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$p$-adic Maass--Shimura operators on $\mu$-ordinary Igusa varieties

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Pith's one-line read One algebra unifies p-adic differential and Hecke operators

desk verdict 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. read the letter →

arxiv 2607.07427 v1 pith:QL4N6F4P submitted 2026-07-08 math.NT math.AGmath.RT

Andrew Graham , Pol van Hoften , Sean Howe This is my paper
classification math.NTmath.AGmath.RT
keywords actionoperatorsadicalgebradifferentialformsfunctionslocally
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 8 minor

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.

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.

minor comments (8)
  1. §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].
  2. §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.
  3. §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].
  4. §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.
  5. §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.
  6. 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.
  7. §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.
  8. 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.

Simulated Author's Rebuttal

0 responses · 0 unresolved

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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity found. Self-citations to [27] and [38] are used as general machinery and special-case comparison, not as load-bearing premises that reduce to the conclusion.

full rationale

The paper's central claim (Theorem A / Theorem 4.2.6) rests on a genuine computation in Theorem 4.1.13: that the derivative of the formal group action dv(∂_t) equals the pullback of the algebraic Maass-Shimura vector field v*∂_w. This computation proceeds by (1) reducing to the Siegel case via a closed immersion argument (Claim 3.2.7), (2) lifting to an fpqc cover with an explicit Dieudonné ring S = Ẑ_p[[ε^{1/p^∞},...]]/(ε², y_i - εe_i), and (3) a diagram chase using the quasi-logarithm (Lemma 4.1.6, citing [69, Lemma 3.5.1]) and crystalline-Gauss-Manin compatibility (§2.2.13, citing [3, Prop. V.3.6.4] = Berthelot-Ogus). None of these inputs are defined in terms of the conclusion. The formal group H is constructed canonically from the filtered Dieudonné module (W(O_E), Ad(b_μ)∘σ, Fil⁰W(O_E)) via Proposition 2.2.12 (Grothendieck-Messing), not postulated or fitted. The identification Lie H ≅ u_μ^N (Lemma 4.2.2) is a direct group-theoretic computation. The extension to O(T_p H^∨) (Corollary 5.1.4) uses [27, Proposition 7.0.2] (Howe's p-adic Fourier theory) as a black box, but [27] is a general framework for p-divisible groups and their duals — it does not assume anything about Maass-Shimura operators or Igusa varieties. The recovery of [38] (Howe's GL₂ construction) in Example 1.2.8 serves as an independent cross-check, not a load-bearing premise. Proposition 4.5.9 provides another independent check via compatibility with Eischen-Mantovan operators. The self-citations to Howe's prior work [27, 38] are normal mathematical practice and do not create a circular dependency: the main theorem's derivation chain is self-contained against these external benchmarks.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The paper introduces no free parameters (the construction is canonical given the Shimura datum) and no ad hoc axioms. The formal group H is constructed from standard Dieudonné theory, not postulated. All background results are standard and properly cited.

assumptions (5)
  • standard math Crystalline Dieudonné theory for QPRS rings (Proposition 2.2.12, citing [69, Theorem A])
    §2.2.7-2.2.12: The full faithfulness of the Dieudonné module functor and the Grothendieck-Messing classification are standard results used throughout §4.
  • standard math Existence of canonical integral models of Hodge type Shimura varieties (§3.1.2, citing [45, 43])
    The smooth integral model S_K over O_{E,(v)} and the embedding into Siegel moduli space are foundational results used to define P_dR and the Igusa varieties.
  • standard math p-adic Fourier theory of [27] (Corollary 5.1.4, citing [27, Proposition 7.1.12, Theorem 1])
    §5.1-5.2: The isomorphism F_H: O(H)* → O(T_p H^∨) and the γ-locally analytic function theory are used to extend the group action to function algebras.
  • standard math Product formula for Igusa varieties (§3.4.8, citing [30, Theorem 6.8])
    The map π_∞: Ig_b × M_{G,b,[μ]} → (Ŝ_K)^{[b]} is used to construct the lift Ig_b → (Ŝ_K)^{[b]} and the map to P_dR.
  • standard math The Gauss-Manin connection agrees with the crystalline connection (§2.2.13, citing [3, Proposition V.3.6.4])
    This identification is used in the proof of Theorem 4.1.13 to compute the Gauss-Manin lift s(du(t)) via crystalline Dieudonné theory.
invented entities (1)
  • The p-divisible formal group H independent evidence
    purpose: Integrates the Maass-Shimura operators on Ig_M; its Tate module dual provides the interpolation algebra
    H is not postulated ad hoc but constructed canonically from the filtered Dieudonné module (W(O_E), Ad(b_μ)∘σ, Fil⁰W(O_E)) via Proposition 2.2.12. Its Lie algebra is identified with u_μ^N (Lemma 4.2.2), and its Tate module is computed explicitly in §4.4 (Corollary 4.4.11). The construction is verified to be compatible with the algebraic Maass-Shimura operators (Theorem 4.2.6) and with Hecke operators (Theorem C).

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Pith. "Pith review of $p$-adic Maass--Shimura operators on $\mu$-ordinary Igusa varieties." pith.science (2026). https://pith.science/paper/QL4N6F4P

@misc{pith2026260707427,
  author       = {Pith},
  title        = {Pith review of: $p$-adic Maass--Shimura operators on $\mu$-ordinary Igusa varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QL4N6F4P}},
  note         = {Machine review of arXiv:2607.07427}
}
abstract

For Shimura varieties of Hodge type, we optimally extend algebraic Maass--Shimura differential operators on $p$-integral nearly holomorphic automorphic forms to differential operators on $\mu$-ordinary Mantovan Igusa varieties. We then show that the rank one operators can be integrated to an action of an explicit formal group. Via $p$-adic Fourier theory, this provides a $p$-adic interpolation by extending the action of a symmetric algebra of differential operators to the algebra of functions on the Tate module of the dual $p$-divisible group. Passing to the generic fiber, we obtain an action of an explicit algebra of $p$-adic locally analytic functions, and we show that the action of the subalgebra of locally constant functions is equivalent to a natural Hecke action and thus preserves classical forms. In the ordinary case, we further show that the locally analytic action extends to nearly overconvergent automorphic forms. Our results extend, clarify, and recover prior constructions.

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