{"id":"2a4535aa-70a1-4bd6-af5d-b7ab9a824c16","arxiv_id":"2508.11099","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Shimura varieties of Hodge type, locally analytic completed cohomology is naturally isomorphic to cohomology of flag varieties with locally analytic sheaves from perfectoid covers.","lead":"This paper proves that for Shimura varieties of Hodge type, a p-adic cohomology built from perfectoid covers matches cohomology of flag varieties. It generalizes earlier results for modular curves and unitary Shimura curves, giving a tool for p-adic automorphic forms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.8 asserts the key exactness after locally analytic vectors without proving that locally analytic vectors commute with tensor products by finite projective modules.","rationale":"The reader's weakest_assumption identifies Proposition 5.8, and I agree this is the correct locus of the main unresolved step. I disagree with the precise mechanism as stated: a surjection of finite projective O(V)-modules remains surjective after tensoring with any O(V)-module, so flatness of O(eU)^la is not the issue. The real unsupported step is the commutation of G(Qp)-locally analytic vectors with tensor products by the finite projective modules appearing in (5.20). Proposition 5.8 is the unique bridge from perfectoid cohomology on the Shimura variety to locally analytic sheaves on the flag variety, and Theorem 5.9 invokes it directly; hence this is genuinely load-bearing. The gap is likely repairable: finite direct sums and G-equivariant idempotents should reduce the claim to standard facts about locally analytic vectors of Banach representations, which supports conditional acceptance rather than rejection. I also noted a probable swapped reference in the proof of Lemma 4.14, where condition (2) appears to use the unitary decomposition that belongs to condition (1); this should be corrected in revision, but I do not rest the verdict on it. Overall, the reader's conditional verdict remains appropriate, with the requested expansion of Proposition 5.8 made more precise by specifying the missing commutation isomorphism.","tokens_in":35775,"tokens_out":14303,"duration_ms":156524,"concrete_test":"Write out the missing commutation lemma in the setting of Proposition 5.8: let H be a compact open subgroup of G(Qp), let M be a finite projective O(V)-module with an algebraic H-action, and let B = O(eU) be an H-Banach algebra. Prove or disprove (M ⊗_{O(V)} B)^la ≅ M ⊗_{O(V)} B^la by first checking the case M = O(V)^n with diagonal algebraic action, then descending along a G-equivariant finite projective idempotent. If the isomorphism fails for a rank-one character M = O(V)(χ) and B = C^{an}(H), Proposition 5.8 collapses; if it holds, insert the proof and verify that exactness of (5.17) after taking locally analytic vectors follows. This check settles whether the one-line assertion in §5.2 is valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.9 depends on Proposition 5.8, which claims the pro-Kummer étale cover eU is locally analytic. The proof identifies the exact sequence (5.17) with the sections on eU of (5.20) and then asserts that, because the map (5.22) W(O(N^c_μ)≤1)(V) → n^{c,0,∨}_μ(V) is a surjection of finite projective O(V)-modules, the exact sequence remains exact after taking locally analytic vectors, since tensoring the surjective map (5.22) with O(eU)^la remains surjective. This is not a proof. Surjectivity after tensoring is automatic, so the missing content is a different assertion: for the G(Qp)-equivariant finite projective O(V)-module M = W(O(N^c_μ)≤1)(V), and for its quotient, the natural map (M ⊗_{O(V)} O(eU))^{G(Qp)-la} → M ⊗_{O(V)} O(eU)^la must be an isomorphism, and similarly for n^{c,0,∨}_μ(V). This commutation of locally analytic vectors with tensor products by finite projective modules is exactly what is needed to pass from surjectivity of (5.22) to exactness of the locally analytic vector sequence; it is neither stated nor referenced. Without it, Proposition 5.6(3) may fail and the comparison in Theorem 5.9 loses its key input. The gap is probably fillable by standard distribution-algebra arguments, but it is load-bearing and currently unsupported.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves comparisons between locally analytic completed cohomology of Shimura varieties of Hodge type and coherent cohomology of flag varieties with locally analytic coefficients. The main result, Theorem 5.9, asserts natural G(Qp)-equivariant isomorphisms (eHi(Kp,Qp)⊗Cp)^la ≅ Hi(Fℓμ,O^la_{Kp}) and the compact-support analogue with I^la_{Kp}. A second theorem, Theorem 5.12, gives a τ-locally analytic comparison for unitary Shimura curves. The proof combines embeddings of Hodge-type Shimura varieties into Siegel modular varieties, geometric Sen theory, J-locally analytic vectors, and an analysis of locally analytic coverings of perfectoid Shimura varieties, following and generalizing work of Pan and Qiu-Su.","tokens_in":36132,"tokens_out":6566,"duration_ms":72984,"significance":"If the main theorems are correct, this is a substantial generalization: it reduces the study of locally analytic completed cohomology of Hodge-type Shimura varieties to coherent cohomology on flag varieties, extending Pan's modular-curve result and Qiu-Su's unitary-curve result. The paper is honest about the overlap with Qiu-Su for Theorem 1.3 and does not derive the theorem from itself; it relies on substantial external results of Scholze, Lan, Camargo, Pan, and Qiu-Su. The central new input is the local analyticity of perfectoid covers in Section 5, and the main theorems are clean and concrete. However, the proof of the key Proposition 5.8 contains a load-bearing gap that must be repaired before the main theorem is fully established.","major_comments":[{"comment":"The proof asserts that the surjective map W(O(N^c_μ)≤1)(V) → n^{c,0,∨}_μ(V) in (5.22) remains surjective after tensoring with O(eU)^la and concludes that the exact sequence obtained from sections of (5.20) remains exact after taking locally analytic vectors. This is not justified: surjectivity after tensoring is automatic, but the load-bearing point is the compatibility of locally analytic vectors with tensor products by finite projective modules. One must show that for M = W(O(N^c_μ)≤1)(V), the natural map (M ⊗_{O(V)} O(eU))^{G(Qp)-la} → M ⊗_{O(V)} O(eU)^la is an isomorphism, and similarly for the quotient n^{c,0,∨}_μ(V); otherwise the left-exact functor of locally analytic vectors need not preserve exactness of the sequence (5.17)/(5.19). Since this exactness is exactly condition (3) of Proposition 5.6 and is used in Theorem 5.9, the proof of the main theorem is incomplete. The gap is likely fillable by standard distribution-algebra arguments, but it must be stated and proved or explicitly cited.","section":"§5.2, Proposition 5.8"},{"comment":"The proof of the second isomorphism is not written out. The sentence beginning 'Take an open cover ⋃_i V_i = Fℓμ in analytic topology, where Then each section...' is garbled, and the claim that each section O^{τ-la}_{Kp,E}(V_i) is a smooth K^℘_p-representation follows from Proposition 5.10 is not immediate: Proposition 5.10 gives expansions over the particular U, not over arbitrary V_i. The spectral sequence E^{p,q}_2 = H^p(K^℘_p, H^q(Fℓμ, O^{τ-la}_{Kp,E})) ⇒ H^{p+q}(Fℓμ, O^{τ-la}_{K℘,E}) and its degeneration at E^{0,1}_2 need a detailed justification. This is load-bearing for Theorem 1.3.","section":"§5.4, Theorem 5.12"},{"comment":"The proof of Theorem 5.9 is a citation to Proposition 5.5, Proposition 5.7, Proposition 5.8, [32, Theorem IV.2.1], and the argument of [24, Theorem 4.4.6]. Since Proposition 5.8 is the key step establishing that the perfectoid cover is locally analytic, and its proof is incomplete as explained above, the main theorem is not yet fully supported. The author should either complete the proof or indicate precisely which statement in the literature supplies the missing compatibility of locally analytic vectors with tensor products by finite projective modules.","section":"§5.3, Theorem 5.9"}],"minor_comments":[{"comment":"There are typos: 'Hogde type' in the Section 2 heading and 'Shmura curves' in Section 1.3; these should be corrected.","section":"Title and headings"},{"comment":"In the sentence 'see also [13, §4.2] for te case of the modular curve', 'te' should be 'the'.","section":"§4.2"},{"comment":"The sentence 'Take an open cover ⋃_i V_i = Fℓμ in analytic topology, where Then each section...' is corrupted and should be rewritten as a complete sentence.","section":"§5.3, Theorem 5.12 proof"},{"comment":"The phrase 'We note ote that' is a typo and should read 'We note that'; the notation i^{-1}_{K''pKp''}(U') is also used before being clearly introduced.","section":"§5.1, Proposition 5.1 proof"},{"comment":"The statement 'For any positive integer m>0, there exists a positive integer n' should specify whether n depends on m and on the open U; as written the quantifier order is ambiguous.","section":"§5.4, Proposition 5.10"},{"comment":"The notation (eHi(Kp,Qp) b⊗_{Qp} Cp)^la is used in the introduction but 'la' is only defined in Section 4; a forward reference would help the reader.","section":"Introduction, Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The main new content of the paper is the Hodge-type generalization in Theorem 5.9; Theorem 5.12 overlaps with Qiu-Su [27], as the author acknowledges in Remark 1.4. The gap in Proposition 5.8 is the principal obstacle: it is likely fillable by standard arguments about locally analytic vectors and finite projective modules, but it is load-bearing and must be resolved before the paper can be accepted. If the author supplies the missing proof, the paper should be a solid contribution to the area."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: Theorem 5.9 is a genuine new result—the first flag-variety description of locally analytic completed cohomology for general Hodge-type Shimura varieties—and the paper is an honest, well-organized extension of Pan and Qiu-Su. But the proof as written has a load-bearing gap in Proposition 5.8 that needs to be filled before I'd trust the theorem without checking it myself.\n\nWhat's good: the statement is exactly what you'd expect, the strategy of reducing to Siegel modular varieties via the perfectoid embedding is sound, and the author clearly states which parts are already in Qiu-Su—Theorem 5.12 is flagged as a reproof. The heavy machinery (Camargo's geometric Sen theory, logarithmic adic spaces) is used appropriately, and the paper is readable for someone with Sen theory background.\n\nWhere it's soft: Proposition 5.8 claims the perfectoid cover eU is locally analytic. The proof shows the algebraic surjection (5.22) of finite projective O(V)-modules, then asserts that the exact sequence of sheaves remains exact after taking G(Qp)-locally analytic vectors because tensoring with O(eU)^la preserves surjectivity. That isn't sufficient: surjectivity on the third term is automatic; what's missing is the commutation of locally analytic vectors with tensor products by finite projective modules—an isomorphism (M ⊗_{O(V)} O(eU))^{la} ≅ M ⊗_{O(V)} O(eU)^la, and the same for the quotient. Without that, you can't conclude the kernel is O(eU)^la, and Proposition 5.6(3) doesn't follow. I suspect this is a standard distribution-algebra argument and the gap is fillable, but it's exactly the step that makes Theorem 5.9 depend on it. The stress-test note puts this correctly.\n\nTwo smaller issues: the proof of Theorem 5.12 contains an incomplete sentence—'where Then each section...'—which suggests the write-up was rushed. And Theorem 5.9's proof is essentially a pointer to [24, Theorem 4.4.6] after citing four propositions; that's fine if the propositions are solid, but the chain is long.\n\nBottom line: for people working on p-adic Langlands and eigenvarieties, this is worth reading and worth citing once the gap is addressed. I'd send it to a serious referee, conditional on the author expanding Proposition 5.8. The paper is not fatally flawed; it's a promising preprint that needs one key proof written out.","headline":"Theorem 5.9 is a plausible and genuinely new generalization, but Proposition 5.8 has an unsupported exactness assertion that needs a real proof before the main result is solid.","tokens_in":36642,"tokens_out":3579,"would_cite":true,"duration_ms":37562,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G18","14G35","11F85","14F30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Locally analytic completed cohomology of Hodge-type Shimura varieties equals coherent cohomology on a flag variety.","keywords":["locally analytic completed cohomology","Shimura varieties of Hodge type","Hodge-Tate period map","perfectoid Shimura varieties","geometric Sen theory","flag varieties","unitary Shimura curves","p-adic cohomology"],"falsifier":"Compute the first Tor term of the tensor product in (5.22) over $\\mathcal{O}(V)$ with $\\mathcal{O}(\\widetilde{U})^{\\mathrm{la}}$ for a flag-variety open $V$; if this Tor term is nonzero, the sequence would not remain exact after taking locally analytic sections, contradicting Proposition 5.8 and invalidating the proof of Theorem 5.9.","tokens_in":35580,"feed_emoji":"📐","tokens_out":8584,"duration_ms":79184,"temperature":0.7,"pith_summary":"This paper claims that the p-adic locally analytic completed cohomology of a Hodge-type Shimura variety can be computed on a flag variety. More precisely, after tensoring completed cohomology with $\\mathbb{C}_p$ and taking locally analytic vectors for the action of $G(\\mathbb{Q}_p)$, the result is isomorphic to sheaf cohomology on the Hodge-Tate flag variety with coefficients in a sheaf of locally analytic sections coming from the perfectoid cover. This matters because the flag-variety side is algebraic, while completed cohomology is a rich representation-theoretic object. The theorem extends earlier results from modular curves and unitary Shimura curves to all Hodge-type Shimura varieties, and the paper gives a matching $\\tau$-locally analytic statement for unitary Shimura curves.","feed_headline":"Locally analytic cohomology reduced to flag varieties","feed_subtitle":"For Hodge-type Shimura varieties, completed cohomology becomes sheaf cohomology on a Hodge-Tate flag variety.","key_machinery":"The load-bearing object is the Hodge-Tate period map from the perfectoid Shimura variety, together with its toroidal compactification, to the flag variety $\\mathcal{F}\\ell_\\mu$, where $\\mu$ is the Hodge cocharacter. The coefficients $\\mathcal{O}_{K^p}=\\pi^{\\mathrm{tor}}_{\\mathrm{HT},*}(\\widehat{\\mathcal{O}}_{\\mathcal{S}\\mathfrak{h}})$ and $\\mathcal{I}_{K^p}=\\pi^{\\mathrm{tor}}_{\\mathrm{HT},*}(\\widehat{\\mathcal{I}}_{\\mathcal{S}\\mathfrak{h}})$ are sheaves of topological algebras on the flag variety, and $\\mathcal{O}^{\\mathrm{la}}_{K^p}$, $\\mathcal{I}^{\\mathrm{la}}_{K^p}$ are their locally analytic subsheaves. The proof also relies on three mechanisms: the embedding of Hodge-type Shimura varieties into Siegel modular varieties, which transfers surjectivity of maps of locally analytic sections; geometric Sen theory, which computes group cohomology through Sen operators and Lie algebra cohomology; and the local analyticity of the perfectoid cover, proved by showing that Faltings-extension exact sequences stay exact after taking locally analytic vectors. The exactness of those sequences is what ultimately makes completed cohomology on the Shimura variety agree with coherent cohomology on the flag variety.","core_discovery":"The central claim is Theorem 5.9: for any $i\\geq 0$ and any compact open subgroup $K^p$ away from $p$, there are natural $G(\\mathbb{Q}_p)$-equivariant isomorphisms\n$$(\\widetilde{H}^i(K^p,\\mathbb{Q}_p)\\widehat{\\otimes}_{\\mathbb{Q}_p}\\mathbb{C}_p)^{\\mathrm{la}}\\simeq H^i(\\mathcal{F}\\ell_\\mu,\\mathcal{O}^{\\mathrm{la}}_{K^p})$$\nand\n$$(\\widetilde{H}^i_c(K^p,\\mathbb{Q}_p)\\widehat{\\otimes}_{\\mathbb{Q}_p}\\mathbb{C}_p)^{\\mathrm{la}}\\simeq H^i(\\mathcal{F}\\ell_\\mu,\\mathcal{I}^{\\mathrm{la}}_{K^p}),$$\nwhere $\\mathcal{O}^{\\mathrm{la}}_{K^p}$ and $\\mathcal{I}^{\\mathrm{la}}_{K^p}$ are subsheaves of locally analytic sections of the sheaves obtained by pushing forward the completed structure sheaf of the perfectoid Shimura variety along the Hodge-Tate period map $\\pi^{\\mathrm{tor}}_{\\mathrm{HT}}:\\mathcal{S}\\mathfrak{h}^{\\mathrm{tor}}_{K^p}\\to \\mathcal{F}\\ell_\\mu$. The first sheaf uses the full structure sheaf; the second uses the kernel supported on the boundary of a toroidal compactification. The proof transfers the problem to Siegel modular varieties, establishes local analyticity of the perfectoid cover through geometric Sen theory and Faltings-extension sequences, and then identifies cohomology on the Shimura variety with cohomology on the flag variety. For unitary Shimura curves, the same method gives Theorem 5.12, a $\\tau$-locally analytic version for $H^1$.","pith_inferences":["By analogy with the unitary-curve case, one expects the $\\tau$-locally analytic method to extend to other p-adic Lie group components and to higher-degree cohomology, not just $H^1$.","A natural testable extension is to check Hecke equivariance of the isomorphisms; if compatible, the flag-variety description could serve as a site for interpolating classical automorphic forms into p-adic families.","The paper's reliance on the Siegel embedding suggests that a direct proof of Proposition 5.8 using only the Hodge-Tate period map would simplify the argument and likely extend it to all Shimura varieties."],"forward_implications":["If the main theorem is right, locally analytic completed cohomology of any Hodge-type Shimura variety is a purely geometric invariant computed on the Hodge-Tate flag variety, with no direct reference to the Shimura variety's moduli structure.","The compactly supported variant sees only the boundary contribution encoded by $\\mathcal{I}^{\\mathrm{la}}_{K^p}$, so boundary phenomena in completed cohomology are isolated on the flag variety.","For unitary Shimura curves, $\\tau$-locally analytic completed $H^1$ is isomorphic to $H^1$ of the flag variety $\\mathbb{P}^1$ with a $\\tau$-locally analytic coefficient sheaf, and $K^p$-level cohomology agrees with $K_\\wp$-level cohomology after taking invariants.","All the isomorphisms are $G(\\mathbb{Q}_p)$-equivariant, so the p-adic representation theory carried by completed cohomology is faithfully reflected in the equivariant structure of the flag-variety sheaves."],"supporting_citations":[{"why":"Supplies the modular-curve case and the method of proving the main theorem by comparing cohomology on the Shimura variety with flag-variety cohomology.","marker":"[24, Theorem 4.4.6]"},{"why":"Gives the unitary Shimura curve case and the expansion-along-Borel technique used in Theorem 5.12.","marker":"[27, §3.5]"},{"why":"Provides the Hodge-Tate period map and the perfectoid Shimura varieties at infinite level on which the coefficient sheaves are built.","marker":"[32, Theorem IV.2.1]"},{"why":"Establishes the locally analytic completed cohomology comparison for general Shimura varieties, whose proof strategy is adapted here.","marker":"[5, Theorem 6.2.6]"},{"why":"Geometric Sen theory used to prove surjectivity of maps of locally analytic sections and local analyticity of perfectoid covers.","marker":"[4, Theorem 2.4.3]"},{"why":"Almost purity theorem that makes key exact sequences of completed structure sheaves exact.","marker":"[11, Theorem 5.4.3]"},{"why":"Constructs the perfectoid Shimura varieties of Hodge type and their toroidal compactifications used as the domain of the period map.","marker":"[20, Proposition 6.1]"},{"why":"Provides the logarithmic Riemann-Hilbert and de Rham comparison underlying the Hodge-Tate filtration and the period map.","marker":"[10, Theorem 5.3.1]"}],"fun_headline_variants":["Hodge-type Shimura cohomology maps to flag varieties","Locally analytic cohomology: Shimura varieties meet flag varieties","Completed cohomology reduced to flag sheaves for Hodge type","Hodge-type Shimura: analytic cohomology via perfectoid covers","Shimura varieties: analytic cohomology identified with flag cohomology"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the surjective map of algebraic representations in (5.22) remains surjective after tensoring with locally analytic sections; the paper asserts this without proof, and the main theorem collapses if this exactness fails.","fun_headline_variants_meta":{"raw":{"variants":["Hodge-type Shimura cohomology maps to flag varieties","Locally analytic cohomology: Shimura varieties meet flag varieties","Completed cohomology reduced to flag sheaves for Hodge type","Hodge-type Shimura: analytic cohomology via perfectoid covers","Shimura varieties: analytic cohomology identified with flag cohomology"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000888,"raw_usage":{"total_tokens":3838,"prompt_tokens":956,"completion_tokens":2882,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":572,"completion_tokens_details":{"reasoning_tokens":2790}},"tokens_in":572,"tokens_out":2882,"duration_ms":21923,"temperature":1.0,"reasoning_tokens":2790,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:27:54.117518+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the first Tor term of the tensor product in (5.22) over $\\mathcal{O}(V)$ with $\\mathcal{O}(\\widetilde{U})^{\\mathrm{la}}$ for a flag-variety open $V$; if this Tor term is nonzero, the sequence would not remain exact after taking locally analytic sections, contradicting Proposition 5.8 and invalidating the proof of Theorem 5.9.","supporting_citations":[],"review_version":2}