On the Special Values of Certain Rankin–Selberg L-Functions and Applications to Odd Symmetric Power L-Functions of Modular Forms

Raghuram, A.

doi:10.1093/imrn/rnp127

Cited by 49 publications

(98 citation statements)

References 36 publications

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“…Clearly, both (3.4.2) and (3.4.3) can be stated in a Galois-equivariant fashion. Using results in [32] and [34], we may proceed in this fashion, and by induction, get algebraicity results for L(1, Sym 2m (π)), which are unconditional for m ≤ 4, and depend on unproven instances of Langlands's functoriality for higher even symmetric powers. Let V = V 1 ⊕ V 2 andṼ =Ṽ 1 ⊕Ṽ 2 be decompositions of V andṼ into subspaces that respect the pairing above (i.e., V 1 ⊥Ṽ 2 and V 2 ⊥Ṽ 1 ).…”

Section: 3)mentioning

confidence: 99%

Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field

Balasubramanyam¹,

Raghuram²

2017

American Journal of Mathematics

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Introduction 1 2. Special values of the adjoint L-function 4 2.1. Rankin-Selberg integrals for GL n × GL n 4 2.2. An integral representation of L(1, Ad 0 , π) 5 2.3. Ramified calculations 7 3. Whittaker models and automorphic cohomology 8 3.1. The basic set-up to study the cohomology of arithmetic groups 8 3.2. Betti-Whittaker periods 9 3.3. Cohomological pairing and the main theorem on adjoint L-values 11 3.4. On the criticality of the adjoint L-function at s = 1 14 4. Discriminant calculations and cohomological congruences 15 4.1. Some linear algebra 15 4.2. Discriminants and congruence modules 16 4.3. The main theorem on congruences 19 5. The non-vanishing property 20 References 25

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Section: 3)mentioning

confidence: 99%

Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field

Balasubramanyam¹,

Raghuram²

2017

American Journal of Mathematics

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“…The map Θ 0 . This section is in analogy with Mahnkopf [35], Raghuram-Shahidi, [41], Raghuram [40], [39] and Grobner-Raghuram [17]. As a first step, we choose and fix generators of the one-dimensional spaces H bn (g, K, W (Π ∞ ) ⊗ E µ ) and…”

Section: A Diagrammentioning

confidence: 99%

Whittaker Periods, Motivic Periods, and Special Values of Tensor Product -Functions

Grobner¹,

Harris

2015

J. Inst. Math. Jussieu

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Abstract. Let K be an imaginary quadratic field. Let Π and Π ′ be irreducible generic cohomological automorphic representation of GL(n)/K and GL(n − 1)/K, respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, the other is given in terms of the Whittaker model. The ratio between these rational structures is called a Whittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if Π is cuspidal and the weights of Π and Π ′ are in a standard relative position, the critical values of the Rankin-Selberg product L(s, Π × Π ′ ) are essentially algebraic multiples of the product of the Whittaker periods of Π and Π ′ . We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal Π can be given a motivic interpretation, and can also be related to a critical value of the adjoint L-function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin-Selberg and adjoint L-functions are compatible with Deligne's conjecture. 1. Introduction L-functions can be attached both to automorphic representations and to arithmetic objects such as Galois representations or motives, and one implication of the Langlands program is that L-functions of the second kind are examples of L-functions of the first kind. Very few results of arithmetic interest can be proved about the second kind of L-functions until they have been identified with automorphic L-functions. For example, there is an extraordinarily deep web of conjectures relating the values at integer points of arithmetic (motivic) L-functions to cohomological invariants of the corresponding geometric (motivic) objects. In practically all the instances 1 where these conjectures have been proved, automorphic methods have proved indispensable. At the same time, there is a growing number of results on special values of automorphic L-functions that make no direct reference to arithmetic. Instead, the special values are written as algebraic multiples of complex invariants defined by means of representation theory. Examples relevant to the present paper ′ ) of a cuspdial automorphic representation Π of GL n and a cuspidal or abelian automorphic representation Π ′ of GL n−1 , where the general linear groups are over an imaginary quadratic field K. Here, the notion "abelian automorphic" refers to a representation, which is a tempered Eisenstein representation induced from a Borel subgroup of GL n−1 . In view of Raghuram's recent preprint [39], which we received after writing a first version of this paper, the inclusion of such automorphic representations Π ′ is the new feature of this result. The theorem applies in particular when Π and Π ′ are obtained by base change from cohomological cuspidal representations π and π ′ of unitary groups. As in [35], [40] and [39] the critical values of these L-functions are then expressed in terms of Whittaker periods, which are p...

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“…Since σ(L( 1 2 , Π v )) = L( 1 2 , σ Π v ) = 0 for all v ∈ S\S ∞ , cf. [28], Prop. 3.17, and recalling once more that S = S(Π, ψ) = S( σ Π, ψ), we may rewrite this by…”

Section: 3mentioning

confidence: 99%

“…Guided by the above methods, meanwhile, there is a growing number of results that have been proved about the rationality of special values of certain automorphic L-functions attached to GL n . As a selection of examples, relevant to the present paper, we refer to Raghuram [28,27], Harder-Raghuram [16], Grobner-Harris [10]; Grobner-Raghuram [13], Grobner-Harris-Lapid [11] and Balasubramanyam-Raghuram [2].…”

Section: Introductionmentioning

confidence: 99%

Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

Grobner

2017

Math. Ann.

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Let G = GL 2n over a totally real number field F and n ≥ 2. Let be a cuspidal automorphic representation of G(A), which is cohomological and a functorial lift from SO(2n + 1). The latter condition can be equivalently reformulated that the exterior square L-function of has a pole at s = 1. In this paper, we prove a rationality result for the residue of the exterior square L-function at s = 1 and also for the holomorphic value of the symmetric square L-function at s = 1 attached to . As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L-functions and a product of Gauß sums of Hecke characters.

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On the Special Values of Certain Rankin–Selberg L-Functions and Applications to Odd Symmetric Power L-Functions of Modular Forms

Cited by 49 publications

References 36 publications

Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field

Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field

Whittaker Periods, Motivic Periods, and Special Values of Tensor Product -Functions

Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

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