{"id":"a22c478b-604b-4a1f-bc4c-aba7846fcbc9","arxiv_id":"2506.02643","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The full four-step Eichler-Shimura decomposition for GSp4 is interpolated along p-adic weight families, and near any nice-enough eigenvariety point it splits with Hodge-Tate-Sen weights (-3, κ2-2, κ1-1, κ1+κ2).","lead":"This paper constructs p-adic families of morphisms, called overconvergent Eichler-Shimura morphisms, that interpolate the four-term cohomology decomposition of Faltings and Chai for genus-two Siegel modular forms as the weight varies p-adically. If correct, this organizes all finite-slope Siegel modular forms on a single eigenvariety with explicit Hodge-Tate-Sen weights and yields new determinant-free Galois representations.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step 6 of Theorem 5.5.2 cites [Kis03, Prop. 2.3] to split the ES filtration, but the hypotheses (Sen operator over R_U b⊗ C_p with determinant in R_U) are unverified; the full decomposition claim is not yet established.","rationale":"The reader's weakest_assumption points primarily to Assumption 5.1.2 (multiplicity-one) and the small-slope condition, with Step 6's Sen-operator citation as a secondary issue. I agree that multiplicity-one is fragile and restricts the theorem's scope, but it is an explicit, honestly stated hypothesis of the nice-enough definition, so it limits rather than invalidates the claim. The more urgent concern is Step 6, which is required for the direct-sum decomposition and its specialization to Faltings–Chai. There the manuscript invokes [Kis03, Prop. 2.3] without verifying that the Sen operator for N_i is R_U-linear with determinant in R_U, nor that the determinant is nonzero after specialization, nor that the relevant H^1 vanishes in the family-coefficient setting. Since the theorem's headline contribution is exactly the p-adic interpolation of the entire decomposition, this omitted verification is load-bearing. The proposed test would settle whether the cited lemma applies; if it does not, the theorem remains true only at the level of filtrations, and the abstract's claim about interpolating the entire decomposition is premature. This does not change the reader's CONDITIONAL verdict, but it sharpens the conditions that a revision must satisfy.","tokens_in":86401,"tokens_out":7697,"duration_ms":85351,"concrete_test":"Re-derive Step 6 with explicit coefficient ring R = R_U b⊗ C_p. Check [Kis03, Prop. 2.3] against N_i = Hom_R(Gr^i_{ES,V}, Fil^{i-1}_{ES,V}): (1) prove φ_Sen,i is an R-linear endomorphism of N_i and det φ_Sen,i ∈ R; (2) compute its specialization at x_Π and verify nonvanishing; (3) verify H^1(Gal_Qp, N_i[1/det]) = 0 using the cited proposition's exact hypotheses. Also check that the Sen weights of Gr^i and Fil^{i-1} remain distinct on V so the unique Galois splitting is Hecke-stable. If (1)–(3) hold, the concern is resolved; if not, Step 6 needs a replacement or the theorem should be weakened to the filtration statement.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central decomposition in Theorem 5.5.2(iv) depends on Step 6, where the authors split each short exact sequence 0 → Fil^{i-1}_{ES,V} → Fil^i_{ES,V} → Gr^i_{ES,V} → 0 by applying [Kis03, Prop. 2.3] to N_i = Hom_{R_U}(Gr^i_{ES,V}, Fil^{i-1}_{ES,V}). They assert that 0 ≠ det φ_Sen,i ∈ R_U kills H^1(Gal_Qp, N_i). This is load-bearing because without it the theorem gives only a filtration with graded isomorphisms, not the direct-sum decomposition that specializes to Faltings–Chai. But the cited proposition is not checked in the required setting: N_i is a finite free module over the Banach affinoid algebra R_U b⊗ C_p with a semilinear Galois action, and no proof is supplied that a Sen operator φ_Sen,i exists that is R_U-linear 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 supplementary 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. The paper itself labels this step a sketch, so the missing hypotheses are a genuine gap in the proof of the main decomposition.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":86641,"tokens_out":4796,"duration_ms":54946,"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":[{"comment":"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.","section":"§5.5, Step 6 of the proof of Theorem 5.5.2"},{"comment":"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.","section":"Assumption 5.1.2 and Definition 5.1.5"},{"comment":"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.","section":"Proposition 5.2.4 and the edge map from spectral sequence (51)"}],"minor_comments":[{"comment":"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.","section":"§5.3, diagrams (54) and (57)"},{"comment":"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.","section":"§2.3"},{"comment":"The section title reads 'Overconvergent Eicher–Shimura decomposition'; 'Eichler' is missing the letter 'l'. Please fix the typo.","section":"§5.5 title"},{"comment":"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.","section":"Proposition 5.4.1"}],"recommendation":"major_revision","confidential_remarks":"The central construction of the Eichler–Shimura morphisms is detailed and convincing, and the comparison with the classical Faltings–Chai diagram via Proposition 5.3.1 is a real strength. The main risk is the sketched splitting in Step 6 of Theorem 5.5.2, which is not a routine citation as written; if that step cannot be justified, the direct-sum decomposition and the étaleness corollary would need to be weakened to a filtered statement. The multiplicity-one hypothesis is also a serious limitation, but it is explicitly stated and can be treated as a standing hypothesis. I would be supportive of publication after the Step 6 argument is supplied or replaced by a precise lemma with verified hypotheses."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing you should know: this is a real paper, not a stunt. It extends the authors' earlier GSp4 H^0 work to a full four-row overconvergent Eichler–Shimura diagram, with displayed Tate twists mechanically matched to Hodge–Tate–Sen weights, and it checks the new morphisms against the classical Faltings–Chai diagram at classical weights. The pro-Kummer étale cohomology-with-supports formalism in Appendix A is a genuine new tool and is written in enough detail to be usable. I found no circularity: the interpolation is benchmarked against an external classical theorem, and the authors do not fit parameters.\n\nWhat is actually new: Theorem 5.2.5 constructs the diagram for families, and Theorem 5.5.2 gives a splitting at nice-enough points on the eigenvariety, yielding big Galois representations without determinants. The genuinely new part is the w0, w1, w2 strata and the explicit cyclotomic twists. That part looks internally consistent.\n\nSoft spots, in proportion. The decomposition in Theorem 5.5.2 is conditional twice over. First, it requires Assumption 5.1.2 (multiplicity-one in all four coherent degrees), which is not a theorem for general GSp4 and which the authors admit can fail for CAP representations. That honestly limits the scope, and the abstract overstates the result by omitting the nice-enough caveat. Second, Step 6 of Theorem 5.5.2 is the piece I would send back. The authors split each short exact sequence by invoking [Kis03, Prop 2.3] on N_i = Hom(Gr^i, Fil^{i-1}). The stress-test note is right: they do not verify that a Sen operator exists over R_U µ⊗ C_p with nonzero determinant in R_U, nor that the determinant kills H^1 after base change to the family. If the determinant only lies in C_p, localizing at it is not a family operation and the induction breaks. Without Step 6, the theorem still gives a filtration with graded isomorphisms—that is already substantial—but it does not give the direct-sum decomposition specializing to Faltings–Chai. That is a gap in the proof of the headline claim, not a hole in the whole construction.\n\nAlso, Theorem 4.4.4 and Proposition 5.4.1 are deferred with sketch-level references; probably acceptable for a first pass, but a referee should ask for fuller arguments. Non-neat levels are only a strategy.\n\nBottom line: this deserves a serious referee. The main construction is careful and likely correct conditional on multiplicity-one; the splitting needs to be either proved or stated as a hypothesis. I would not desk-reject.","headline":"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.","tokens_in":87397,"tokens_out":2329,"would_cite":true,"duration_ms":26459,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F46","11F80","11F33","14G35","14F30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["overconvergent Siegel modular forms","Eichler-Shimura morphisms","GSp4","p-adic families","pro-Kummer etale cohomology with supports","eigenvarieties","Faltings-Chai decomposition","Hodge-Tate-Sen weights"],"falsifier":"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.","tokens_in":85944,"feed_emoji":"🔢","tokens_out":10765,"duration_ms":95671,"temperature":0.7,"pith_summary":"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.","feed_headline":"Four-step Eichler–Shimura decomposition varies p-adically in weight","feed_subtitle":"New Eichler–Shimura morphisms interpolate all four graded pieces and build Galois-representation families.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the classical four-step Eichler–Shimura decomposition for $\\mathrm{GSp}_4$ that the paper p-adically interpolates.","marker":"[FC90]"},{"why":"Provides the higher Coleman theory used throughout: overconvergent automorphic sheaves, classicality theorems, slope theory, and the $U_p$ dynamics on strata.","marker":"[BP20]"},{"why":"Provides the higher Coleman theory on modular curves that the paper names as a key new input for interpolating the higher-degree part.","marker":"[BP22]"},{"why":"Earlier work by the same authors constructing the $H^0$-part and the overconvergent automorphic sheaf whose global sections are overconvergent Siegel modular forms.","marker":"[DRW21]"},{"why":"Constructs the perfectoid Siegel modular variety and the Hodge–Tate period map that define the strata and carry the sheaf pullbacks.","marker":"[PS16]"},{"why":"Supplies the pro-Kummer étale site and the cohomology-with-supports formalism used in Section 4 and the appendix.","marker":"[DLLZ23]"},{"why":"Supplies the construction of the middle-degree eigenvariety and the control theorem used to identify the two eigenvarieties.","marker":"[Han17]"},{"why":"Provides the Sen-operator criterion used in Step 6 of Theorem 5.5.2 to split the filtration after localising.","marker":"[Kis03]"}],"fun_headline_variants":["Full p-adic Eichler-Shimura decomposition for GSp4 Siegel forms","All four Eichler-Shimura graded pieces now vary p-adically","Entire Eichler-Shimura decomposition interpolated p-adically for genus two","From H^0 to H^3: full p-adic Eichler-Shimura for Siegel modular forms","Four-step p-adic Eichler-Shimura morphisms for GSp4 families"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Full p-adic Eichler-Shimura decomposition for GSp4 Siegel forms","All four Eichler-Shimura graded pieces now vary p-adically","Entire Eichler-Shimura decomposition interpolated p-adically for genus two","From H^0 to H^3: full p-adic Eichler-Shimura for Siegel modular forms","Four-step p-adic Eichler-Shimura morphisms for GSp4 families"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000709,"raw_usage":{"total_tokens":3192,"prompt_tokens":946,"completion_tokens":2246,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":2111}},"tokens_in":562,"tokens_out":2246,"duration_ms":16383,"temperature":1.0,"reasoning_tokens":2111,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:22:14.004823+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}