{"id":"dcb7a7fb-84ee-4947-b24c-58fa29123a3c","arxiv_id":"2501.00259","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A proof that quasi-semisimple mod p G-valued Galois representations admit crystalline lifts with arbitrary fixed crystalline abelianization, for split reductive groups and tame quasi-split groups.","lead":"This paper proves that every quasi-semisimple mod p Galois representation valued in a split reductive group admits a crystalline lift with any prescribed crystalline abelianization. It extends earlier results for GL_m and for lifts without fixed abelianization, and gives a partial analogue for L-parameters of quasi-split tame groups.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.5 and the abstract's quasi-split tame claim rest on an unproved section i: G^ab -> S of S -> G^ab; tame non-split tori need not admit one (the norm Res_{E/K}G_m -> G_m is an obstruction), so the theorem needs the splitting as a hypothesis or a proof that Lin's S splits.","rationale":"The split-group part (Theorem 4.6) is well supported: Lemma 4.5 gives the required section for a split torus, and the proof is detailed. The same support is missing in Theorem 5.5. The norm obstruction is not hypothetical: it occurs already in GL_2 with the tame torus Res_{E/K}G_m, so an algebraic section cannot be assumed for arbitrary tame maximal tori. Since the proof uses the section to pass from H^1_K(G^ab) ⊕ H^1_K(S′) to H^1_K(S), the construction collapses if no such section exists. The theorem statement does list i as a datum, but the abstract states the quasi-split tame result unconditionally, and no argument is given that Lin's torus satisfies the condition. This matches the reader's weakest assumption. I would keep the verdict CONDITIONAL: the first theorem appears sound and novel, while the second theorem requires either a proof of existence of the section for the torus supplied by Theorem 5.3 or an explicit hypothesis in both theorem and abstract.","tokens_in":13641,"tokens_out":23923,"duration_ms":244976,"concrete_test":"Run the proof of Theorem 5.5 on the example K = Q_p (p odd), E the unramified quadratic extension, G = GL_2, S = Res_{E/K}G_m ⊂ GL_2, and ρ = Ind_{Gal_E}^{Gal_K} χ for a mod p character χ of Gal_E. Verify: (1) ρ is semisimple and factors through LS(F_p); (2) the map S → G^ab = G_m is the norm, so Hom_K(G_m, S) consists only of t ↦ t^n with N∘i = t^{n[E:K]}, hence no section i exists; (3) the proof's isomorphism S ≅ G^ab × S′ fails at this point. If the construction cannot be repaired by choosing a different S with a section, Theorem 5.5 needs the splitting as an explicit hypothesis; if the authors can exhibit such an S for this ρ, they still owe a proof that one always exists.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing point is the section i in Theorem 5.5. The proof uses it at the sentence 'By the existence of the right inverse of the map S → G^ab, we have S ≅ G^ab × S′', and then decomposes H^1_K(S) as H^1_K(G^ab) ⊕ H^1_K(S′) to form ρ′ = (ψ, [ρ_S′]). For K-tori, a surjection need not have a K-algebraic section. Concrete obstruction: take G = GL_2, E/K a tame quadratic extension, and S = Res_{E/K}G_m embedded via multiplication on E. The determinant, i.e. the map S → G^ab = G_m, is the field norm N. Every K-homomorphism i: G_m → S is a diagonal cocharacter t ↦ t^n, and N(i(t)) = t^{n[E:K]}; this equals id_{G_m} only if n[E:K] = 1, impossible over Z. So no right inverse exists. This S is a tame maximal torus of the kind allowed in Theorem 5.5, and a semisimple mod p L-parameter factoring through LS(F_p) can be obtained by inducing a character of Gal_E. Theorem 5.5 lists i as a datum, but it neither proves existence for the torus supplied by Theorem 5.3 nor restricts the choice of S; the abstract omits the condition entirely. The quasi-split tame theorem is therefore at best conditional, not the unconditional result announced.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies lifting problems for mod p Galois representations valued in reductive groups. For a split reductive group G over O_K, Theorem 4.6 asserts that every quasi-semisimple representation \\bar{\\rho}: Gal_K -> G(k_L) with inertia in a fixed split maximal torus admits a crystalline lift with regular Hodge-Tate weights and with abelianization equal to any prescribed crystalline lift of \\bar{\\rho}^{ab}. Theorem 5.5 makes an analogous assertion for potentially crystalline lifts of semisimple mod p L-parameters of quasi-split tame groups, using a torus S supplied by a factorization theorem of Lin. The first theorem is proven in detail using Lin's methods combined with norm computations from Böckle-Iyengar-Paškūnas. The second theorem is proven assuming the existence of a right inverse i: G^ab -> S of the natural surjection S -> G^ab, an assumption that is not proved and is not satisfied for arbitrary tame tori.","tokens_in":13901,"tokens_out":8915,"duration_ms":89469,"significance":"If Theorem 4.6 holds, it is a substantial generalization of earlier results by Lin and by Böckle-Iyengar-Paškūnas, and it is a useful step toward Zariski density of crystalline points on G-valued deformation rings. The proof is explicit and appears sound. The quasi-split tame theorem would be a natural analog, but as stated it depends on an unproved splitting condition. The paper is honest in citing external theorems, and I saw no circularity or fitted parameters. The central issue is therefore the status of Theorem 5.5 and its advertised unconditional claim.","major_comments":[{"comment":"The theorem assumes, without proof, the existence of a right inverse i: G^ab -> S of the map S -> G^ab. This hypothesis is used in the proof at the sentence \"By the existence of the right inverse of the map S → G^ab, we have S ≅ G^ab × S′,\" and the subsequent decomposition H^1_K(S) ≅ H^1_K(G^ab) ⊕ H^1_K(S′). For non-split tame tori such a section need not exist. For example, if G = GL_2 and S = Res_{E/K} G_m embedded via the regular representation of a tame quadratic extension E/K, the map S -> G^ab = G_m is the field norm; every K-homomorphism G_m -> S has the form t ↦ t^n, and its norm is t^{n[E:K]}, which is never the identity. Thus the decomposition used to form ρ′ = (ψ, [ρ_{S′}]) may be unavailable. The theorem is therefore conditional, and the proof does not establish the claim as stated.","section":"§5, Theorem 5.5"},{"comment":"The abstract states unconditionally that \"We also show analogous results in the case that G is a quasi-split tame group,\" and Theorem 1.2 is phrased without any hypothesis on the existence of the section i. Since Theorem 5.3 only guarantees that some tame K-torus S factors ρ and does not guarantee that S → G^ab admits a section, the advertised theorem is not established. The author should either prove that the torus S from Theorem 5.3 always admits such a section, or add the existence of i as an explicit hypothesis in Theorem 1.2/5.5 and adjust the abstract and introduction accordingly.","section":"Abstract and Theorem 1.2"}],"minor_comments":[{"comment":"The definition of τ^ab says it is induced by \"the composite Gab --i--> S ↪ G ↠ Gab,\" but this composite is the identity on Gab and does not define a map LS(A) -> LG^ab(A). The intended map is presumably induced by the quotient S -> G^ab; this should be corrected.","section":"§5, Theorem 5.5"},{"comment":"The notation ψ is used first for the mod p abelianization \\bar{ρ}^{ab} and then for its potentially crystalline lift; this is confusing and should be resolved, for instance by writing \\bar{ψ} for the reduction.","section":"§5, Theorem 5.5"},{"comment":"The symbol v is used both for the mod p inertia representation ρ|_IK and for a chosen lift in M^0_{T,crys}; this ambiguity should be clarified.","section":"§4, Step 3 of the proof of Theorem 4.6"},{"comment":"The term \"co-labeled Hodge-Tate characters\" is unusual; if it is translated from Lin's paper, a brief explanatory gloss would help the reader.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The split-group theorem appears correct and is a nice contribution. The main issue is the quasi-split tame theorem's missing section hypothesis. If the author can either prove existence of i for Lin's torus or revise the statement honestly, the paper would be publishable. I recommend asking for a revision that addresses the gap in Theorem 5.5 and the abstract."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The split-group part of this paper is the reason to read it. Aoki removes the GL_m restriction in the fixed-abelianization lifting problem with a global construction: for quasi-semisimple rho valued in a split reductive group, Theorem 4.6 produces a crystalline lift with regular Hodge-Tate weights and prescribed abelianization. The proof builds on Lin and Bockle-Iyengar-Paskunas honestly, with the norm calculations done in the text. Lemma 4.3 is a clean way to get regular weights without leaving the abelianization fixed, and Lemma 3.5's treatment of infinite-order Weyl elements fills a real gap in Lin's Lemma 4. The citation pattern is fine; the main result depends on external theorems, not on its own conclusion, and the one self-citation is only in a remark.\n\nThe soft spot is exactly the one the stress-test identifies. Theorem 5.5 assumes a right inverse i: G^ab -> S of the torus quotient, but the proof uses that hypothesis to conclude S is isomorphic to G^ab times S' and then splits H^1_K(S). That conclusion already requires S to be split or at least to admit such a decomposition as K-tori. For a tame non-split torus the section need not exist. The norm map Res_{E/K}G_m -> G_m is a standard obstruction: every cocharacter t -> t^n of the source composes to t^{n[E:K]}, which is never the identity. So the proof of Theorem 5.5 collapses for tori of exactly the kind Lin's factorization theorem 5.3 can supply. The theorem lists i as a datum, but it neither proves existence for the S from Theorem 5.3 nor restricts S; the abstract omits the condition entirely. That makes the advertised quasi-split tame result conditional, not the unconditional analogue promised. I agree with the reader: make the splitting a hypothesis or prove Lin's S splits.\n\nEverything else is competent. The paper is notationally dense but not impenetrable. Theorem 4.6 is a solid contribution and deserves referee time even though Theorem 5.5 needs revision. I would send it to peer review, ask the authors to fix the statement and proof of Section 5, and let Theorem 4.6 carry the paper.","headline":"The split reductive theorem is a genuine step forward and mostly convincing; the quasi-split tame theorem is conditional on an unproved splitting of S -> G^ab that the abstract omits.","tokens_in":14539,"tokens_out":2432,"would_cite":true,"duration_ms":26083,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F80","11S20","14L15"],"pacs":[],"model":"deepseek-v4-flash","headline":"Mod p Galois representations into split reductive groups admit crystalline lifts with any prescribed abelian part.","keywords":["crystalline lift","Galois representation","quasi-semisimple","abelianization","split reductive group","mod p L-parameter","Hodge-Tate weights","local class field theory"],"falsifier":"Take the quotient map from a non-split torus to the abelianized part, for example the norm map from the restriction of scalars of the multiplicative group to the multiplicative group itself for a tame extension, and check whether it has an algebraic section; if it does not and this map is the one supplied by the L-parameter factorization theorem, the theorem's construction cannot be applied as stated.","tokens_in":13342,"feed_emoji":"💎","tokens_out":11634,"duration_ms":98538,"temperature":0.7,"pith_summary":"This paper proves that the abelian part of a mod $p$ Galois representation can be fixed independently when lifting to characteristic zero. For a quasi-semisimple representation $\\rho$ with abelianization $\\rho^{\\mathrm{ab}}$, any crystalline lift $\\psi$ of $\\rho^{\\mathrm{ab}}$ is realized as the abelianization of a crystalline lift $\\rho$ of $\\rho$ with regular Hodge-Tate weights, after a finite extension of the coefficient field. The same statement is proved for potentially crystalline lifts of semisimple mod $p$ $L$-parameters of quasi-split tame groups, conditional on a section of the torus quotient. These results extend earlier constructions for general linear groups and for split reductive groups without fixed determinants, replacing case-by-case determinant adjustments with a uniform reciprocity argument. A reader should care because prescribing the determinant is often the hardest step in constructing lifts into non-abelian groups.","feed_headline":"Mod p Galois reps admit crystalline lifts with prescribed abelian part","feed_subtitle":"A single construction fixes the determinant and the Hodge-Tate weights at once, for split and tame quasi-split groups.","key_machinery":"The carrying mechanism is a norm computation in local class field theory. Given a lift $R'$ of an inertia-type character on a torus $T$, the proof shows that $(\\Xi R')^{\\mathrm{ab}}(\\mathrm{rec}_{K_f}(y)) = \\psi(\\mathrm{rec}_{K_f}(y))$ for all $y \\in K_f^\\times$, using the identity $\\mathrm{rec}_{K_f}(\\mathrm{Nm}_{K_f/K}(y)) = \\mathrm{rec}_K(y)$ up to the inclusion of Weil groups; this forces the abelianization of the extended representation to agree with $\\psi$ on $\\mathrm{Gal}_{K_f}$ and hence on all of $\\mathrm{Gal}_K$. The extension from $T$ to the normalizer $N_G(T)$ is controlled by Lemma 3.5, which characterizes when an inertia representation extends with prescribed Frobenius via the vanishing of $(\\zeta(w) \\otimes 1 - 1 \\otimes \\Phi_K)v$. A second mechanism is the co-labeled Hodge-Tate character $\\mathrm{HT}(\\rho) = (\\mathrm{HT}(\\rho)_\\sigma)_{\\sigma \\in \\Sigma_L}$, used to enforce regular Hodge-Tate weights by adding a carefully chosen crystalline torus-valued character with trivial abelianization.","core_discovery":"The central claim is that quasi-semisimplicity is the only condition needed to couple a mod $p$ representation to an arbitrary crystalline abelian lift. Theorem 4.6 states that for a connected split reductive group $G$ over $\\mathcal{O}_L$ and a quasi-semisimple representation $\\rho\\colon \\mathrm{Gal}_K \\to G(k_L)$ with $\\rho(I_K)$ contained in $T(k_L)$ and $\\rho(\\mathrm{Gal}_K)$ contained in $N_G(T)(k_L)$, a crystalline lift $\\psi$ of $\\rho^{\\mathrm{ab}}$ extends to a crystalline lift $\\rho$ of $\\rho$ with regular Hodge-Tate weights satisfying $\\rho^{\\mathrm{ab}} = \\psi$, up to a finite extension $L'/L$. Theorem 5.5 gives the analogous potentially crystalline statement for semisimple mod $p$ $L$-parameters of a connected quasi-split tame group, provided a right inverse $G^{\\mathrm{ab}} \\to S$ to the torus projection is fixed. The proof combines a reduction to elliptic representations and normalizers of tori with a reciprocity-law norm computation that forces the abelianization of the lifted representation to equal $\\psi$.","pith_inferences":["Beyond the paper's statements, the same norm/reciprocity technique should apply to lift-theoretic problems with a prescribed abelian part, such as constructing crystalline lifts with prescribed inertial types inside parahoric subgroups.","The paper's expectation that Theorem 4.6 is a first step toward Zariski density of crystalline points on $G$-valued framed deformation rings suggests that a fixed-determinant density theorem on each irreducible component is now within reach.","A concrete testable extension is to run the construction for small groups such as $\\mathrm{GSp}_4$ and check whether the regular-weight step can be done over an unramified extension without increasing ramification.","If the section $G^{\\mathrm{ab}} \\to S$ is genuinely obstructed for some tame non-split torus, a repair of Theorem 5.5 would need to replace the product decomposition $S \\cong G^{\\mathrm{ab}} \\times S'$ by a direct norm-compatible construction on $S$."],"forward_implications":["For any split reductive $G$, a crystalline lift with prescribed abelianization exists after a finite extension of coefficients, and without regularity the extension can be chosen unramified.","For $G = \\mathrm{GL}_m$, the theorem supplies crystalline lifts with an arbitrary fixed determinant, recovering the earlier determinant-fixing results as a special case.","For quasi-split tame groups, semisimple mod $p$ $L$-parameters admit potentially crystalline lifts with regular Hodge-Tate weights and prescribed abelianization, provided the torus section exists.","The combined lifting and determinant-matching step removes the need for a separate twisting argument, so the resulting lift can be made regular without disturbing the fixed abelianization."],"supporting_citations":[{"why":"supplies the reduction of quasi-semisimple representations to elliptic representations and the extension lemma for inertia representations on normalizers of tori.","marker":"[9]"},{"why":"provides the reciprocity-law norm computation and the crystalline-character adjustment used to force the abelianization of the lift to equal psi.","marker":"[4]"},{"why":"supplies the Langlands-Shelstad factorization for semisimple mod p L-parameters, the torus S in Theorem 5.3, and the definition of semisimplicity used in Theorem 5.5.","marker":"[10]"},{"why":"provides the p-adic local Langlands correspondence for tori that converts the prescribed abelianization into an L-parameter on the torus.","marker":"[3]"},{"why":"gives the description of crystalline characters as twists of unramified characters by powers of fundamental characters, used to construct the components of psi with prescribed inertial reduction.","marker":"[6]"}],"fun_headline_variants":["Quasi-semisimplicity is enough: crystalline lifts with chosen abelian part","Galois reps get crystalline lifts: abelian part can be fixed arbitrarily","Regular Hodge-Tate weights and any abelianization: crystalline lift existence","For split and tame groups: mod p lifts to crystalline with prescribed determinant"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem for L-parameters rests on the unproved assumption that one can split the quotient map from the auxiliary torus back to the abelianized part; if that splitting does not exist, the construction that produces the lift falls apart.","fun_headline_variants_meta":{"raw":{"variants":["Quasi-semisimplicity is enough: crystalline lifts with chosen abelian part","Galois reps get crystalline lifts: abelian part can be fixed arbitrarily","Regular Hodge-Tate weights and any abelianization: crystalline lift existence","For split and tame groups: mod p lifts to crystalline with prescribed determinant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000515,"raw_usage":{"total_tokens":2542,"prompt_tokens":1030,"completion_tokens":1512,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":646,"completion_tokens_details":{"reasoning_tokens":1430}},"tokens_in":646,"tokens_out":1512,"duration_ms":13881,"temperature":1.0,"reasoning_tokens":1430,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:56:56.483580+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the quotient map from a non-split torus to the abelianized part, for example the norm map from the restriction of scalars of the multiplicative group to the multiplicative group itself for a tame extension, and check whether it has an algebraic section; if it does not and this map is the one supplied by the L-parameter factorization theorem, the theorem's construction cannot be applied as stated.","supporting_citations":[{"cited_title":"Lin, Crystalline lifts and a variant of the Steinberg-Winter theor em, Doc","cited_arxiv_id":null,"evidence_quote":"supplies the reduction of quasi-semisimple representations to elliptic representations and the extension lemma for inertia representations on normalizers of tori."},{"cited_title":"Böckle, A","cited_arxiv_id":null,"evidence_quote":"provides the reciprocity-law norm computation and the crystalline-character adjustment used to force the abelianization of the lift to equal psi."},{"cited_title":"Birkbeck, On the p-adic Langlands correspondence for algebraic tori , J","cited_arxiv_id":null,"evidence_quote":"provides the p-adic local Langlands correspondence for tori that converts the prescribed abelianization into an L-parameter on the torus."},{"cited_title":"Conrad, Lifting global representations with local properties , preprint, available at https://math.stanford.edu/~conrad/papers/locchar.pdf","cited_arxiv_id":null,"evidence_quote":"gives the description of crystalline characters as twists of unramified characters by powers of fundamental characters, used to construct the components of psi with prescribed inertial reduction."}],"review_version":1}