{"id":"dd1750a8-93fa-46aa-a6c9-3577aa930a70","arxiv_id":"2509.00434","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Blasius-Deligne conjecture is proved for standard L-functions of symplectic type for GL_{2n}, for all n at least 1, assuming a nonvanishing condition when the base field has complex places.","lead":"This paper proves the Blasius-Deligne conjecture for critical values of standard L-functions of symplectic type for GL_{2n}, for every n at least 1. The result establishes Galois-invariant period relations and algebraicity of these L-values, completing a project begun in 2019.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claimed unconditional proof is conditional: Theorem 1.4 requires Assumption 1.3 to define Shalika periods; no proof or known theorem establishes it.","rationale":"The reader's weakest_assumption identifies Assumption 1.3 as the central fragile point, and I agree. The paper itself explicitly states that without Assumption 1.3 a canonical family of Shalika periods is unavailable when k has a complex place, and Definition 10.3 is made under Assumption 1.3. Since Theorem 1.4 states its reciprocity law using those periods, the theorem is conditional even though the abstract says 'unconditional'. The proposed check directly tests whether Assumption 1.3 can be removed; if it cannot, the original verdict must remain CONDITIONAL. The reader already returned CONDITIONAL, so no change is needed.","tokens_in":52024,"tokens_out":7960,"duration_ms":107271,"concrete_test":"Re-derive Lemma 10.2 and the definition of Omega_epsilon in Section 10.2 without invoking Assumption 1.3, using only the known nonvanishing of Archimedean modular symbols [JST19, Theorem 3.11] and the rationality of the quotient H(sigma Pi_infty)[epsilon]/Ker sigma(℘_epsilon). If the kernel-invariance argument fails for complex places, then Assumption 1.3 is a genuine load-bearing condition and the theorem must be labeled conditional; if it succeeds, the unconditional claim can be restored.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive issue is not any local computation but the existence of the period family that normalizes the central identity. Assumption 1.3 (Section 1) postulates some sigma' in Aut(C) and a balanced algebraic Hecke character chi' with chi'_sharp = epsilon and L(1/2, sigma' Pi tensor sigma' chi') != 0. Definition 10.3 then defines Omega_epsilon(sigma Pi, sigma eta) = 1 / sigma(℘_epsilon)(sigma kappa_epsilon), and the proof of Lemma 10.2 uses Assumption 1.3 to show that sigma preserves the codimension-one kernel Ker ℘_epsilon. The authors explicitly say that without Assumption 1.3 such canonical periods are currently unavailable when k has a complex place. Thus Theorem 1.4 is not an unconditional theorem as claimed in the abstract; it is conditional on an unproved nonvanishing statement. If Assumption 1.3 fails, then for every balanced chi with chi_sharp = epsilon the central value L(1/2, sigma Pi tensor sigma chi) is zero, so the reciprocity statement is vacuous (at least if k contains no CM field), but the period Omega_epsilon in the denominator is not defined, and the proof of Theorem 1.4 cannot even be formulated. This is not a matter of exposition: the normalization in the main identity is undefined in the residual case.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves a Blasius–Deligne reciprocity law for the central critical values of the standard L-functions of symplectic type for GL_{2n}, n ≥ 1, with general algebraic Hecke characters. The central result is Theorem 1.4: for a regular algebraic cuspidal automorphic representation Π of symplectic type, the ratio L(1/2, Π⊗χ)/(Ω_{μ,χ♮} G(χ)^n Ω_ε(Π,η)) is Aut(C)-equivariant and lies in the rationality field Q(Π,η,χ). The proof is built on a large local theory: Archimedean Jacquet–Shalika integrals, modifying factors for exterior-square and Friedberg–Jacquet integrals, refined Archimedean period relations via translation functors, and a global modular-symbol formalism. Shalika periods are defined in Definition 10.3, subject to Assumption 1.3 when k has a complex place.","tokens_in":52333,"tokens_out":5995,"duration_ms":75725,"significance":"If the proof is completed as written, this is a major result: it gives the first general treatment of Blasius–Deligne periods for GL_{2n}-standard L-functions of symplectic type with arbitrary algebraic Hecke characters, not only finite-order characters as in [JST19]. The local results are substantial and interesting in their own right: Theorem 2.2 establishes the Archimedean theory of Jacquet–Shalika integrals for principal series, and Theorems 2.6 and 2.15 provide modifying factors with arithmetic predictions in the style of Coates–Perrin-Riou. The paper is careful and largely self-contained, and it explicitly identifies its main assumption. However, the advertised unconditional status is not supported, and two local arguments are asserted rather than proved.","major_comments":[{"comment":"The abstract claims an unconditional proof, but Theorem 1.4 is conditional as stated. Assumption 1.3 postulates the existence of σ′ ∈ Aut(C) and an algebraic Hecke character χ′ with χ′♮ = ε and L(1/2, σ′Π⊗σ′χ′) ≠ 0. Definition 10.3 defines the Shalika periods Ω_ε(σΠ,ση) only under Assumption 1.3 when k has a complex place, and Lemma 10.2 uses Assumption 1.3 to show that σ preserves Ker ℘_ε°. The text explicitly says that without Assumption 1.3 such a canonical period family is currently unavailable when k has a complex place. Thus the denominator in (1.5) is undefined in the residual case; the observation that all relevant central values would vanish does not repair the formulation. The theorem should be restated as conditional on Assumption 1.3, or the assumption should be proved and removed.","section":"§1 (Assumption 1.3), §10 (Definition 10.3), Theorem 1.4"},{"comment":"The proof of Theorem 2.4 (FE′_m) contains two omitted arguments in the odd case. In §5.1 the meromorphic continuation for m = 2n+1 is dismissed as 'similar' with details omitted. In §5.3, Lemma 5.2—which is essential for the odd-case functional equation (2.17)—is stated with the proof omitted ('can be verified directly'). These are load-bearing: Theorem 2.4 is an induction hypothesis for the proofs of Theorems 2.2 and 2.6. A similar omission occurs in Proposition 3.7, where the proof explicitly treats only the even case. Please provide full proofs or precise references for these steps.","section":"§5.1 and §5.3 (Lemma 5.2, odd-case continuation)"},{"comment":"The statement of Theorem 1.4 does not list Assumption 1.3 among its hypotheses; it refers only to 'the family of Shalika periods in Definition 10.3.' Since Definition 10.3 is conditional on Assumption 1.3 when k has a complex place, the statement of the main theorem is ambiguous about its hypotheses. This is a presentation issue, but it is directly connected to the unconditional claim and should be fixed in the revision.","section":"Theorem 1.4 vs. Definition 10.3"}],"minor_comments":[{"comment":"The title has a line break artifact 'ST ANDARD', and Section 8 is headed 'Archimdedean period relations' instead of 'Archimedean period relations'.","section":"Title and Section 8 heading"},{"comment":"Lemma 10.5 shows that a different choice of κ_ε changes the Shalika period by a scalar in Q(σΠ,ση)^×. For the exact reciprocity identity (1.5), it would be clearer to state that the scalar is the image under σ of the corresponding scalar in Q(Π,η)^×, so that the family transforms coherently under Aut(C).","section":"§10.2, Lemma 10.5"},{"comment":"The notation P^◦_∞⊗P^◦_f in the commutative diagram is not defined explicitly; it should be labeled as the product of the normalized Archimedean and non-Archimedean modular symbols.","section":"§9.2.4, diagram (9.7)"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely correct as a conditional theorem, and the local theory is a substantial contribution. The main issue is the gap between the abstract's 'unconditional' claim and the actual Assumption 1.3. Since this is a load-bearing hypothesis and not merely an expository choice, I recommend major revision rather than acceptance. The authors should either prove the nonvanishing assumption (unlikely in this framework) or restate the main theorem as conditional and adjust the abstract accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dihua and coauthors have done the real thing: Theorem 1.4 is the first reciprocity law for Blasius-Deligne in this setting with general algebraic Hecke characters, going beyond the finite-order case of [JST19]. The Archimedean period relations (Theorem 2.16) obtained via Zuckerman translation functors and modifying factors are a genuine methodological step forward, and the uniform Jacquet-Shalika local theory (Theorem 2.2) is a substantial stand-alone contribution. The paper is long, but the organization is clear and the body is candid about what it assumes.\n\nThe candid part matters, because the abstract says 'unconditional' and that is not quite true. Theorem 1.4 requires Assumption 1.3 (existence of a twist with nonzero central value) in order to define the Shalika periods in Definition 10.3 when k has a complex place. If the assumption fails, the normalized identity in (1.5) is not defined — this is not cosmetic. The authors themselves state that without it no canonical period family is currently available, and Remark 10.4 flags the issue as open. So the correct summary is: a conditional theorem in the general case, unconditional in the totally real case. The paper should say that in the abstract, and the referee should make it a required revision.\n\nThe other soft spots are smaller. Lemma 5.2 and the odd-case continuation are asserted without full detail — common in this literature, but it means the local chain is not fully checkable from the printed page. The heavy use of prior work by the same group ([JST19], [LLS24], [LLSS23], [LS25]) is not circular: the Shalika periods are defined through integrals and modular symbols, not as reciprocals of the L-values under study. I read the circularity concern in the stress-test note as resolved on inspection.\n\nBottom line: with the conditional framing made honest, this is a major result that deserves careful refereeing. It is written for specialists in automorphic forms and period relations; most of the benefit will be felt in p-adic L-function work. I would send it to a top journal and would cite it, with the Assumption 1.3 caveat, in my own work. Push the authors to either prove or clearly isolate Assumption 1.3, and to supply the omitted local proofs.","headline":"The reciprocity law for general algebraic Hecke characters is new and important, but the abstract's 'unconditional' framing is not backed by the body: Assumption 1.3 is needed to define the Shalika periods and is itself unproved. Still deserves a serious referee.","tokens_in":52844,"tokens_out":3512,"would_cite":true,"duration_ms":39991,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["22E50","43A80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a Galois reciprocity law for normalized central L-values of regular algebraic cuspidal automorphic representations of GL(2n) of symplectic type, yielding algebraicity in explicit rationality fields.","keywords":["Blasius-Deligne conjecture","critical L-values","symplectic type","Shalika periods","Jacquet-Shalika integrals","Friedberg-Jacquet integrals","modifying factors","Galois reciprocity"],"falsifier":"Take a concrete Pi of symplectic type over a number field with a complex place, already at n=1 a Hilbert modular form of CM type, choose epsilon and a balanced chi satisfying Assumption 1.3, and compute both sides of (1.5) for a nontrivial sigma in Aut(C) using two different auxiliary classes kappa_epsilon, kappa'_epsilon. If the normalized values depend on this choice beyond a factor in Q(sigma Pi, sigma eta)^x, or if the normalized value fails to lie in Q(Pi, eta, chi), the period family is not canonical and the theorem fails.","tokens_in":51930,"feed_emoji":"🔢","tokens_out":11633,"duration_ms":126002,"temperature":0.7,"pith_summary":"The paper proves the Blasius-Deligne conjecture for the standard L-functions of symplectic type on GL_{2n}: central critical values, once normalized by a canonical family of Shalika periods, Gauss sums, and a local factor, are algebraic numbers in explicit rationality fields and satisfy a reciprocity law under Galois automorphisms. This is the first treatment of these L-functions with arbitrary algebraic Hecke characters rather than only finite-order characters, and it completes a project begun in earlier work. The reciprocity identity says that applying any automorphism of the complex numbers to the normalized value at Pi tensor chi gives the same normalized value at the twisted representation. The proof is carried by a new archimedean local theory: Jacquet-Shalika integrals with their functional equations, open-orbit modifying factors, and refined period relations for Friedberg-Jacquet integrals. The only auxiliary hypothesis is used to define the Shalika periods when the base field has a complex place; without it the reciprocity statement would be vacuous in the relevant cases.","feed_headline":"Normalized GL(2n) L-values of symplectic type are Galois-invariant","feed_subtitle":"A reciprocity identity pins down the algebraicity of these critical L-values for every algebraic Hecke character.","key_machinery":"The central object is the family of Shalika periods Omega_epsilon(sigma Pi, sigma eta) of Definition 10.3. Each period is the value of an archimedean modular symbol on a distinguished cohomology class kappa_epsilon, chosen so that the non-archimedean modular symbol is rational; the quotient space H(Pi_infinity)[epsilon] / Ker is one-dimensional and, by Lemma 10.2, defined over the rationality field. The proof that these periods transform correctly under Aut(C) is carried by the refined archimedean period relations (Theorem 2.16), obtained from local zeta integrals: the archimedean theory of Jacquet-Shalika exterior-square integrals, the open-orbit comparison producing modifying factors, and","core_discovery":"The central result is Theorem 1.4. For a regular algebraic cuspidal automorphic representation Pi of GL_{2n}(A) of symplectic type and each epsilon, the identity sigma( L(1/2, Pi tensor chi) / (Omega_{mu,chi_sharp} G(chi)^n Omega_epsilon(Pi, eta)) ) = L(1/2, sigma Pi tensor sigma chi) / (Omega_{mu,chi_sharp} G(sigma chi)^n Omega_epsilon(sigma Pi, sigma eta)) holds for every sigma in Aut(C) and every algebraic Hecke character chi whose archimedean type is F_mu-balanced with quadratic part epsilon. Hence the normalized value lies in Q(Pi, eta, chi). The family of Shalika periods Omega_epsilon(sigma Pi, sigma eta) is defined canonically under Assumption 1.3; the proof uses the archimedean perio","pith_inferences":["If Lemma 10.2 can be proved without Assumption 1.3, as the authors expect, the same theorem would hold with no auxiliary existence hypothesis even over fields with complex places; this is a concrete open step.","The same architecture of open-orbit integrals, modifying factors, and translation functors is likely to yield analogous reciprocity laws for other spherical-pair L-functions, suggesting a uniform mechanism rather than a case-by-case phenomenon.","The local modifying factors identify a canonical normalization for p-adic interpolation; one can test this by constructing the corresponding p-adic L-function outside the nearly ordinary case and checking that its special values recover the algebraic numbers of Theorem 1.4."],"forward_implications":["Every normalized central value L(1/2, Pi tensor chi) / (Omega_{mu,chi_sharp} G(chi)^n Omega_epsilon(Pi, eta)) is algebraic, explicitly in the rationality field Q(Pi, eta, chi).","If L(1/2, Pi tensor chi) is nonzero, then L(1/2, sigma Pi tensor sigma chi) is nonzero for every sigma in Aut(C): vanishing of the central value is a Galois-invariant property.","The archimedean Jacquet-Shalika integrals are now known to converge, admit meromorphic continuation, satisfy functional equations with the expected Artin factors, and have nonvanishing normalized values under mild hypotheses, for principal series over every local field.","The modifying factors obtained by comparing Jacquet-Shalika and open-orbit integrals match the p-adic predictions, so the same local periods should serve as normalizations for p-adic L-functions of exterior-square and standard type.","The theorem covers all n >= 1 and all algebraic Hecke characters, extending the earlier finite-order-character case."],"supporting_citations":[{"why":"Predecessor that proved the algebraicity for finite-order characters and supplied the nonvanishing of archimedean modular symbols on which the period definition relies.","marker":"[JST19]"},{"why":"Introduced the Friedberg-Jacquet linear periods whose local zeta integrals define the L-function and the periods.","marker":"[FJ93]"},{"why":"Introduced the Jacquet-Shalika exterior square integrals whose archimedean theory the paper develops.","marker":"[JS90]"},{"why":"Established the analogous period relations for the GL(n) x GL(n-1) convolution via translation functors and modifying factors, the template for the refined archimedean relations here.","marker":"[LLS24]"},{"why":"Gave the open-orbit comparison and modifying factors for the GL(n) x GL(n) convolution, used to prove the Jacquet-Shalika analogues.","marker":"[LLSS23]"},{"why":"Supplies the analytic framework for archimedean local zeta integrals: convergence, meromorphic continuation, and finite-order growth.","marker":"[BP21]"},{"why":"Supplies the archimedean theory of standard sections and convolution integrals used in the inductive proof.","marker":"[J09]"},{"why":"Provides the matrix-coefficient zeta integrals for GL_n used to relate the two local integrals in Theorem 2.15.","marker":"[GJ72]"},{"why":"Defines regular algebraic automorphic representations, coefficient systems, and the cohomological setup.","marker":"[Cl90]"}],"fun_headline_variants":["Galois invariance of critical L-values for symplectic GL(2n) proven","Cuspidal GL(2n) symplectic L-values rational under Galois action","Unconditional proof of Blasius-Deligne for symplectic L-functions","Blasius-Deligne conjecture settled for GL(2n) symplectic type","Critical L-values of symplectic GL(2n) are Galois-invariant"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is Assumption 1.3: that some Galois twist of Pi has a nonzero central L-value for some F_mu-balanced Hecke character with quadratic part epsilon, because without it the canonical Shalika periods of Definition 10.3 are not defined when k has a complex place; the proof also depends on the nonvanishing of the archimedean modular symbols established in [JST19].","fun_headline_variants_meta":{"raw":{"variants":["Galois invariance of critical L-values for symplectic GL(2n) proven","Cuspidal GL(2n) symplectic L-values rational under Galois action","Unconditional proof of Blasius-Deligne for symplectic L-functions","Blasius-Deligne conjecture settled for GL(2n) symplectic type","Critical L-values of symplectic GL(2n) are Galois-invariant"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000662,"raw_usage":{"total_tokens":2809,"prompt_tokens":639,"completion_tokens":2170,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":383,"completion_tokens_details":{"reasoning_tokens":2056}},"tokens_in":383,"tokens_out":2170,"duration_ms":16785,"temperature":1.0,"reasoning_tokens":2056,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T13:34:40.910266+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete Pi of symplectic type over a number field with a complex place, already at n=1 a Hilbert modular form of CM type, choose epsilon and a balanced chi satisfying Assumption 1.3, and compute both sides of (1.5) for a nontrivial sigma in Aut(C) using two different auxiliary classes kappa_epsilon, kappa'_epsilon. If the normalized values depend on this choice beyond a factor in Q(sigma Pi, sigma eta)^x, or if the normalized value fails to lie in Q(Pi, eta, chi), the period family is not canonical and the theorem fails.","supporting_citations":[],"review_version":1}