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Period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$

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Pith's one-line read The paper proves rationality and Aut(C)-equivariant period relations for critical values of Rankin-Selberg L-functions for GL(n)×GL(n) over any number field containing a CM field.

desk verdict Strong Eisenstein cohomology work with a real finite-place gap in Theorem 1.2 as stated; worth refereeing, but needs revision. read the letter →

arxiv 2506.20942 v4 pith:JCNQBZUZ submitted 2025-06-26 math.NT math.RT

Yubo Jin , Jian-Shu Li , Dongwen Liu , Binyong Sun This is my paper · ORCID
classification math.NTmath.RT MSC 11F6711F7011F7522E45
keywords Deligne'sconjectureRankin-SelbergconvolutionperiodrelationEisensteincohomologyseriesmodularsymbolsL-functionspecialvaluesGL(n)
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

This paper establishes a period relation for critical values of Rankin-Selberg L-functions for GL(n)×GL(n), over any number field that contains a CM field. The main theorem says that for a regular algebraic representation Π=Σ⊠Σ′ with Σ tamely isobaric and Σ′ cuspidal, and for every balanced algebraic Hecke character χ, the value L(0,Π×χ) becomes rational after dividing by an explicit product: a period of the central character, a power of a Gauss sum, an archimedean constant, and a Whittaker period. The resulting quotient is compatible with all field automorphisms of the complex numbers, so Galois conjugation of the data is mirrored by Galois conjugation of the quotient. This gives the most general period relation currently available for the square Rankin-Selberg case, and the paper points out that the new elements of the method also improve the earlier GL(n)×GL(n−1) results.

What carries the argument

The central mechanism is the Eisenstein cohomology map Eis_η from the degree-c_n cohomology of the degenerate principal series J_η to the sheaf cohomology of the Borel-Serre compactification of the symmetric space, where η=ω_Πχ^n. This map is built from the Eisenstein series attached to the induced representation I_η, normalized so that it is holomorphic and embedded at s=0. The modular symbol P_{Π,χ} is obtained by pairing the cohomology class of Π with this Eisenstein class and the character class, and then integrating over the fundamental class. The proof of Galois equivariance reduces to Theorem 1.3, whose argument uses the constant-term formula for the Eisenstein series, the explicit structure of the archimedean cohomology space, and a rationality statement for normalized intertwining operators.

What would settle it

Take n=2 over an imaginary quadratic field, choose a concrete regular algebraic Π from a CM elliptic curve and a balanced character χ, and compute both sides of the Aut(C)-equivariance law (1.8) under complex conjugation; a single mismatch between the conjugated quotient and the quotient for the conjugated data would falsify the theorem, while agreement would only confirm this special case.

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Extended reading notes

Core claim

The load-bearing assertion is Theorem 1.2. For Π=Σ⊠Σ′, regular algebraic with coefficient system F_μ⊠F_ν, with Σ tamely isobaric and Σ′ cuspidal, and for every χ in B(μ,ν)+, the quotient in (1.6) lies in the compositum Q(Π,χ) and obeys the Aut(C)-equivariance law (1.8). The proof is a modular-symbol argument: the global Rankin-Selberg integral factorizes into a normalized archimedean period and a normalized non-archimedean period, and the genuinely new ingredient is the rationality of the Eisenstein cohomology for degenerate principal series (Theorem 1.3). That theorem states that the Eisenstein map on the bottom-degree cohomology commutes with every field automorphism of the complex numbers, and it is proved by identifying the boundary restriction of the Eisenstein class with a sum of normalized intertwining operators and checking rationality on an explicit archimedean generator.

Load-bearing premise

The proof leans on a previously established archimedean period relation: the archimedean modular symbol is nonzero on the chosen generator and transforms exactly as prescribed under field automorphisms; if that failed, the global commutativity diagram would not close even though the Eisenstein-cohomology theorem is new.

Editorial extensions

If this is right

  • The quotient in (1.6) is well defined up to field automorphisms, so a rationality field Q(Π,χ) can be attached to every critical value in the admissible range.
  • The same archimedean and non-archimedean improvements, dropping a restrictive integral-twist condition and allowing a broader class of isobaric representations, carry over to the GL(n)×GL(n−1) period relations.
  • Because Σ itself may be induced from cuspidal pieces, the theorem supplies a new base case for inductively proving period relations for more general Rankin-Selberg convolutions.
  • Specializing n=1 recovers Deligne's conjecture for Hecke characters, and the Eisenstein-cohomology theorem gives an independent proof of that conjecture.
  • For n=2 the result overlaps with previously known GL(2)×GL(2) period relations, while the theorem works for all n.

Reading between the lines

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

  • Pith inference: One could numerically test the archimedean period relation in the smallest new case, n=2 over an imaginary quadratic field, using a CM elliptic curve and a Hecke character; a computed match would calibrate the constants, and a mismatch would indicate a normalization error.
  • Pith inference: The Eisenstein-cohomology method may extend to period relations for other Langlands pairs where the degenerate principal series has one-dimensional bottom cohomology, such as hermitian or symplectic groups, once the archimedean nonvanishing is available.
  • Pith inference: The Aut(C)-equivariant modular symbol could be used to construct p-adic interpolation of the normalized critical values, since the rationality statement provides a canonical lattice to interpolate in the character direction.
  • Pith inference: The broader tamely isobaric class, allowing multi-component induced representations without additional regularity, is likely to be the right category for reducing period relations of arbitrary isobaric representations to cuspidal ones.
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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 proves an Aut(C)-equivariant rationality and period relation for L(0, Π×χ) for Rankin-Selberg convolutions GL(n)×GL(n) over number fields containing a CM field, for Π=Σ⊠Σ′ with Σ tamely isobaric and Σ′ cuspidal and χ in the set B(µ,ν)+. The proof combines a new theorem on rationality of Eisenstein cohomology (Theorem 1.3) with archimedean period relations from [JLS25] and non-archimedean relations adapted from [LLS24]. Sections 2–5 give a detailed construction of the Eisenstein map, explicit generators for the bottom-degree cohomology, and an Aut(C)-equivariance proof via boundary cohomology. Section 7 assembles local periods into the global modular symbol and derives the stated period relation.

Significance. If the main theorem is correct, it is a substantial advance: it gives the first general-n period relation for GL(n)×GL(n) of this kind, extending the GL(n)×GL(n−1) results and providing new Eisenstein cohomology rationality results. The proof of Theorem 1.3 is a genuine new ingredient, with explicit generators and Aut(C)-equivariance diagrams, and the paper improves on [LLS24] by dropping the ♮ restriction and broadening the class of tamely isobaric representations. However, the stated domain B(µ,ν)+ is not handled by the proof as written: the assertion Iη=Jη is false without a finite-place condition, so the central theorem has a gap that needs to be addressed.

major comments (3)
  1. [Section 7, opening paragraph; Section 1.2] The assertion 'the assumption χ∈B(µ,ν)+ implies that Iη=Jη' is not justified by the definition of B(µ,ν)+. The conditions (1.3)–(1.5) are archimedean or global-algebraic conditions and impose no restriction on η_v at finite places. By the results cited in Section 1.3, Iη,v is reducible of length 2 when η_v=1, and Jη,v is then the unique proper subrepresentation. For instance, over k=Q(i) with n=2, take Π=Σ′∨⊠Σ′ with ωΠ=1, choose integers a,c with c≥a and a+c+1≡0 mod 4, and let χ be an algebraic Hecke character of infinity type (−a,c+1) with χ_v=1 at a chosen finite place v. Then η^ι=2(−a)≤0, η^ι̅=2(c+1)≥2, so (1.5) and η^ι+η^ι̅≥2 hold, and the balanced condition (1.3) holds because Fη⊗Fχ contains the Cartan component of highest weight µ+ν; yet η_v=1 and Iη,v≠Jη,v. Consequently H(Iη)≠H(Jη), the modular symbol (7.8) is not defined via Eisη on all of Iη, and the top square of (7.27) does not go through for such χ. The theorem needs a finite-place hypothesis on η (e.g., η_v≠1 for all finite v) or a separate argument for the reducible finite places.
  2. [Section 7.2, Proposition 7.1] The proof of Proposition 7.1 is omitted with a reference to [LLS24, Proposition 7.2]. This proposition is load-bearing: it identifies the global Rankin-Selberg integral as the product of the archimedean modular symbol, the non-archimedean normalized period, and L(s,Π×χ)/L(s,ωΠχ^n)·δ(s,η)|_{s=0}. Without a proof or a precise statement of the imported hypotheses, the Euler-product step in Theorem 1.2 is not fully supported. The authors should include this proof or make the reduction to [LLS24] explicit.
  3. [Section 7.3, diagram (7.16)] The non-archimedean period relation (7.16) is asserted to follow from [LLS24, Proposition 5.1†, Proposition 7.4] with the computations omitted. This is the finite-place Aut(C)-equivariance input to the top square of (7.27). Since a corrigendum is appended to the referenced Proposition 5.1, the authors should state the corrected version and verify that the Gauss-sum factor G(χ)^{n(n-1)/2} has the correct normalization under σ. A proof or a detailed reference should be provided.
minor comments (4)
  1. [Section 2.1, after Eq. (2.15)] The phrase 'vis archimedan' is a typo for 'v is archimedean'.
  2. [Section 7.4, definition of χ^◦_v] The two displayed cases for χ^◦_v are identical ('η^{ι_v}≥n' in both lines); the second should presumably be 'η^{ι_v}≤0'.
  3. [Section 7.5.1, Lemma 7.2] The proof of Lemma 7.2 is omitted with 'We omit the details'; since this lemma supplies the Aut(C)-equivariant normalization of the Deligne period c^+(ξ), a reference or short proof should be supplied.
  4. [Section 7.3, footnote †] The corrigendum to [LLS24, Proposition 5.1] should be stated in the main text rather than as a footnote, because the corrected Haar-measure normalization affects the proof of (7.16).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Theorem 1.2 combines independent published local period theorems (JLS25, LLS24) with a new Eisenstein cohomology result; the Section 7 claim that B(µ,ν)+ implies Iη=Jη is a proof gap, not a circular reduction.

full rationale

The central derivation is not circular. Theorem 1.2's proof assembles three independent ingredients: the archimedean period relation and nonvanishing of the archimedean modular symbol imported from [JLS25, Theorem 1.6] (used in diagram (7.27): 'the top horizontal arrow is surjective by non-vanishing of the archimedean modular symbol ([JLS25, Theorem 1.6(b)])'), the non-archimedean period relations and Whittaker periods from [LLS24, Proposition 5.1, Proposition 7.4, Section 6], and the new Aut(C)-equivariant Eisenstein cohomology (Theorem 1.3) proved in this paper. These prior inputs are published, peer-reviewed, and concern local or GL(n)xGL(n-1) statements that do not contain Theorem 1.2; their use is real evidence, not a circular reduction. The quotient (1.6) is not defined in terms of itself: c(ωΠχn), G(χ), Ω∞(Π,χ), and Ω(Π) are constructed independently of the L-value being studied, and no fitted parameter is renamed as a prediction. Flagged per the reviewing rule: Section 7 asserts 'the assumption χ∈B(µ,ν)+ implies that Iη=Jη', but B(µ,ν)+ is an archimedean condition (1.4)-(1.5), while at a finite place with η_v=1 or |·|^n, Iη,v is reducible of length 2 and Jη,v is its proper subrepresentation; this is a correctness gap in the proof as written, not a circularity, and it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data. The periods Omega_infty, Omega, c, and G are canonical normalizations, not fitted constants. The theorem relies on several deep prior results, some from the same authors' recent papers; these are published and not obviously circular.

assumptions (5)
  • standard math Delorme's lemma computes relative Lie algebra cohomology of induced representations ([BW00, Theorem III.3.3]).
    Used throughout Sections 3 and 6 to identify cohomology spaces and their dimensions.
  • standard math Langlands' formula for normalized intertwining operators, equation (2.8).
    Used to compute constant terms of degenerate Eisenstein series in Section 2.
  • domain assumption Archimedean period relation and nonvanishing of the archimedean modular symbol ([JLS25, Theorem 1.6]).
    A load-bearing input in Section 7.4; published in Advances in Mathematics with overlapping authorship.
  • domain assumption Deligne's conjecture for Hecke characters, proved in [Ku24], equation (7.21).
    Used to normalize Deligne periods c+(xi) and to prove Lemma 7.2.
  • domain assumption Harder's rationality theorem for critical Hecke L-values and Waldspurger's rationality of intertwining operators ([Har87], [Wal03]).
    Used in Proposition 5.8 and in the Aut(C)-equivariance of non-archimedean intertwining operators.

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Pith. "Pith review of Period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$." pith.science (2026). https://pith.science/paper/JCNQBZUZ

@misc{pith2026250620942,
  author       = {Pith},
  title        = {Pith review of: Period relations for Rankin-Selberg convolutions for $\mathrmGL(n)\times\mathrmGL(n)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JCNQBZUZ}},
  note         = {Machine review of arXiv:2506.20942}
}
abstract

Using the modular symbol approach, we prove the rationality and period relations for certain critical values of Rankin-Selberg L-functions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$ over any number field that contains a CM field.

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Works this paper leans on

19 extracted references · 17 canonical work pages

  1. [1]

    Blasius,On the critical values of Hecke L-series, Ann

    [Bla86] D. Blasius,On the critical values of Hecke L-series, Ann. Math. (2) 124 (1986), no. 1, 23–63. [Bla97] D. Blasius,Period relations and critical values of L-functions, Pacific J. Math. 1997, 53–83. [BW00] A. Borel and N. Wallach,Continuous cohomology, discrete subgroups, and represen- tations of reductive groups, volume 67 of Mathematical Surveys an...

    work page 1986

  2. [9]

    Hida,On the critical values ofL-functions ofGL(2)andGL(2)×GL(2), Duke Math

    [Hi94] H. Hida,On the critical values ofL-functions ofGL(2)andGL(2)×GL(2), Duke Math. J. 74 (1994), 431–529. [HL99] R. Howe and S. T. Lee,Degenerate principal series representations ofGL n(C)and GLn(R), J. Funct. Anal., 166(2):244–309,

    work page 1994

  3. [18]

    [vE58] van Est,A generalization of the Cartan-Leray spectral sequence II, Indagationes Math

    xvi+624 pp. [vE58] van Est,A generalization of the Cartan-Leray spectral sequence II, Indagationes Math. 20 (1958), 406–413. [Wa88] N. R. Wallach,Real reductive groups. I, Pure and Applied Mathematics, 132, Academic Press, Boston, MA,

    work page 1958

  4. [19]

    Waldspurger,La formule de Plancherel pour les groupesp-adiques (d’apr` es Harish- Chandra), J

    [Wal03] J.-L. Waldspurger,La formule de Plancherel pour les groupesp-adiques (d’apr` es Harish- Chandra), J. Inst. Math. Jussieu 2 (2003), no. 2, 235–333. [Xue18] H. Xue,Homogeneous distributions on finite dimensional vector spaces, J. Lie Theory 28(2018), no. 1, 33–41. [Zel80] A. V. Zelevinsky,Induced representations of reductivep-adic groups. II. On irr...

    work page 2003

  5. [1967]

    [Lan76] R. P. Langlands,On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer, Berlin-New York, 1976; MR0579181 [Lan79] R. P. Langlands,On the notion of an automorphic representations. a supplement to the preceeding paper, In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math....

    work page 1976

  6. [1981]

    Kasten and C.-G

    [KS13] H. Kasten and C.-G. Schmidt,On critical values of Rankin–Selberg convolutions, Int.J.Number Theory 9 (2013), 205–256. [KMS00] D. Kazhdan, B. Mazur and C.-G. Schmidt,Relative modular symbols and Rankin–Selberg convolutions, J. Reine Angew. Math. 519 (2000), 97–141. [Kim05] H. Kim,On local L-functions and normalized intertwining operators, Canad. J. ...

    work page 2013

  7. [1987]

    [Har90] G. Harder,Some results on the Eisenstein cohomology of arithmetic subgroups ofGL n, In Cohomology of arithmetic groups and automorphic forms (Luminy-Marseille, 1989), Lecture Notes in Math. 1447, pages 85–153. Springer, Berlin,

    work page 1989

  8. [1988]

    [Sch93] C.-G

    xvi+160 pp. [Sch93] C.-G. Schmidt,Relative modular symbols and p-adic Rankin-Selberg convolutions, Invent. Math. 112 (1993), no. 1, 31–76. [Sch83] J. Schwermer,Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen. Lect. Notes in Math. 988, Berlin–Heidelberg–New York

    work page 1993

Show all 19 references
  1. [1990]

    Deligne,Valeurs de fonctions L et p´ eriodes d’int´ egrales, In Automorphic forms, rep- resentations and L-functions (Proc

    [Del79] P. Deligne,Valeurs de fonctions L et p´ eriodes d’int´ egrales, In Automorphic forms, rep- resentations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corval- lis, Ore., 1977), Part 2, volume XXXIII of Proc. Sympos. Pure Math., pages 313–346. Amer. Math...

    1977

  2. [1995]

    [Mui07] Goran Mui´ c.On certain classes of unitary representations for split classical groups, Canad. J. Math., (1) 59 (2007), 148–185. [PR V67] K. R. Parthasarathy, R. Ranga Rao; V. S. Varadarajan,Representations of complex semi-simple Lie groups and Lie algebras, Ann. of Mat...

    2007

  3. [1999]

    Ishii and T

    [IM22] T. Ishii and T. Miyazaki,Calculus of archimedean Rankin-Selberg integrals with recur- rence relations, Represent. Theory 26 (2022), 714–763. [JLS25] Y. Jin, D. Liu and B. Sun,Archimedean period relations for Rankin-Selberg convolutions, Advances in Mathematics 478 (2025...

    2022

  4. [2000]

    Chen,Algebraicity of ratios of Rankin-Selberg L-functions and applications to Deligne’s conjecture, arXiv:2205.15382

    [Che22] S.-Y. Chen,Algebraicity of ratios of Rankin-Selberg L-functions and applications to Deligne’s conjecture, arXiv:2205.15382. [Clo90] L. Clozel,Motifs et formes automorphes: applications du principe de fonctorialit´ e, In Automorphic forms, Shimura varieties, and L-funct...

    1988 arXiv

  5. [2005]

    Kings and J

    [KS25] G. Kings and J. Sprang,Eisenstein-Kronecker classes, integrality of critical values of Hecke L-functions and p-adic interpolation, Ann. of Math. (2) 202 (2025), no. 1, 1–109. [Kos61] B. Kostant,Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math....

    2025 arXiv

  6. [2008]

    Li and B

    [LS19] J.-S. Li and B. Sun,Low degree cohomologies of congruence groups, Sci. China Math. 62 (2019), no. 11, 2287–2308. [LLS24] J.-S. Li, D. Liu, and B. Sun,Period relations for Rankin-Selberg convolutions for GL(n)×GL(n−1), Compos. Math., 160(8):1871–1915,

    2019

  7. [2010]

    Shimura,The special values of the zeta functions associated with cusp forms, Comm

    [Sh76] G. Shimura,The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976), no. 6, 783–804. [Sh78] G. Shimura,The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679. [...

    1976

  8. [2015]

    Harder,Period integral of Eisenstein cohomology classes and special values of some L-functions, Number theory related to Fermat’s last theorem (Cambridge, Mass., 1981), Progr

    [Har82] G. Harder,Period integral of Eisenstein cohomology classes and special values of some L-functions, Number theory related to Fermat’s last theorem (Cambridge, Mass., 1981), Progr. Math., vol. 26, pp. 103-142. Birkh¨ auser, Boston,

    1981

  9. [2016]

    Grobner, M

    [GHL21] H. Grobner, M. Harris and J. Lin,Deligne’s conjecture for automorphic motives over CM-fields, arXiv:1802.02958. [GL21] H. Grobner and J. Lin,Special values of L-functions and the refined Gan-Gross-Prasad conjecture, Amer. J. Math., 143(3):859–937,

    arXiv

  10. [2017]

    Greni´ e,Critical values of automorphic L-functions forGL(r)×GL(r), Manuscripta Math

    [Gr03] L. Greni´ e,Critical values of automorphic L-functions forGL(r)×GL(r), Manuscripta Math. 110 (2003), no. 3, 283–311. [Gro18] H. Grobner,Rationality for isobaric automorphic representations: the CM-case, Monatsh. Math., 187(1):79–94,

    2003

  11. [2020]

    Harder and N

    [HS85] G. Harder and N. Schappacher,Special values of Hecke L-functions and abelian integrals, Workshop Bonn 1984 (Bonn, 1984), 17–49. Lecture Notes in Math. 1111, Springer-Verlag, Berlin,

    1984

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