{"id":"6d07dd3a-fdb1-436b-b862-8deb32e74722","arxiv_id":"2607.07427","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors construct a canonical formal group action on μ-ordinary Igusa varieties for Hodge type Shimura data that integrates Maass-Shimura operators and, via p-adic Fourier theory, yields a unified algebra action encoding both p-adic differential operators and Hecke operators.","lead":"This paper constructs p-adic Maass-Shimura differential operators on Igusa varieties for general Hodge type Shimura varieties, extending them from the GL₂ case. It shows these operators integrate to a formal group action that, via p-adic Fourier theory, simultaneously interpolates differential operators and Hecke operators — a unification useful for constructing p-adic L-functions.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The Siegel reduction and Dieudonné computation in Theorem 4.1.13 follow established methods, and independent checks exist via recovery of [38] in the ordinary case and compatibility with Eischen–Mantovan operators (Prop. 4.5.9).","rationale":"The reader correctly identifies the Siegel reduction and Dieudonné computation as the most delicate part of the argument. However, upon careful examination, the argument is sound: each ingredient is standard, the diagram chases are correct, and fpqc descent is properly applied. The paper also provides independent cross-checks (recovery of [38] in the ordinary case, compatibility with Eischen–Mantovan in Proposition 4.5.9). The reader's verdict of ACCEPT with MODERATE confidence is appropriate. The correctness risk remains 'unknown' in the sense that no machine-checked proof exists, but the argument is well-constructed and builds on established machinery. No verdict adjustment is needed.","tokens_in":74657,"tokens_out":3290,"duration_ms":186470,"concrete_test":"Verify Lemma 4.1.6 independently in the case of an ordinary elliptic curve (G = GL_2, H = Ĝ_m): explicitly compute D(α_t̃) in the ring S = Ẑ_p[[ε^{1/p^∞}, x^{1/p^∞}]]/(ε², y - ε) and confirm that its image in gr^{-1}W(S) = Ẑ_p[[ε^{1/p^∞}]]/(ε) equals the class of t = ε·(d/dy). This is the simplest non-trivial instance of the quasi-logarithm computation and should recover the classical logarithm on Ĝ_m. If this check fails, the main theorem would be in jeopardy.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader correctly identifies the Siegel reduction in Theorem 4.1.13 as the most delicate step. Having examined the argument carefully, I do not find a significant objection that lands. The reduction proceeds in two stages: (1) the closed immersion Ig_b(G,X) → Ig_b(G_V, H_V) induces an injection on tangent bundles, so it suffices to verify in the Siegel case (Claim 3.2.7 style argument, using equal ranks); (2) in the Siegel case, the key equality dv(t) = w·s(du(t)) (equation 4.1.3) is verified by lifting to the fpqc cover Spf S × Ig_b with S = Ẑ_p[[ε^{1/p^∞}, x_i^{1/p^∞}]]/(ε², y_i - εe_i), and using: (a) the quasi-logarithm identification (Lemma 4.1.6, citing [69, Lemma 3.5.1]), (b) the compatibility of crystalline and Gauss-Manin connections (§2.2.13, citing [3, Prop. V.3.6.4]), and (c) the commutativity of the diagram relating α_t̃, ρ̃_S, and ψ̃_S. Each ingredient is standard: the quasi-logarithm result is from [69], the crystalline-Gauss-Manin compatibility is foundational (Berthelot-Ogus), and the diagram chase is straightforward by construction. The fpqc descent from Ig_b^S to Ig_b[ε] is valid. Moreover, the paper provides two independent cross-checks: the ordinary case recovers [38, Theorem 5.3.1] (Example 1.2.8), and Proposition 4.5.9 verifies compatibility with Eischen–Mantovan operators. The p-adic Fourier theory application (Corollary 5.1.4) relies on [27, Proposition 7.0.2] for the differentiation-Hodge-Tate compatibility, which is established in Howe's prior work. Theorem C's identification of locally constant functions with Hecke operators (Theorem 5.3.8(2)) uses the functoriality of the Fourier transform and is clean. I do not identify a load-bearing concern that would undermine the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper constructs p-adic Maass-Shimura differential operators on μ-ordinary Mantovan Igusa varieties for Hodge type Shimura varieties, and integrates them to an action of an explicit p-divisible formal group H. Via p-adic Fourier theory (building on [27]), the symmetric algebra action extends to an algebra of functions on the Tate module of the dual p-divisible group, and on the generic fiber to an algebra of γ-locally analytic functions. The authors show that the locally constant subalgebra corresponds to Hecke operators (Theorem C), so a single algebra action simultaneously interpolates differential operators and Hecke operators. In the ordinary case, the action extends to nearly overconvergent forms (Theorem B). The main technical result (Theorem 4.1.13) reduces the key derivative computation to the Siegel case via a closed immersion argument and verifies it there using crystalline Dieudonné theory and Grothendieck-Messing theory.","tokens_in":75219,"tokens_out":1157,"duration_ms":119266,"significance":"The paper makes a substantial contribution to the p-adic theory of automorphic forms on higher-dimensional Shimura varieties. The coordinate-free construction of the formal group action integrating Maass-Shimura operators is new and conceptually cleaner than prior approaches. The simultaneous interpolation of differential operators and Hecke operators within a single algebra action (Theorem C) is a notable structural insight with clear potential for p-adic L-function constructions. The paper provides falsifiable predictions and explicit examples (§5.5, §5.6, §7), and the comparison with Eischen-Mantovan operators (Prop. 4.5.9) and recovery of [38] in the ordinary case serve as independent cross-checks. The extension to nearly overconvergent forms (Theorem 6.1.1) with an improved coordinate choice (Prop. 6.2.4, Remark 6.2.6) strengthens the ordinary case beyond prior work.","major_comments":[],"minor_comments":[{"comment":"§3.1.6, Remark 3.1.6: The remark flags a potential error or convention mismatch in Lovering's [55] regarding cohomology vs. homology. While the authors handle this correctly, a brief footnote or parenthetical clarifying the exact discrepancy would help readers cross-referencing [55].","section":null},{"comment":"§4.1.12 (proof of Theorem 4.1.13): The notation for the quasi-isogeny ψ and its lift ψ̃_S is introduced somewhat abruptly in the proof. A sentence explicitly defining ψ as the quasi-isogeny A_{Igb[ε]} → B_{Igb[ε]} induced by the action mod p would improve readability.","section":null},{"comment":"§4.5.14, line containing 'modulop k−1(p−1)': The congruence condition appears to be missing spaces ('modulo p^{k-1}(p-1)'). This occurs in the discussion comparing with [16, Prop. 6.3.9].","section":null},{"comment":"§5.5.4, Remark 5.5.4: The remark states it is 'not clear to us how Ω_ζ is related to the Lubin-Tate period of [65, Appendix].' Since this period automorphism is load-bearing for the generic fiber description (Cor. 1.2.11), even a partial clarification or a pointer to where the relationship should be checkable would be valuable.","section":null},{"comment":"§6.2.6, Remark 6.2.6: The asymptotic formula n(ε) = -[log_p(ε)] + O(1) is stated without full justification of the O(1) term. A reference or one-line justification for the bounded error would strengthen this useful remark.","section":null},{"comment":"Appendix A.1.1: The disclosure of ChatGPT usage is appropriate and transparent. No issue here, but the proof of Lemma A.1.7 appears truncated in the manuscript text ('...this is a direct consequence o'). This should be completed.","section":null},{"comment":"§7: The outlook on p-adic L-functions is informal. While the authors clearly label it as speculative, a brief statement of what would be needed to make Conjecture 6.4.2 tractable (e.g., explicit descriptions of O(T_p H^∨) sections) would help orient future work.","section":null},{"comment":"Notation: The symbol ⊗■ for the solid tensor product is used before its definition is referenced. A forward pointer to [68] or [27, §2] at first occurrence (around §1.2.10) would help.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is long and technically dense, but well-organized. The reliance on [27] (Howe's p-adic Fourier theory) and [38] (Howe's GL₂ construction) is substantial, and Howe is a co-author of both this paper and those references. However, the central constructions here are independent geometric derivations, and the paper provides two independent cross-checks (ordinary case recovery, Eischen-Mantovan compatibility). I do not see a circularity concern. The Siegel reduction in Theorem 4.1.13 is the most delicate step, but the argument is standard and each ingredient is properly cited. The paper is appropriate for a serious journal in arithmetic geometry."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and generous report. The referee recommends minor revision and raises no major comments. We address the report below.","responses":[],"tokens_in":74030,"tokens_out":44,"duration_ms":44596,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"This paper constructs formal group actions on mu-ordinary Igusa varieties that integrate Maass-Shimura differential operators, for general Hodge type Shimura data. The main results are: Theorem A gives the formal group H acting on Ig_M whose differentiation recovers the Maass-Shimura operators; Corollary 5.1.4 extends this via p-adic Fourier theory to an action of O(T_p H^vee) on O(Ig_M); and Theorem C shows the locally constant subalgebra corresponds to Hecke operators, so a single algebra action interpolates both differential and Hecke operators simultaneously. The ordinary case recovers Howe's earlier GL_2 construction, and the authors verify compatibility with Eischen-Mantovan operators (Proposition 4.5.9). The coordinate-free parallelization of T P_dR (Proposition 3.2.6) is a genuine simplification over prior sheaf-theoretic approaches and makes the Lie algebra structure transparent. The unification of Hecke and differential operators under one algebra action is the most conceptually interesting output and should matter for p-adic L-function constructions. The stress-test concern about the Siegel reduction in Theorem 4.1.13 being load-bearing is legitimate in the sense that the whole paper hinges on it, but the argument itself is sound. The reduction uses that the closed immersion into the Siegel Igusa variety induces an injection on tangent bundles (Claim 3.2.7), and the Siegel case verification proceeds via a standard Dieudonné theory computation using the quasi-logarithm (Lemma 4.1.6, citing [69]) and the crystalline-Gauss-Manin compatibility (Berthelot-Ogus). Each ingredient is standard and the diagram chase is straightforward. The two cross-checks — recovery of [38] in the ordinary case and compatibility with Eischen-Mantovan — provide independent confirmation. I do not see a gap here. The self-citation to [27, 38] is appropriate: the central construction is independent of those results, which supply the Fourier theory machinery. One minor concern: the paper does not provide an independent cross-check of the compatibility between the crystalline connection and the Gauss-Manin connection used in the key identification, but this is foundational material. The mu-ordinary generalization of Theorem B (near overconvergence) is left as a conjecture (6.4.2), which is honest. This paper is for specialists in p-adic automorphic forms and p-adic L-functions. It deserves a serious referee who can check the Dieudonné theory computation in section 4.1 carefully.","headline":"Extends p-adic Maass-Shimura interpolation to Hodge type Shimura varieties in the mu-ordinary setting, with a clean coordinate-free approach and a unified algebra action for Hecke and differential operators.","tokens_in":75863,"tokens_out":631,"would_cite":true,"duration_ms":56095,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"One algebra unifies p-adic differential and Hecke operators","keywords":[],"falsifier":"An error in the compatibility between the crystalline connection and the Gauss–Manin connection used in the Siegel-case Dieudonné theory computation would break the identification dv(∂_t) = v*∂_w, which is the equality that makes differentiation of the formal group action recover the Maass–Shimura operators.","tokens_in":74888,"feed_emoji":"🔧","tokens_out":843,"duration_ms":193030,"temperature":0.7,"pith_summary":"This paper proves that the classical Maass–Shimura differential operators on nearly holomorphic automorphic forms can be p-adically interpolated on Igusa varieties for arbitrary Hodge type Shimura varieties. The authors construct an explicit p-divisible formal group H whose Lie algebra matches the rank-one part of the Maass–Shimura operators, and whose action on the Igusa variety integrates them. Applying p-adic Fourier theory then extends the polynomial differential operators to a full algebra of continuous p-adic functions, and on the generic fiber to an algebra of locally analytic functions. The key unifying result is that the subalgebra of locally constant functions in this algebra corresponds exactly to Hecke operators, so a single algebra action simultaneously interpolates differential operators and Hecke operators. In the ordinary case, the authors further show the action extends to nearly overconvergent automorphic forms.","feed_headline":"One algebra unifies p-adic differential and Hecke operators","feed_subtitle":"Maass–Shimura operators on Hodge type Shimura varieties are integrated to a formal group action whose p-adic Fourier transform also encodes","key_machinery":"The p-divisible formal group H built from the filtered Dieudonné module associated to the center of U_mu; the Gauss–Manin parallelization of the tangent bundle of the de Rham torsor P_dR; p-adic Fourier theory converting formal group actions to algebra actions; and the reduction to the Siegel case for the key derivative computation.","core_discovery":"The central object is an explicit p-divisible formal group H over the ring of integers of the local reflex field, constructed from the center of the unipotent radical U_mu via Dieudonné theory. Theorem A shows that H acts on the Igusa variety Ig_M in a way that differentiating its action recovers the algebraic Maass–Shimura operators. The Lie algebra of H is equivariantly identified with the N-invariant part of u_mu, where N is the intersection of the Levi M_mu with the unipotent radical. Via p-adic Fourier theory, this integration of differential operators extends to an algebra action of functions on the Tate module of the dual p-divisible group, and on the generic fiber to an algebra of so","pith_inferences":[],"forward_implications":["The simultaneous interpolation of differential and Hecke operators in a single algebra could simplify the construction of p-adic L-functions, especially in non-ordinary settings where the mu-ordinary locus differs from the ordinary locus.","The identification of p-depletions (traditionally described via Hecke operators) with simple indicator functions in the locally analytic algebra may clarify choices of test vectors in existing p-adic L-function constructions.","The framework extends to arbitrary Hodge type Shimura varieties, potentially enabling constructions of p-adic L-functions in settings (such as unitary GGP with p inert) where previous coordinate-dependent methods were unavailable.","The extension to nearly overconvergent forms in the ordinary case provides the analytic control needed for p-adic L-function constructions beyond the level of classical forms."],"fun_headline_variants":["Formal group action integrates p-adic Maass–Shimura operators on Igusa varieties","p-divisible formal group recovers Maass–Shimura operators via differentiation","Maass–Shimura operators integrate to formal group action on Igusa varieties","p-adic Fourier theory extends differential operators to Hecke-compatible action","Dieudonné-built formal group unifies p-adic differential and Hecke operators"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The proof reduces the key derivative computation to the Siegel case by checking that a closed immersion of Igusa varieties induces an injection on tangent bundles. The Siegel case itself is verified by an explicit but intricate Dieudonné theory computation, and the paper does not provide an independent cross-check of it.","fun_headline_variants_meta":{"raw":{"variants":["Formal group action integrates p-adic Maass–Shimura operators on Igusa varieties","p-divisible formal group recovers Maass–Shimura operators via differentiation","Maass–Shimura operators integrate to formal group action on Igusa varieties","p-adic Fourier theory extends differential operators to Hecke-compatible action","Dieudonné-built formal group unifies p-adic differential and Hecke operators"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":662,"prompt_tokens":572,"completion_tokens":90,"prompt_tokens_details":null},"tokens_in":572,"tokens_out":90,"duration_ms":107287,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T11:10:36.697388+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"An error in the compatibility between the crystalline connection and the Gauss–Manin connection used in the Siegel-case Dieudonné theory computation would break the identification dv(∂_t) = v*∂_w, which is the equality that makes differentiation of the formal group action recover the Maass–Shimura operators.","supporting_citations":[],"review_version":1}