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Overconvergent Eichler-Shimura morphisms for $\mathrm{GSp}_4$

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Pith's one-line read This paper constructs explicit Hecke- and Galois-equivariant Eichler–Shimura morphisms that carry the p-adic overconvergent cohomology of Siegel threefolds to the four strata cohomology groups of overconvergent automorphic sheaves, and…

desk verdict A serious, technically substantial paper that plausibly constructs the full overconvergent Eichler–Shimura diagram for GSp4, but the direct-sum decomposition at eigenvariety points still rests on an unproven splitting and a multiplicity-one assumption. read the letter →

arxiv 2506.02643 v1 pith:4O352LVK submitted 2025-06-03 math.NT math.AG

Hansheng Diao , Giovanni Rosso , Ju-Feng Wu This is my paper
classification math.NTmath.AG MSC 11F4611F8011F3314G3514F30
keywords overconvergentSiegelmodularformsEichler-ShimuramorphismsGSp4p-adicfamiliespro-KummeretalecohomologywithsupportseigenvarietiesFaltings-ChaidecompositionHodge-Tate-Senweights
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

The paper claims that the entire Eichler–Shimura decomposition for Siegel modular forms of genus two, not just the $H^0$-part, can be interpolated $p$-adically in families of weights. The construction produces explicit Hecke- and Galois-equivariant morphisms from the pro-Kummer étale cohomology of Ash–Stevens distribution sheaves to the cohomology of overconvergent automorphic sheaves on the four strata of the Siegel threefold. These morphisms assemble into a four-step filtration, and around a 'nice-enough' point of the middle-degree eigenvariety the filtration splits into a direct sum. If the paper is right, the classical Faltings–Chai decomposition of étale cohomology varies in $p$-adic families with Hodge–Tate–Sen weights $(-3, \kappa_{U,2}-2, \kappa_{U,1}-1, \kappa_{U,1}+\kappa_{U,2})$, and the weight map of the eigenvariety is étale at such points.

What carries the argument

The load-bearing object is the family of explicit morphisms $\mathrm{ES}^{w,r}_{\kappa_U}$, given by integrating the highest-weight vector $e^{\mathrm{hst}}_{\kappa_U}$ against Ash–Stevens distributions $\mu \in D^r_{\kappa_U}$ after evaluating at matrices depending on the coordinate $z$ on the flag-variety stratum. These morphisms are pulled back along the Hodge–Tate period map $\pi_{\mathrm{HT}}\colon X^{\mathrm{tor}}_{\Gamma(p^\infty)}\to \mathbb{F}\ell$ and descend to the pro-Kummer étale site of the toroidal compactification. The surrounding machinery is the Bruhat stratification of the flag variety with $w$-loci, the overconvergent automorphic sheaves $\omega^{\bullet}_{n,r}$, pro-Kummer étale cohomology with supports, and the support conditions $Z_{n,w}$ that control the $U_p$-dynamics on each stratum.

What would settle it

Find a $p$-stabilised automorphic representation $\Pi$ of $\mathrm{GSp}_4$ that has small slope but does not satisfy the multiplicity-one condition: for some $w\in W^H$, the Hecke eigenspace $H^{3-\ell(w)}(X^{\mathrm{tor}}_n,\omega^{w_3^{-1}wk+k_w})_{m_\Pi}$ has dimension different from $1$. At such a point Theorem 5.5.2 cannot hold, because its proof uses rank-one freeness of every graded piece, so the decomposition and the étaleness of the weight map would fail there.

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Extended reading notes

Core claim

The central claim is Theorem 5.2.5: the sheaf morphisms $\mathrm{ES}^{w,r}_{\kappa_U}\colon \mathcal{O}D^r_{\kappa_U}\to \widehat{\omega}^{\,w_3^{-1}w\kappa_U}_{n,r}(w\kappa_U^{\mathrm{cyc}})$ are Hecke- and Galois-equivariant and induce a natural diagram linking $H^3_{\mathrm{prok\'et}}(X^{\mathrm{tor}}_n,\mathcal{O}D^r_{\kappa_U})^{\mathrm{fs}}$ to the four support-cohomology groups $H^{3-\ell(w)}_{Z_{n,w}}(X^{\mathrm{tor},u_p}_{n,w},\omega^{w_3^{-1}w\kappa_U+k_w}_{n,r})^{\mathrm{fs}}(w\kappa_U^{\mathrm{cyc}}-\ell(w))$. The key improvement over the $H^0$ case is that every stratum contributes. Under Assumption 5.1.2 (multiplicity one) and the small-slope condition, Theorem 5.5.2 proves that at a nice-enough point the finite-slope part splits as the direct sum of the four graded pieces, specialising to the classical Faltings–Chai decomposition. Consequently the Hodge–Tate–Sen weights of the $p$-adic family are read off from the explicit Tate twists rather than from a BGG resolution or a comparison theorem.

Load-bearing premise

The whole local splitting rests on the assumption that, at the chosen automorphic representation, each of the four relevant coherent cohomology spaces is exactly one-dimensional, a multiplicity-one property that is known only in special cases and can fail for CAP representations.

Editorial extensions

If this is right

  • The full four-step Faltings–Chai decomposition, not merely the $H^0$-part, has a $p$-adic family version whose graded pieces live on the individual Bruhat strata of the flag variety.
  • The middle-degree eigenvariety is equidimensional of dimension two, and its weight map is étale at every nice-enough point (Corollary 5.5.3).
  • Every nice-enough family carries a Galois representation of $\mathrm{Gal}_{\mathbb{Q}}$ with prescribed Hecke polynomial away from $Np$ and, at $p$, a filtration with Hodge–Tate–Sen weights $(-3,\kappa_{U,2}-2,\kappa_{U,1}-1,\kappa_{U,1}+\kappa_{U,2})$; this construction avoids Galois determinants (Corollary 5.5.4).
  • The same morphisms admit cuspidal and interior versions (Theorem 5.2.6), so the decomposition is also available for interior cohomology.
  • The authors expect the constructions to extend to Shimura varieties of PEL type and to support new $p$-adic $L$-functions over these eigenvarieties.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If Theorem 5.5.2 is correct, the Hodge–Tate–Sen weight list gives a purely combinatorial signature for eigenvariety points: the four weights are determined by the two weight characters and the Weyl element, so two different families can be compared by their weight maps alone.
  • The support-condition machinery suggests a practical way to compute slope decompositions on Siegel threefolds: replace global overconvergent cohomology by the four stratum complexes $R\Gamma_{Z_{n,w}}(\cdot)$, and test the equality $E^{\mathrm{oc}}\cong E^{\mathrm{aut}}$ numerically by comparing $U_p$ eigenvalues on the two sides.
  • If the multiplicity-one assumption fails, the filtration built in Theorem 5.2.5 should still exist, but the graded pieces may have rank larger than one; the splitting and étaleness would then fail exactly at such points, so Theorem 5.5.2 carves out the locus where the eigenvariety has a local product structure.
  • The pro-Kummer étale cohomology-with-supports formalism is not tied to $\mathrm{GSp}_4$; the same stratification-by-Weyl-representatives argument could be run on any Shimura variety with a Bruhat decomposition of its flag variety, with the Tate twists read off from the Hodge cocharacter.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a p-adic interpolation, for GSp_4, of the Faltings–Chai Eichler–Shimura decomposition of H^3 of Siegel threefolds. The authors construct Hecke- and Galois-equivariant morphisms ES_{w,r}^{κ_U}: O D^r_{κ_U} → bω^{w_3^{-1}w κ_U}_{n,r}(w κ_U^{cyc}) on pro-Kummer étale sites, assemble them into an overconvergent Eichler–Shimura diagram (Theorem 5.2.5), and then prove a decomposition of the slope-≤h part of H^3_{prokét}(X_n^{tor}, O D^r_{κ_U}) around a 'nice-enough' eigenvariety point into the four strata cohomology groups of overconvergent automorphic sheaves (Theorem 5.5.2). The paper also draws consequences for étaleness of the weight map and for big Galois representations without Galois determinants.

Significance. If the main results are correct, this is a substantial step: it gives a family version of the full four-step Eichler–Shimura decomposition for genus-two Siegel modular forms, determines the Hodge–Tate–Sen weights of the family, and yields a new construction of big Galois representations. The paper is careful and largely self-contained about the geometric and analytic foundations: the coordinate computations on the flag variety, the two constructions of overconvergent automorphic sheaves, and the comparison with the classical diagram in Proposition 5.3.1 are concrete and checkable. The interpolation is measured against an independent external theorem (Faltings–Chai), so there is no circularity. The main caveats are the reliance on the explicit multiplicity-one hypothesis in Assumption 5.1.2 and an unverified splitting step in the proof of Theorem 5.5.2; both are load-bearing for the decomposition claim.

major comments (3)
  1. [§5.5, Step 6 of the proof of Theorem 5.5.2] The decomposition eV H^3_{prokét}(X_n^{tor}, O D^r_{κ_U})^{≤h} ≅ ⊕_{i=0}^3 eV H^{3-i}_{Z_{n,w_i}}(X_{n,w_i}^{tor,up}, ω^{w_3^{-1}w_i κ_U+k_{w_i}}_{n,r})^{≤h}(w_i κ_U^{cyc} - i) depends on splitting each short exact sequence 0 → Fil^{i-1}_{ES,V} → Fil^i_{ES,V} → Gr^i_{ES,V} → 0 by applying [Kis03, Proposition 2.3] to N_i = Hom_{R_U}(Gr^i_{ES,V}, Fil^{i-1}_{ES,V}). The cited proposition is not verified in the required setting: N_i is a finite free module over the Banach affinoid algebra R_U b⊗ C_p with a semilinear Gal_Qp-action, and the proof does not show that a Sen operator φ_Sen,i exists with determinant in R_U, nor that det φ_Sen,i is nonzero on the classical fibre. If the determinant only lies in C_p, localizing at it is not an operation inside the family and the induction in Step 6 breaks. The additional assertion that the splitting is Hecke-stable also needs the Sen weights of Gr^i and Fil^{i-1} to remain distinct over V; this is plausible from Corollary 5.1.4 but is not verified after base change to the family. Since the paper itself labels this step a sketch, the missing hypotheses are a genuine gap in the proof of the main decomposition.
  2. [Assumption 5.1.2 and Definition 5.1.5] The rank-one freeness statements in Step 2 of Theorem 5.5.2, and hence the decomposition, graded-piece isomorphisms, and étaleness of the weight map in Corollary 5.5.3, rely on the multiplicity-one hypothesis dim H^{3-l(w)}(X_n^{tor}, ω_{w_3^{-1}w k + k_w})_{m_Π} = 1 for every w ∈ W^H. As Remark 5.1.3 notes, this is not a theorem for general GSp_4 automorphic representations and may fail for CAP representations; the known sufficient cases are generic representations and paramodular forms. This is not an internal inconsistency, but it substantially limits the unconditional scope. The introduction and abstract should state more prominently that the full decomposition theorem is established only under this unproved multiplicity-one assumption, and the paper would be strengthened by a precise discussion of how Assumption 5.1.2 is verified in the examples to which the main theorems are applied.
  3. [Proposition 5.2.4 and the edge map from spectral sequence (51)] The construction of the overconvergent Eichler–Shimura morphisms in cohomology uses the vanishing of the low-degree finite-slope terms of the Leray spectral sequence (51), citing [BP20, Theorem 6.7.3]. The hypotheses of that theorem (including the relevant slope bounds and radius conditions) are not checked in the GSp_4 setting here. Since the spectral sequence edge map is what produces the target H^{3-l(w)}_{Z_{n,w},két}(X_{n,w}^{tor,up}, ω^{...}_{n,r})^{fs}, this is a load-bearing point, though it is likely repairable by a direct verification parallel to [BP20, §6]. The authors should spell out the verification rather than leave it to the cited theorem.
minor comments (4)
  1. [§5.3, diagrams (54) and (57)] The displayed cohomological degrees in the rows for w_2 and w_1 appear to be interchanged: by Proposition 5.2.4 the row for w_2 should target H^1, and the row for w_1 should target H^2, as in Theorem 5.2.5. Please correct the two diagrams.
  2. [§2.3] The notation Fℓ_{w,(m,n)} is used for four different loci in the same displayed block, with closures taken with respect to the analytic topology; this makes the definitions hard to read. Please introduce distinct symbols or an explicit sentence explaining the four variants.
  3. [§5.5 title] The section title reads 'Overconvergent Eicher–Shimura decomposition'; 'Eichler' is missing the letter 'l'. Please fix the typo.
  4. [Proposition 5.4.1] The proof of the isomorphism of eigenvarieties E^{oc} ≅ E^{aut} is a one-paragraph citation to [Han17, Theorem 5.1.2]. Since this comparison is used in Corollary 5.4.2 and in the definition of the point x_Π, the density argument for the very Zariski dense sets of classical small-slope points should be expanded.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the p-adic Eichler-Shimura decomposition is derived from independent inputs and checked at classical weights against Faltings-Chai; the main caveats (Assumption 5.1.2, sketched Step 6) are hypotheses or proof gaps, not circular reductions.

full rationale

The central Theorem 5.5.2 does not define its conclusion into its hypotheses. The filtration Fil^•_ES,V is constructed from images of support cohomology, and the graded pieces are shown to be isomorphic to the overconvergent automorphic cohomology groups via Nakayama, specialization, and classicality (Steps 1-5); the right-hand side is not installed by definition. The interpolation claim is benchmarked externally: Theorem 5.2.5 is checked at classical weights in Proposition 5.3.1 against the classical Eichler-Shimura diagram of Proposition 5.1.6, which itself invokes Faltings-Chai (Theorem 1.2.1/5.1.1), an independent theorem. The Hodge-Tate-Sen weights (-3, κ_{U,2}-2, κ_{U,1}-1, κ_{U,1}+κ_{U,2}) are read off from the explicitly computed Tate twists (Remark 5.2.1, wκ^cyc_U), not fitted to the target. Assumption 5.1.2 and Definition 5.1.5 are stated hypotheses (multiplicity-one and small slope), and Remark 5.1.3 openly notes CAP representations can fail them, so they are not smuggled conclusions. Self-citations appear ([DRW21] for the H^0-part and the top-row morphism; [Han17] for eigenvariety comparison) but they supply constructions and machinery rather than the target decomposition, and are backed by external references such as [BP20] and [FC90]. The one genuinely load-bearing caveat is Step 6 of Theorem 5.5.2, which cites [Kis03, Proposition 2.3] to split the filtration after localizing at det φ_Sen,i without verifying the Sen-operator hypotheses in the R_U ⊗ C_p family setting; this is a proof gap and a correctness risk, not circularity, because the splitting is claimed from an external proposition rather than assumed as input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

The paper fits no data and introduces no fitted constants; every numerical condition (slope bounds h^oc_{i,j,k}, h^sh_{i,k}, h_k in Definition 3.5.13) is defined combinatorially from the weight k rather than chosen to match outputs. The load-bearing inputs are the standing hypotheses and cited theorems listed as axioms: the multiplicity-one/small-slope assumptions that restrict the main theorem, Boxer-Pilloni's classicality and slope theory, Faltings-Chai, Kisin's Sen-operator lemma, Hansen's eigenvariety comparison criterion, and Lan's vanishing and Kodaira-Spencer results. The two new objects (pro-Kummer cohomology with supports, pseudoautomorphic sheaves) are constructed with proofs rather than postulated, so the invented-entities list is short and each entry carries independent mathematical support within the paper.

assumptions (6)
  • ad hoc to paper Multiplicity-one hypothesis (Assumption 5.1.2): dim H^{3-l(w)}(X^{tor}_n, ω_{w_3^{-1} w k + k_w})_{mΠ} = 1 for all w ∈ W^H, together with the small-slope condition (Definition 5.1.5).
    Introduced to make the freeness and rank-one statements in Steps 1-4 of Theorem 5.5.2 work; not proved for general GSp4 representations. Remark 5.1.3 notes CAP representations may fail it; known cases are generic representations via [Sou87] and paramodular via [RW17].
  • domain assumption Boxer-Pilloni higher Coleman theory: classicality, control, and slope-decomposition theorems ([BP20, Theorem 5.12.3, Corollary 6.8.4, Theorem 6.4.3]).
    Quoted as a black box in Theorem 3.5.18, Proposition 3.6.1, and Proposition 4.4.1; the paper extends the framework to GSp4 and the pro-Kummer étale setting but relies on BP20 for classicality and slope decompositions.
  • domain assumption Faltings-Chai p-adic Eichler-Shimura decomposition (Theorem 1.2.1, [FC90, Chapter VI, Theorem 6.2]), including distinctness of the Hodge-Tate weights {w_i k^cyc - i} for k1 ≥ k2 > 0.
    The classical input being interpolated; used in Corollary 5.1.4, Proposition 5.1.6, and Proposition 5.4.1. Distinctness of the weights is what makes the classical filtration unique.
  • domain assumption Kisin's Sen-operator result ([Kis03, Proposition 2.3]): det φ_Sen,i ≠ 0 in R_U kills H^1(Gal_Qp, N_i), used to split the filtration extensions in Step 6 of Theorem 5.5.2.
    The split is claimed Hecke-stable because Galois and Hecke actions commute; the proof is sketched, and the hypotheses of [Kis03, Proposition 2.3] in the R_U b⊗ Cp family setting are not verified in the text.
  • domain assumption Hansen's eigenvariety machine ([Han17, Proposition 3.1.5, Theorem 5.1.2]) and very Zariski density of small-slope classical cuspidal points used to identify E_oc with E_aut (Proposition 5.4.1).
    Proposition 5.4.1 is proved by citing [Han17, Theorem 5.1.2] twice; the density check is asserted, not demonstrated.
  • standard math Lan's vanishing theorem for coherent cohomology of automorphic sheaves ([Lan16, Theorem 4.1]) and the Kodaira-Spencer isomorphism ([Lan12, Theorem 1.41(4)]).
    Used in Construction 3 of §5.1 to obtain edge maps from the Leray spectral sequence and in Proposition 5.2.4 to identify Ω^{log,l(w)} with ω^{k_{w_i}}.
invented entities (2)
  • Pro-Kummer étale cohomology with supports (RΓ_{Z,prokét}, Appendix A) independent evidence
    purpose: Computes the overconvergent cohomology groups by decomposing the Siegel threefold along the stratification X^{tor}_n = X^{tor}_{n,≤w3} ⊋ ... ⊋ X^{tor}_{n,14}, supplying the left column of the main diagram (Theorem 5.2.5) and the spectral sequence in §1.2 and §4.4.
    The theory is constructed and proved in Appendix A (definitions, spectral sequence A.2.1, Banach-sheaf statements A.3.5 and A.3.6); it is a rigorously built object, not a postulate. Its failure would break Theorem 5.2.5's filtration, but the paper supplies proofs.
  • Pseudoautomorphic sheaves A^r_{κU,Fℓw} and Iw^+_{H,n}-torsors IW^+_{H,n,Fℓw} on flag-variety strata independent evidence
    purpose: Intermediate sheaves used to define the overconvergent Eichler-Shimura morphisms ESw,r_{κU} by pullback along the Hodge-Tate period map (Proposition 2.6.2 and §5.2); the w3-twist w_3^{-1}w κU is chosen so that at w3 the sheaf reduces to the known overconvergent Siegel automorphic sheaf of [DRW21].
    Explicitly constructed in §2.4-2.6 with a comparison theorem (Theorem 3.4.3) against the perfectoid construction; no external falsifiable prediction is attached, but the object is provably constructed and specializes correctly at classical weights (Remark 3.3.4 and Proposition 5.3.1).

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Pith. "Pith review of Overconvergent Eichler-Shimura morphisms for $\mathrm{GSp}_4$." pith.science (2026). https://pith.science/paper/4O352LVK

@misc{pith2026250602643,
  author       = {Pith},
  title        = {Pith review of: Overconvergent Eichler-Shimura morphisms for $\mathrmGSp_4$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4O352LVK}},
  note         = {Machine review of arXiv:2506.02643}
}
abstract

We construct explicit Eichler-Shimura morphisms for families of overconvergent Siegel modular forms of genus two. These can be viewed as $p$-adic interpolations of the Eichler-Shimura decomposition of Faltings-Chai for classical Siegel modular forms. In particular, we are able to $p$-adically interpolate the entire decomposition, extending our previous work on the $H^0$-part. The key new inputs are the higher Coleman theory of Boxer-Pilloni and a theory of pro-Kummer \'etale cohomology with supports.

Figures

Figures reproduced from arXiv: 2506.02643 by the authors.

Figure 1
Figure 1. Stratification of X tor n Thirdly, we need the notion of families of p-adic weights. Let W be the weight space which parameterises p-adic weights (cf. §2.5). Then, by a family of p-adic weights, we mean an affinoid open U = Spa(RU, R◦ U ) ,→ W; we denote by (RU, κU) (or just κU) the corresponding weight character. Now, we are ready to p-adically interpolate the objects on both sides of (1). On the side of coherent c… view at source ↗

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    Available at: https://arxiv.org/abs/2003.05960,

    [LZ20] David Loeffler and Sarah Livia Zerbes, On the Bloch–Kato conjecture forGSp(4), Preprint. Available at: https://arxiv.org/abs/2003.05960,

    2003 arXiv

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Reviewed August 7, 2026 · model on record in the stance chip above.