{"id":"1ecfeab0-aece-49ff-937c-4de7a1ca13b3","arxiv_id":"2412.17795","paper_version":2,"verdict":"REJECT","confidence":"LOW","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Aomoto dilogarithms on modular arrangements are claimed to be expressible as rational combinations of Bloch-Wigner dilogarithms at algebraic numbers.","lead":"The authors construct explicit differential forms on the square of the modular curve with poles along Hecke curves, and state that certain integrals (Aomoto dilogarithms) reduce to combinations of classical dilogarithms at algebraic numbers. The result is a new computational tool for periods in mixed Hodge theory, but the proof is only a sketch.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1 does not justify the Stokes boundary reduction to Rudenko's reciprocity: the divisor terms in (18) are never shown to satisfy the tame-symbol hypothesis (14), so the central equality is not established even if the constructed forms are closed.","rationale":"The reader's weakest_assumption is the unproved vanishing of Zagier's omega function in Section 3.5. I agree that this is a genuine gap: if that assertion is false, CoRes2(D) is not closed and the Aomoto integral loses its cohomological meaning. However, that assertion is cited to [Sa19] and is plausible in the harmonic Maass form framework. The more central missing link is the Stokes-to-Rudenko step in the proof of Theorem 1: even granting closedness, the paper does not derive its main formula. The proof is a sketch exactly where the arithmetic conclusion (rational coefficients at algebraic arguments) is produced, and the boundary terms from identity (18) are never controlled. This concern supports the reader's REJECT verdict, and I would not change it, but it identifies a different primary obstruction than the reader's weakest_assumption. No ad hominem is intended; the issue is completeness of the argument, not the authors' competence.","tokens_in":10497,"tokens_out":24926,"duration_ms":242894,"concrete_test":"Choose the smallest admissible Hauptmodule triangle for which the three Hecke curves intersect pairwise at distinct CM points (e.g., levels (2,3,6) if such a triangle exists; otherwise any small triple with genus-zero levels). Compute the pulled-back functions F, f, G, g on the fiber product Y^(2)(n',n) from (17), list all divisors C with nonzero ord_C(F) or ord_C(f), and evaluate the full Stokes boundary sum obtained from (18), including the delta-current corrections from (15). Check whether this sum equals the tame-symbol data required by Rudenko's theorem (14). If it does not match, Rudenko's reciprocity cannot be applied and Theorem 1 is unsupported; if it does match, the reduction mechanism is confirmed in a concrete non-trivial case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Load-bearing concern: the proof of Theorem 1 skips the step where the claimed reduction to Rudenko's reciprocity actually happens. After pulling back to the fiber product Y^(2)(n',n), the paper writes identity (18), whose right-hand side contains boundary currents supported on divisors C with ord_C(F) != 0 or ord_C(f) != 0. Integrating (18) and applying Stokes produces a sum of one-dimensional integrals of the form ∫_C r2(G,g) ∧ ∂log|f|^2 and ∫_C r2(G,g) ∧ ∂log|F|^2, together with point-current corrections from (12)/(15). To invoke Rudenko's theorem (14), the paper must prove that, after summing over all such divisors C and over the three sides of each triangle, these boundary terms combine into exactly the tame-symbol data required by (14). No such verification appears; the sentence 'by the Stokes formula, we reduce the integral to one-dimensional... now we apply result of Rudenko' skips precisely the point where the rational combination of dilogarithms is produced. If the boundary terms do not cancel or do not assemble into the tame-symbol relation, the central formula does not follow even if the forms are closed. This concern is independent of, and in addition to, the reader's flagged assertion about the vanishing of Zagier's omega function in Section 3.5.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs meromorphic differential forms with logarithmic singularities along Hecke curves in the square X×X of the modular curve, using the modular Cauchy kernel from the authors' previous work and pullbacks along the maps λ×ρ from X0(n)×X0(n). For a triangular arrangement of Hecke curves, it defines a 2-form CoRes2(D) by a signed sum of pullbacks of forms built from a Hauptmodul, asserts that this form is closed, and states Theorem 1: for an admissible pair of 'Hauptmodule modular arrangements' D and D′, the Aomoto dilogarithm integral of CoRes2(D′)∧CoRes2(D) equals a rational combination of values of the Bloch–Wigner dilogarithm (2πi)²D₂ at algebraic numbers. The proof is a sketch that reduces the integral by Stokes' formula to one-dimensional integrals and then invokes Rudenko's reciprocity law.","tokens_in":10795,"tokens_out":6893,"duration_ms":67237,"significance":"If the theorem is proved, it would give an explicit regulator formula for certain modular arrangements, a concrete test of the conjectural framework of mixed Hodge structure periods. The construction is interesting: it combines the modular Cauchy kernel, pullbacks from level structures, and Rudenko's strong Suslin reciprocity. The intended reduction is plausible, and the paper connects several nontrivial inputs from the literature. However, the proof as written leaves two load-bearing steps unverified: the closedness of the constructed form and the boundary-term analysis in the Stokes reduction. Both concerns raised in the stress-test note are substantiated by the text, so the central claim is not yet established.","major_comments":[{"comment":"The reduction to Rudenko's theorem is the central step and is not demonstrated. Starting from Eq. (18), applying Stokes' formula produces boundary currents supported on divisors C with ord_C(F)≠0 or ord_C(f)≠0; these give sums of one-dimensional integrals of the form ∫_C r₂(G,g)∧∂log|f|² and ∫_C r₂(G,g)∧∂log|F|², together with point-current terms from Eqs. (12)/(15). To invoke Rudenko's reciprocity (14), one must show that, after summing over all such divisors C and over the three sides of each triangle, these boundary terms combine exactly into the tame-symbol data required by (14). The sentence 'by the Stokes formula, we reduce the integral to one-dimensional ... now we apply result of Rudenko' skips precisely that verification. Without it, the claimed rational combination of dilogarithms does not follow even if the forms are closed.","section":"§4.3, proof of Theorem 1, Eq. (18)"},{"comment":"The asserted ∂-closedness of CoRes₂(D) rests on the unproved statement that Zagier's function ω(w,z̄) is a cusp form of weight 2 for the full modular group and therefore vanishes identically, following the displayed limit formula for lim_{s→1}(d/dz̄)Ξ(w,z,s). This assertion is load-bearing: if it fails, CoRes₂(D) is not closed and the Stokes argument in the proof of Theorem 1 collapses. The text says only 'It is easy to show' and gives no proof or precise reference; it is not demonstrated that the cited [Sa19] contains this statement. A complete proof of the vanishing (or an exact citation with the statement) must be supplied before the theorem can be accepted.","section":"§3.5, closedness of CoRes2(D)"},{"comment":"The conclusion 'at algebraic number' is not justified by the proof. Rudenko's theorem produces arguments ηⱼ that are values of the pulled-back functions at points on the curves; to conclude that the ηⱼ are algebraic, the proof must show that the J-values J_{Γ₀(n)}(a), J_{Γ₀(n)}(b), etc. at the CM intersection points are algebraic. This is presumably a complex-multiplication statement, but it is neither stated nor proved in Section 4.3.","section":"§4.3, algebraicity in Theorem 1"},{"comment":"The proof passes to the fiber product Y^(2)(n′,n) and then to a finite covering 'by enlarging level' without specifying the algebro-geometric setting. The modular curves here are orbifolds or stacks, and the Hecke correspondences may not be smooth in the coarse moduli space; the current identities (12) and (15) and Stokes' theorem require a precise statement about the smooth domain, the normal-crossing property of the divisors, and the convergence of the integrals. Without this, the formal manipulation with currents in Eq. (18) is not fully rigorous.","section":"§4.3, orbifold and current setup"}],"minor_comments":[{"comment":"There are repeated typos and terminological inconsistencies: 'Hautmodule' should be 'Hauptmodule'; 'Bellow' should be 'Below'; 'Maas' should be 'Maass'; 'Stocks' should be 'Stokes'; 'reponds' should be 'responds'; and the author line 'N. Sakharov A' in the header is garbled.","section":"Throughout"},{"comment":"The series defining Ξ_{0(n)}(z,w,s) in (1) is written with s appearing only in the exponents; it would help to state the domain of convergence and the precise meaning of the limit s→1, especially since this is an analytic continuation result quoted from [Sa15].","section":"§3.3, Definition 2"},{"comment":"The field F in Isog(F;a,b) is not defined; it should be F = O_Δ ⊗ Q, and the index function I(μ) should be defined before it is used.","section":"§2.3"},{"comment":"Equation (12) appears to have a parenthesis mismatch in the point-current term: '(ord_p(f₂)(φ₁(p)φ₃(p)) − ord_p(f₁)(φ₂(p)φ₃(p)))' should be checked and the line-breaking clarified.","section":"§4.2, Eq. (12)"},{"comment":"The notation h and h* appears in the residue computation without definition; the proof would be clearer if the representatives and the action of Γ₀(n) on the indices were made explicit.","section":"§3.4, proof of Lemma 1"},{"comment":"The term 'Hauptmodule modular arrangement' is used in Theorem 1 but is never formally defined; the informal sentence restricting to genus-zero correspondences should be turned into a precise definition.","section":"§4.3, after Definition 4"},{"comment":"The labels in Figure 1 (p_mn, p_ml, p_nl, ~p_mn, ~p_nl) are not introduced in the caption, making it hard to verify the geometry of the triangle and the preimages used in Definition 3.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is a very compressed note whose main theorem is presented as a proof sketch. The two load-bearing gaps – the unproved vanishing of ω(w,z̄) in §3.5 and the missing boundary-term analysis in the Stokes reduction in §4.3 – are both local and may be fixable, but until they are filled the central claim is not substantiated. I recommend major revision rather than outright rejection because the construction is plausible and the missing steps are identifiable; however, if the boundary terms do not assemble into Rudenko's tame-symbol data, the theorem would require substantial reformulation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper has a real idea: it builds explicit 2-forms CoRes2(D) on Y×Y with log poles along Hecke-curve triangles, and claims their Aomoto integrals evaluate to Bloch-Wigner dilogarithms at algebraic points. If Theorem 1 is true, it is a useful bridge between modular-curve periods and classical dilogarithms, and the construction of the forms themselves is new — it extends the authors' modular Cauchy kernel work and applies Rudenko's reciprocity. That part is worth reading.\n\nThe problem is that the proof of Theorem 1 is a sketch with two load-bearing holes. First, closedness of the form CoRes2(D) (Definition 4) is made to depend on the assertion in Section 3.5 that Zagier's ω(w, \\bar z) is a cusp form of weight 2 for the full modular group and therefore vanishes. That vanishing is likely true — there are no nonzero cusp forms of weight 2 for SL2(Z) — but the paper just says 'It is easy to show' and gives no argument. Since the ∂bar-closedness is what makes the integral well-defined, this needs a proof, not a shrug.\n\nSecond, and more serious, the reduction to Rudenko's theorem in the proof of Theorem 1 skips the actual boundary bookkeeping. Equation (18) writes a differential identity with boundary currents supported on divisors where F or f has zeros/poles. Applying Stokes leaves you with one-dimensional integrals like ∫_C r2(G,g) ∧ ∂log|f|^2. To apply Rudenko's reciprocity (14) you need to show that, after summing over all such C and all triangle sides, these terms assemble into exactly the tame-symbol relation the theorem requires. The proof jumps from (18) to 'now we apply Rudenko' without that verification. That is not a cosmetic gap; without it, the rational combination of dilogarithms doesn't follow. The stress-test note is right about this.\n\nThere are also smaller issues: the 'Hauptmodule' restriction is used but not sharpened, and the convergence of the Aomoto integral is asserted rather than demonstrated. The paper trusts its earlier work [Sa15, Sa19] for the Cauchy kernel, which is fine since those are published, independent results.\n\nOverall: the construction is credible and the intended result is plausible, but the main theorem is not established in this version. This is a paper an editor should send to a knowledgeable referee — it deserves serious engagement — but it needs a complete proof of closedness and the Stokes/Rudenko step before it can be accepted. I would not rely on Theorem 1 yet.","headline":"The construction is novel and the direction is right, but the main theorem is not proved: closedness of CoRes2(D) and the Stokes/Rudenko reduction are asserted, not shown.","tokens_in":11289,"tokens_out":3208,"would_cite":false,"duration_ms":30101,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G35","14C30","11F11","19F27"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for admissible pairs of Hauptmodule modular arrangements, the Aomoto dilogarithm pairing of two co-residue 2-forms on the square of a modular curve equals a rational combination of Bloch-Wigner dilogarithm values at…","keywords":["modular curves","Hecke correspondences","Aomoto dilogarithm","Bloch-Wigner dilogarithm","logarithmic forms","mixed Hodge structures","Hauptmodule arrangements","regulators"],"falsifier":"Compute the first Fourier coefficient of $\\omega(w,\\bar z)$ as a function of $w$ at a cusp: any nonzero coefficient refutes the asserted vanishing and invalidates closedness of $\\operatorname{CoRes}_2(D)$; alternatively, numerically integrate $d\\operatorname{CoRes}_2(D)$ over a cycle in the complement of the triangle and look for a nonzero result.","tokens_in":10292,"feed_emoji":"📐","tokens_out":8145,"duration_ms":74180,"temperature":0.7,"pith_summary":"This paper tries to establish an explicit regulator formula: for certain finite arrangements of Hecke curves in the square of a modular curve, the pairing integral of two logarithmic differential forms reduces to a rational combination of classical Bloch-Wigner dilogarithms evaluated at algebraic numbers. Such a formula matters because these pairings are periods of mixed Hodge structures attached to the modular square, and classical dilogarithms are the simplest transcendental building blocks for regulators. The paper constructs closed meromorphic 2-forms with prescribed residues on the Hecke curves, then uses Stokes' theorem and a reciprocity law for tame symbols to cut the four-dimensional integral down to one-dimensional integrals and finally to dilogarithm values.","feed_headline":"Modular arrangement periods reduce to classical dilogarithms","feed_subtitle":"Pairings of Hecke-curve arrangements become rational combinations of Bloch-Wigner dilogarithms","key_machinery":"The load-bearing object is the modular Cauchy kernel $C_{0(n)}(z,w)$, a $(1,0)$-form with logarithmic singularities along the Hecke curve $T_n$; for genus-zero level it takes the explicit form $\\partial \\log|J_{\\Gamma_0(n)}(z)-J_{\\Gamma_0(n)}(w)|^2$. This kernel, pushed forward along the maps $\\lambda\\times\\rho: Y_0(n)\\times Y_0(n)\\to Y\\times Y$, produces the co-residue forms $\\operatorname{CoRes}_2(n,a,b)$ whose residues are differences of Cauchy kernels at the chosen CM points. The final reduction rests on the identity expressing $\\operatorname{CoRes}_2(n,a,b)$ as the pullback of a wedge product of two such single-logarithm forms, on a fiber-product cover that makes both arrangements pull back to one surface, and on a reciprocity law for tame symbols that converts the resulting one-dimensional integrals into Bloch-Wigner dilogarithm values.","core_discovery":"The central claim is Theorem 1: for an admissible pair $(D,D')$ of Hauptmodule modular arrangements, the Aomoto dilogarithm\n$$\\int_{$Y^{2}$} \\operatorname{CoRes}_2(D') \\wedge \\operatorname{CoRes}_2(D)$$\nequals a rational linear combination of values $(2\\pi i)^2 D_2(\\eta_j)$ with each $\\eta_j$ algebraic. The proof realizes each side of the triangle arrangement as the pullback, under the maps $\\lambda\\times\\rho$ from a product of level-$n$ modular curves, of the single-logarithm form $\\partial \\log|J(z)-J(w)|^2$, rewrites $\\operatorname{CoRes}_2(D)$ as an alternating sum of three such terms attached to the three vertices of the triangle, passes to a finite fiber-product cover, applies Stokes' theorem to reduce the integral to curves, and invokes a reciprocity law to express the remaining curve integrals as sums of Bloch-Wigner dilogarithms.","pith_inferences":["Inference: the algebraic arguments $\\eta_j$ appearing in the final combination should be the CM intersection points of the chosen Hecke curves, which would make the theorem numerically checkable by computing both sides at small levels to high precision.","Inference: the same mechanism may extend to arrangements whose components have positive genus if the Cauchy-kernel single-logarithm expression is replaced by a suitable regularized kernel, though the Hauptmodule assumption is what makes the formulas fully explicit.","Inference: if the unproved vanishing of the auxiliary series $\\omega(w,\\bar z)$ fails, one could still attempt to repair the theorem by proving closedness of the assembled combination $\\operatorname{CoRes}_2(D)$ directly, rather than of each summand."],"forward_implications":["For every admissible Hauptmodule pair, the Aomoto dilogarithm is not a new transcendental but a rational combination of dilogarithm values at algebraic arguments.","The same construction applies to any cyclic chain, or polygon, of Hecke curves, so the reduction is not tied to the specific triangle arrangement.","Because the square of the modular curve is the moduli space of split abelian surfaces, the formula computes explicit periods of mixed Hodge structures arising from Hecke-curve arrangements in that moduli space.","The co-residue forms are claimed to be closed and to have the prescribed logarithmic residues, giving explicit representatives for cohomology classes of the complement of the arrangement."],"supporting_citations":[{"why":"Supplies the construction and convergence of the Zagier-type series $\\Xi_{0(n)}(z,w,s)$ at $s=1$ that defines the modular Cauchy kernel.","marker":"[Sa15]"},{"why":"Proves the single-logarithm form $\\partial \\log|J-J|^2$ for the modular Cauchy kernel and is cited for the assertion that $\\omega(w,\\bar z)$ vanishes.","marker":"[Sa19]"},{"why":"Provides the reciprocity law for tame symbols that converts the one-dimensional integrals obtained after Stokes' theorem into Bloch-Wigner dilogarithm values.","marker":"[Ru15]"},{"why":"Gives the regulator formalism with the forms $r_2$ and $r_3$ and the dilogarithm identities used to rewrite the pairings.","marker":"[Go02]"},{"why":"Introduces the auxiliary function $\\omega(w,\\bar z)$ whose vanishing is the load-bearing analytic input for closedness of the co-residue form.","marker":"[Za75]"}],"fun_headline_variants":["Modular arrangement periods collapse to Bloch-Wigner dilogs","Square of modular curve yields rational dilogarithm sums","Aomoto dilogs on modular squares become Bloch-Wigner terms","Hecke-curve pairings on modular square reduce to dilogs","Rational combinations of dilogs from modular arrangement integrals"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on the unproved assertion that the auxiliary series $\\omega(w,\\bar z)$ vanishes identically; without that vanishing, the co-residue form is not $\\bar\\partial$-closed and the Stokes reduction in the proof of Theorem 1 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Modular arrangement periods collapse to Bloch-Wigner dilogs","Square of modular curve yields rational dilogarithm sums","Aomoto dilogs on modular squares become Bloch-Wigner terms","Hecke-curve pairings on modular square reduce to dilogs","Rational combinations of dilogs from modular arrangement integrals"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000603,"raw_usage":{"total_tokens":2775,"prompt_tokens":864,"completion_tokens":1911,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":480,"completion_tokens_details":{"reasoning_tokens":1829}},"tokens_in":480,"tokens_out":1911,"duration_ms":13230,"temperature":1.0,"reasoning_tokens":1829,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:08:23.159465+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the first Fourier coefficient of $\\omega(w,\\bar z)$ as a function of $w$ at a cusp: any nonzero coefficient refutes the asserted vanishing and invalidates closedness of $\\operatorname{CoRes}_2(D)$; alternatively, numerically integrate $d\\operatorname{CoRes}_2(D)$ over a cycle in the complement of the triangle and look for a nonzero result.","supporting_citations":[],"review_version":1}