adic groups","Single Iwahori or torus decides depth-zero character property","Wild p-adic groups: depth-zero definitions now coincide","Depth-zero characters: equivalence extends to wild cases","Pro-unipotent radical test unifies depth-zero for p-adic groups"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that a certain technical comparison between the filtration of the group and that of its Lie algebra, originally proved for tamer settings, remains valid when the group is wildly ramified; if that comparison fails anywhere, the equivalence between the torus and parahoric tests for depth zero breaks down.","fun_headline_variants_meta":{"raw":{"variants":["Depth-zero: one check suffices for wildly ramified p-adic groups","Single Iwahori or torus decides depth-zero character property","Wild p-adic groups: depth-zero definitions now coincide","Depth-zero characters: equivalence extends to wild cases","Pro-unipotent radical test unifies depth-zero for p-adic groups"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000485,"raw_usage":{"total_tokens":2338,"prompt_tokens":837,"completion_tokens":1501,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":453,"completion_tokens_details":{"reasoning_tokens":1414}},"tokens_in":453,"tokens_out":1501,"duration_ms":13549,"temperature":1.0,"reasoning_tokens":1414,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T15:06:20.352724+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The claim would be refuted by exhibiting a wildly ramified torus $T$ and a smooth character $\\chi$ trivial on $T_{0+}$ whose Langlands parameter is nontrivial on the wild inertia group $P_F$, or by a reductive group $G$ with a character $\\chi$ trivial on $G_{sc}$ that satisfies one of the four depth-zero conditions and violates another.","supporting_citations":[],"review_version":1}