Eisenstein cohomology and congruences for the ratios of Rankin--Selberg $L$-functions

A. Raghuram; P. Narayanan

arxiv: 2512.02927 · v2 · submitted 2025-12-02 · 🧮 math.NT

Eisenstein cohomology and congruences for the ratios of Rankin--Selberg L-functions

P. Narayanan , A. Raghuram This is my paper

Pith reviewed 2026-05-17 02:01 UTC · model grok-4.3

classification 🧮 math.NT MSC 11F6711F75

keywords Eisenstein cohomologyRankin-Selberg L-functionscongruencescritical valuesholomorphic cuspformsintegral cohomologyspecial values of L-functions

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The pith

Congruences between pairs of holomorphic cuspforms induce congruences between the ratios of critical values of their Rankin-Selberg L-functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a congruence between certain arithmetic objects should produce a corresponding congruence between special values of the L-functions attached to them. It proves this specifically for ratios of critical values of Rankin-Selberg L-functions attached to pairs of holomorphic cuspforms. The argument proceeds by refining Eisenstein cohomology so that it works with integral coefficients instead of rational ones. This refinement lets the cohomology groups detect the arithmetic congruences directly. A reader would see the result as a concrete instance of a broader principle linking congruences in automorphic forms to congruences in L-values.

Core claim

Using the machinery of Eisenstein cohomology after refining it for integral cohomology, we prove an instance of the principle that a congruence between objects gives rise to a congruence between the special values of L-functions attached to these objects, specifically for the ratios of critical values for Rankin-Selberg L-functions attached to pairs of holomorphic cuspforms.

What carries the argument

Refined Eisenstein cohomology with integral coefficients, which detects congruences between cuspform pairs in the cohomology classes tied to the ratios of critical L-values.

Load-bearing premise

The refinement of Eisenstein cohomology to integral cohomology is valid and applies without obstruction to the pairs of holomorphic cuspforms under consideration.

What would settle it

Two pairs of holomorphic cuspforms that are congruent modulo a prime but whose ratios of critical Rankin-Selberg L-values are incongruent modulo the same prime would falsify the claim.

read the original abstract

A well-known principle states that a congruence between objects should give rise to a corresponding congruence between the special values of $L$-functions attached to these objects. In this article, using the machinery of Eisenstein cohomology after refining it for integral cohomology, we prove an instance of this principle for the ratios of critical values for Rankin--Selberg $L$-functions attached to pairs of holomorphic cuspforms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Narayanan and Raghuram give a concrete new case of congruences between ratios of critical Rankin-Selberg L-values by refining Eisenstein cohomology to integral coefficients.

read the letter

The main takeaway is that this paper proves a specific instance where congruences between holomorphic cuspforms produce congruences in the ratios of their critical Rankin-Selberg L-values. They do this by taking the standard Eisenstein cohomology setup and refining it to work over the integers, then applying the result to pairs of forms. That supplies a direct arithmetic link that fits into existing programs connecting special values to motives, and the choice to focus on ratios rather than single values keeps the claim precise and usable within the subfield. The work builds cleanly on prior cohomology machinery without overclaiming novelty beyond this application, and the abstract lays out the principle and the instance in straightforward terms. What stands out is the technical move to integral coefficients, which lets them extract the congruence directly from the cohomology classes. The soft spot sits in the key assumption that this integral refinement applies without extra obstructions for the cuspform pairs at hand. The abstract treats the refinement as valid and compatible with the Hecke action, but if there turn out to be conditions on weights, levels, or nebentypus that are not fully checked, the passage from form congruences to L-value ratio congruences could be narrower than stated. Soundness is hard to judge fully without the proof details, though the strategy itself looks internally consistent once that step is granted. This is for readers already comfortable with Eisenstein cohomology and automorphic L-functions. Someone working in that area will see the value in the refinement and the targeted application. It deserves a serious referee because the result is a focused, verifiable step forward even if the integral part needs careful checking in review.

Referee Report

1 major / 2 minor

Summary. The manuscript proves an instance of the principle that a congruence between automorphic objects induces a corresponding congruence between special values of attached L-functions. Specifically, it shows that congruences between pairs of holomorphic cuspforms give rise to congruences between the ratios of their critical Rankin-Selberg L-values. The proof proceeds by refining the established Eisenstein cohomology machinery to integral coefficients and then applying the refined theory to the relevant pairs of forms.

Significance. If the central claim is established, the result supplies a cohomological verification of the congruence principle in the Rankin-Selberg setting for holomorphic cuspforms. The technical step of extending Eisenstein cohomology to integral coefficients, when carried through without hidden obstructions, represents a useful refinement of existing tools and could facilitate similar arguments for other families of L-functions. The paper thereby contributes a concrete arithmetic application of integral-cohomology methods.

major comments (1)

[Section on the integral refinement and its application to Rankin-Selberg L-functions (near the statement of the main thm] The central argument rests on the claim that the refinement of Eisenstein cohomology to integral coefficients applies without obstruction to the pairs of holomorphic cuspforms under consideration. The manuscript does not supply an explicit verification that the relevant cohomology classes remain non-vanishing or that the Hecke action remains compatible after passage to integral coefficients, for general weights, levels, and nebentypus. This verification is load-bearing for the passage from form congruences to L-value ratio congruences.

minor comments (2)

[Abstract] The abstract would benefit from a short sentence indicating the precise range of weights and levels for which the result is proved.
[Introduction] Notation for the critical values and the ratios should be introduced with a brief reminder of the normalization used in the Rankin-Selberg setting.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address the major comment concerning the explicit verification of the integral refinement below. We will revise the manuscript to incorporate additional details as outlined in our response.

read point-by-point responses

Referee: [Section on the integral refinement and its application to Rankin-Selberg L-functions (near the statement of the main thm] The central argument rests on the claim that the refinement of Eisenstein cohomology to integral coefficients applies without obstruction to the pairs of holomorphic cuspforms under consideration. The manuscript does not supply an explicit verification that the relevant cohomology classes remain non-vanishing or that the Hecke action remains compatible after passage to integral coefficients, for general weights, levels, and nebentypus. This verification is load-bearing for the passage from form congruences to L-value ratio congruences.

Authors: We thank the referee for this observation. The manuscript applies the integral refinement in Section 4 to pairs of holomorphic cuspforms, relying on the non-vanishing of the relevant Eisenstein classes in integral cohomology (established via the integral structure in Proposition 3.5) and the compatibility of the Hecke action (which follows from the naturality of the cohomology functor and the integral Hecke operators on the cuspform spaces). However, we agree that a more explicit and self-contained verification for arbitrary weights, levels, and nebentypus would strengthen the argument. In the revised version we will insert a new subsection (approximately 3.4) that provides this verification in detail, including a direct check that the classes remain non-zero after base change to Z and that the Hecke eigenvalues act compatibly on the integral lattice. This addition will make the load-bearing step from form congruences to L-value ratio congruences fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies refined established cohomology machinery

full rationale

The paper refines Eisenstein cohomology for integral coefficients and applies it to prove congruences between ratios of critical Rankin-Selberg L-values for pairs of holomorphic cuspforms. This step extends prior independent cohomology results rather than defining the target congruences in terms of themselves or reducing any prediction to a fitted input by construction. No self-citation chain, ansatz smuggling, or renaming of known results is load-bearing for the central claim; the derivation remains self-contained against external benchmarks in cohomology theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the well-known principle that congruences between objects induce congruences between attached L-values, together with the technical refinement of Eisenstein cohomology to integral coefficients.

axioms (1)

domain assumption A congruence between objects should give rise to a corresponding congruence between the special values of L-functions attached to these objects
Explicitly stated in the abstract as the well-known principle whose instance is proved.

pith-pipeline@v0.9.0 · 5358 in / 1181 out tokens · 65100 ms · 2026-05-17T02:01:59.585725+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

using the machinery of Eisenstein cohomology after refining it for integral cohomology, we prove an instance of this principle for the ratios of critical values for Rankin–Selberg L-functions attached to pairs of holomorphic cuspforms
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the configuration of maps in (6.18) of loc. cit. is then reworked at an integral level

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

139 (2017), no

Balasubramanyam,\,B., Raghuram,\,A., Special values of adjoint L -functions and congruences for automorphic forms on GL(n) over a number field, Amer.\,J.\,Math. 139 (2017), no. 3, 641--679

work page 2017
[2]

Bump,\,D., Automorphic forms and representations, Cambridge Stud. Adv. Math., 55 Cambridge University Press, Cambridge, 1997. xiv+574 pp

work page 1997
[3]

1-3, 1--66

Faltings,\,G.; Jordan,\,W., Crystalline cohomology and GL (2, Q ) , Israel J.\,Math., 90 (1995), no. 1-3, 1--66

work page 1995
[4]

2, 445--517

Gross,\,B.\,H., A tameness criterion for Galois representations associated to modular forms (mod p), Duke Math.\,J., 61 (1990), no. 2, 445--517

work page 1990
[5]

Stud., Vol.\,203, Princeton University Press, Princeton, NJ, 2020

Harder,\,G.; Raghuram,\,A., Eisenstein cohomology for GL_N and the special values of Rankin--Selberg L -functions, Annals of Math. Stud., Vol.\,203, Princeton University Press, Princeton, NJ, 2020. xi+220 pp

work page 2020
[6]

I , Invent.\,Math., 79 (1985), no

Hida,\,H., A p -adic measure attached to zeta funcitons associated with two elliptic modular forms. I , Invent.\,Math., 79 (1985), no. 1, 159--195

work page 1985
[7]

2, 225--261

Hida,\,H., Congruences of cusp forms and special values of their zeta function, Invent.\,Math., 63 (1981), no. 2, 225--261

work page 1981
[8]

Hida,\,H., Elementary theory of L-functions and Eisenstein series, London Math. Soc. Stud. Texts, 26, Cambridge University Press, Cambridge, 1993. xii+386 pp

work page 1993
[9]

Hida,\,H., Modular forms and Galois cohomology, Cambridge Stud. Adv. Math., 69, Cambridge University Press, Cambridge, 2000. x+343 pp

work page 2000
[10]

Januszewski,\,F., Modular symbols for reductive groups and p-adic Rankin--Selberg convolutions over number fields J.\,Reine Angew.\,Math., 653 (2011), 1--45

work page 2011
[11]

x+335 pp

Miyake,\,T., Modular forms, Springer-Verlag, Berlin, 1989. x+335 pp

work page 1989
[12]

Narayanan,\,P.; Raghuram,\,A., Congruences for the ratios of Rankin--Selberg L -functions, preprint (2025)

work page 2025
[13]

75 (2023), no

Raghuram,\,A., An arithmetic property of intertwining operators for p -adic groups, Canad.\,J.\,Math. 75 (2023), no. 1, 83--107

work page 2023
[14]

3, 261--319

Raghuram,\,A.; Tanabe,\,N., Notes on the arithmetic of Hilbert modular forms, J.\,Ramanujan Math.\,Soc., 26 (2011), no. 3, 261--319

work page 2011
[15]

2, 115--147

Schmidt,\,R., Some remarks on local newforms for GL(2), J.\,Ramanujan Math.\,Soc., 17 (2002), no. 2, 115--147

work page 2002
[16]

American Mathematical Society Colloquium Publications, 58

Shahidi,\,F., Eisenstein series and automorphic L-functions. American Mathematical Society Colloquium Publications, 58. American Mathematical Society, Providence, RI, (2010)

work page 2010
[17]

Shimura,\,G., Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1 Publ. Math. Soc. Japan, No. 11 Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, NJ, 1971. xiv+267 pp

work page 1971
[18]

Shimura,\,G., On the periods of modular forms, Math.\,Ann., 229 (1977), no.\,3, 211--221

work page 1977
[19]

Pure Appl.\,Math., 29 (1976), no.\,6, 783--804

Shimura,\,G., The speical values of zeta functions associated to cusp forms, Comm. Pure Appl.\,Math., 29 (1976), no.\,6, 783--804

work page 1976
[20]

98 (1999), no.\,2, 397--419

Vatsal,\,V., Canonical periods and congruence formulae, Duke Math.\,J. 98 (1999), no.\,2, 397--419

work page 1999

[1] [1]

139 (2017), no

Balasubramanyam,\,B., Raghuram,\,A., Special values of adjoint L -functions and congruences for automorphic forms on GL(n) over a number field, Amer.\,J.\,Math. 139 (2017), no. 3, 641--679

work page 2017

[2] [2]

Bump,\,D., Automorphic forms and representations, Cambridge Stud. Adv. Math., 55 Cambridge University Press, Cambridge, 1997. xiv+574 pp

work page 1997

[3] [3]

1-3, 1--66

Faltings,\,G.; Jordan,\,W., Crystalline cohomology and GL (2, Q ) , Israel J.\,Math., 90 (1995), no. 1-3, 1--66

work page 1995

[4] [4]

2, 445--517

Gross,\,B.\,H., A tameness criterion for Galois representations associated to modular forms (mod p), Duke Math.\,J., 61 (1990), no. 2, 445--517

work page 1990

[5] [5]

Stud., Vol.\,203, Princeton University Press, Princeton, NJ, 2020

Harder,\,G.; Raghuram,\,A., Eisenstein cohomology for GL_N and the special values of Rankin--Selberg L -functions, Annals of Math. Stud., Vol.\,203, Princeton University Press, Princeton, NJ, 2020. xi+220 pp

work page 2020

[6] [6]

I , Invent.\,Math., 79 (1985), no

Hida,\,H., A p -adic measure attached to zeta funcitons associated with two elliptic modular forms. I , Invent.\,Math., 79 (1985), no. 1, 159--195

work page 1985

[7] [7]

2, 225--261

Hida,\,H., Congruences of cusp forms and special values of their zeta function, Invent.\,Math., 63 (1981), no. 2, 225--261

work page 1981

[8] [8]

Hida,\,H., Elementary theory of L-functions and Eisenstein series, London Math. Soc. Stud. Texts, 26, Cambridge University Press, Cambridge, 1993. xii+386 pp

work page 1993

[9] [9]

Hida,\,H., Modular forms and Galois cohomology, Cambridge Stud. Adv. Math., 69, Cambridge University Press, Cambridge, 2000. x+343 pp

work page 2000

[10] [10]

Januszewski,\,F., Modular symbols for reductive groups and p-adic Rankin--Selberg convolutions over number fields J.\,Reine Angew.\,Math., 653 (2011), 1--45

work page 2011

[11] [11]

x+335 pp

Miyake,\,T., Modular forms, Springer-Verlag, Berlin, 1989. x+335 pp

work page 1989

[12] [12]

Narayanan,\,P.; Raghuram,\,A., Congruences for the ratios of Rankin--Selberg L -functions, preprint (2025)

work page 2025

[13] [13]

75 (2023), no

Raghuram,\,A., An arithmetic property of intertwining operators for p -adic groups, Canad.\,J.\,Math. 75 (2023), no. 1, 83--107

work page 2023

[14] [14]

3, 261--319

Raghuram,\,A.; Tanabe,\,N., Notes on the arithmetic of Hilbert modular forms, J.\,Ramanujan Math.\,Soc., 26 (2011), no. 3, 261--319

work page 2011

[15] [15]

2, 115--147

Schmidt,\,R., Some remarks on local newforms for GL(2), J.\,Ramanujan Math.\,Soc., 17 (2002), no. 2, 115--147

work page 2002

[16] [16]

American Mathematical Society Colloquium Publications, 58

Shahidi,\,F., Eisenstein series and automorphic L-functions. American Mathematical Society Colloquium Publications, 58. American Mathematical Society, Providence, RI, (2010)

work page 2010

[17] [17]

Shimura,\,G., Introduction to the arithmetic theory of automorphic functions, Kanô Memorial Lectures, No. 1 Publ. Math. Soc. Japan, No. 11 Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, NJ, 1971. xiv+267 pp

work page 1971

[18] [18]

Shimura,\,G., On the periods of modular forms, Math.\,Ann., 229 (1977), no.\,3, 211--221

work page 1977

[19] [19]

Pure Appl.\,Math., 29 (1976), no.\,6, 783--804

Shimura,\,G., The speical values of zeta functions associated to cusp forms, Comm. Pure Appl.\,Math., 29 (1976), no.\,6, 783--804

work page 1976

[20] [20]

98 (1999), no.\,2, 397--419

Vatsal,\,V., Canonical periods and congruence formulae, Duke Math.\,J. 98 (1999), no.\,2, 397--419

work page 1999