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On the Blasius-Deligne conjecture for the standard $L$-functions of symplectic type for $\textrm{GL}_{2n}$

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2509.00434 v1 pith:KIE2KAMT submitted 2025-08-30 math.NT math.RT

Dihua Jiang , Dongwen Liu , Binyong Sun , Fangyang Tian This is my paper · ORCID
classification math.NTmath.RT MSC 22E5043A80
keywords Blasius-DeligneconjecturecriticalL-valuessymplectictypeShalikaperiodsJacquet-ShalikaintegralsFriedberg-JacquetmodifyingfactorsGaloisreciprocity
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 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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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].

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

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Referee Report

3 major / 3 minor

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.

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 (3)
  1. [§1 (Assumption 1.3), §10 (Definition 10.3), Theorem 1.4] 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.
  2. [§5.1 and §5.3 (Lemma 5.2, odd-case continuation)] 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.
  3. [Theorem 1.4 vs. Definition 10.3] 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.
minor comments (3)
  1. [Title and Section 8 heading] The title has a line break artifact 'ST ANDARD', and Section 8 is headed 'Archimdedean period relations' instead of 'Archimedean period relations'.
  2. [§10.2, Lemma 10.5] 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).
  3. [§9.2.4, diagram (9.7)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation chain consists of independent local zeta-integral identities and global modular-symbol comparisons, with no equation reducing to itself by construction; the principal limitation is the conditional Assumption 1.3, which is a nonvanishing input rather than a disguised form of the target result.

full rationale

The central reciprocity identity (1.5) is obtained by chasing the commutative diagram (10.7), whose commutativity rests on independent ingredients: the local Jacquet–Shalika and Friedberg–Jacquet zeta-integral identities (Theorems 2.2, 2.6, 2.15), the Archimedean period relation (Theorem 2.16, proved via Zuckerman translation functors and modifying factors in Sections 5–8), and the global–local modular-symbol comparison (Proposition 9.1). The Shalika periods in Definition 10.3 are defined as reciprocals of Archimedean modular-symbol values at an arbitrary class κ_ε not in Ker ℘°_ε; they are not fitted to the final L-values, and Lemma 10.5 shows only that the dependence on κ_ε is up to Q(σΠ,ση)^×, which is a standard period normalization. In the diagram chase the Ω_ε factors cancel, leaving the honest content that L/(Ω·G^n) is σ-equivariant, so the result is not true by construction. The paper does rely on prior work by overlapping authors ([JST19, Theorem 3.11], [LLS24], [LLSS23], [CS20], [SZ12], [S12]); these are cited as proven, parameter-free theorems with stated assumptions that do not include the target reciprocity law, and they are not merely asserted in this paper. The only serious limitation is Assumption 1.3: when k has a complex place, Definition 10.3 and Lemma 10.2 require an unproved nonvanishing statement L(1/2,σ′Π⊗σ′χ′)≠0; without it the period family is currently unavailable, so the abstract's word 'unconditional' is too strong. This is a conditionality/gap, not a circularity: the assumption is an input, not the reciprocal of the target period or a hidden restatement of (1.5). Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof relies on standard background in local Langlands, Casselman-Wallach theory, uniqueness of Shalika and Rankin-Selberg periods, and prior papers by the same authors and collaborators ([JST19], [LLS24], [LLSS23], [LS25]). These are treated as background results, not reproved. Assumption 1.3 is the only non-standard input; it is a domain assumption explicitly stated and needed for complex places. No free parameters are fitted: all constants are fixed by integrals and cohomology. No new physical or formal entities are introduced; the Shalika periods are normalization constants derived from integrals.

assumptions (4)
  • standard math Local Langlands correspondence for GL_m(k) and the associated Artin local factors for exterior square and standard representations exist and are compatible with parabolic induction.
    Invoked in Section 2.1.1 to define L(s, pi, wedge^2 tensor eta^-1), epsilon, gamma factors via [CST17, Sh24]. For GL_n over local fields this is established theory.
  • standard math Shalika model uniqueness and Casselman-Wallach theory for irreducible admissible representations over archimedean fields.
    Used throughout, e.g., Sections 3.1 and 10.1; uniqueness results from [AGJ09], [SZ12]. The paper relies on these for factorization of global Shalika functionals.
  • standard math Godement sections, Jacquet integrals, and the Rankin-Selberg and Godement-Jacquet functional equations for principal series as stated in [J09] and [GJ72].
    These underpin the inductive proof of Theorem 2.2 and Theorem 2.6 in Section 6, and the proof of Theorem 2.15 in Section 7.
  • domain assumption Assumption 1.3: existence of sigma' in Aut(C) and algebraic Hecke character chi' such that chi'_sharp is F_mu-balanced, chi'_sharp = epsilon, and L(1/2, sigma' Pi tensor sigma' chi') is nonzero.
    Required for the definition of the canonical Shalika periods in Definition 10.3 when the base field k has a complex place. Not proved in the paper; if it fails the main identity is not formulated for those fields.

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Pith. "Pith review of On the Blasius-Deligne conjecture for the standard $L$-functions of symplectic type for $\textrm{GL}_{2n}$." pith.science (2026). https://pith.science/paper/KIE2KAMT

@misc{pith2026250900434,
  author       = {Pith},
  title        = {Pith review of: On the Blasius-Deligne conjecture for the standard $L$-functions of symplectic type for $\textrmGL_2n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KIE2KAMT}},
  note         = {Machine review of arXiv:2509.00434}
}
abstract

In this paper we give an unconditional proof of the Blasius-Deligne conjecture for the critical values of the $\textrm{GL}_{2n}$-standard $L$-functions of symplectic type with $n\geq 1$ and complete the project started in [JST19].

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  1. Finite-Sum Realization of Archimedean Asai and Exterior-Square $L$-Factors

    math.NT 2026-08 conditional novelty 7.0 of 10

    The archimedean Asai L-factor for GL_n(C) and the exterior-square L-factor for GL_m(F) are finite sums of Flicker and Jacquet-Shalika local zeta integrals, respectively.

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