Residues of Eisenstein series and the automorphic cohomology of reductive groups

Grobner, Harald

doi:10.1112/s0010437x12000863

Cited by 13 publications

(26 citation statements)

References 17 publications

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“…This improves a result of Rohlfs and Speh [2011] (see also our Remark 4.2) and confirms an idea of Harder. Moreover, it may be viewed as a refinement of one of our own results in [Grobner 2013]. Although we believe that it is interesting in its own right, we hope that it will also be of use in a forthcoming work of Harder and Raghuram on special values of Ranking-Selberg L-functions.…”

Section: Introductionmentioning

confidence: 80%

A cohomological injectivity result for the residual automorphic spectrum of GL_n

Grobner

2014

Pacific J. Math.

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Let … be a cohomological residual automorphic representation of GL n =F , for F an arbitrary number field. Let q min be the lowest degree in which … has nonvanishing cohomology. We prove that the cohomology of … always injects into the cohomology of the corresponding locally symmetric space in degree q min. This extends the well-known result of Borel for cuspidal automorphic representations to all square-integrable automorphic representations in this certain degree. Moreover, we thereby improve a result of Rohlfs and Speh and confirm an idea of Harder.

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Section: Introductionmentioning

confidence: 80%

A cohomological injectivity result for the residual automorphic spectrum of GL_n

Grobner

2014

Pacific J. Math.

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“…The algebra a G ′ B ′ ,C operates trivially onτ . Hence, one may check that (B ′ ,τ , 0, 0) is one of the quadruples, constructed in Grobner [14], 3.3. Let ϕ B ′ be the associate class of unitary cuspidal automorphic represen-…”

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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“…The space of automorphic forms is then defined as in §3.1. See also the original source [ 10 , §1.3] or [ 11 , §2.3]. We obtain the following important result on Eisenstein cohomology:…”

Section: Rationality For Isobaric Automorphic Representations: the Gementioning

confidence: 80%

“…We assume familiarity with the general results of [ 11 ]. In [ 11 , §3.1], following [ 9 ], a filtration of of finite length has been defined.…”

Section: Rationality For Isobaric Automorphic Representations: the Gementioning

confidence: 99%

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Rationality for isobaric automorphic representations: the CM-case

Grobner

2018

Monatsh Math

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In this note we prove a simultaneous extension of the author’s joint result with M. Harris for critical values of Rankin–Selberg L -functions (Grobner and Harris in J Inst Math Jussieu 15:711–769, 2016 , Thm. 3.9) to (i) general CM-fields F and (ii) cohomological automorphic representations which are the isobaric sum of unitary cuspidal automorphic representations of general linear groups of arbitrary rank over F . In this sense, the main result of these notes, cf. Theorem 1.9 , is a generalization, as well as a complement, of the main results in Raghuram (Forum Math 28:457–489, 2016 ; Int Math Res Not 2:334–372, 2010 . https://doi.org/10.1093/imrn/rnp127 ), and Mahnkopf (J Inst Math Jussieu 4:553–637, 2005 ).

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Residues of Eisenstein series and the automorphic cohomology of reductive groups

Cited by 13 publications

References 17 publications

A cohomological injectivity result for the residual automorphic spectrum of GL_n

A cohomological injectivity result for the residual automorphic spectrum of GL_n

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

Rationality for isobaric automorphic representations: the CM-case

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Residues of Eisenstein series and the automorphic cohomology of reductive groups

Cited by 13 publications

References 17 publications

A cohomological injectivity result for the residual automorphic spectrum of GLn

A cohomological injectivity result for the residual automorphic spectrum of GLn

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

Rationality for isobaric automorphic representations: the CM-case

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Product

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A cohomological injectivity result for the residual automorphic spectrum of GL_n

A cohomological injectivity result for the residual automorphic spectrum of GL_n