On the Cuspidal Cohomology of S -Arithmetic Subgroups of Reductive Groups over Number Fields

Borel, A.; Labesse, Jean-Pierre; Schwermer, Joachim

doi:10.1007/978-3-642-41240-0_35

Cited by 31 publications

(51 citation statements)

References 6 publications

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“…The functorial lifts used were base change for GL 2 and the symmetric square lifting of Gelbart and Jacquet. This was generalized for SL n over E, in conjunction with Borel [5]; with the additional input of base change for GL n . (2) Motivated by [26], Clozel [7] used automorphic induction and proved the existence of nontrivial cuspidal cohomology classes for SL 2n over any number field.…”

Section: 2mentioning

confidence: 99%

Functoriality and Special Values of L-Functions

Raghuram

Shahidi

Eisenstein Series and Applications

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Abstract. This is a semi-expository article concerning Langlands functoriality and Deligne's conjecture on the special values of L-functions. The emphasis is on symmetric power L-functions associated to a holomorphic cusp form, while appealing to a recent work of Mahnkopf on the special values of automorphic L-functions. Contents IntroductionLanglands functoriality principle reduces the study of automorphic forms on the adèlic points of a reductive algebraic group to those of an appropriate general linear group. In particular, every automorphic L-function on an arbitrary reductive group must be one for a suitable GL n . One should therefore be able to reduce the study of of special values of an automorphic L-function to those of a principal L-function of Godement and Jacquet on GL n . A. RAGHURAM AND FREYDOON SHAHIDIWhile the integral representations of Godement and Jacquet do not seem to admit a cohomological interpretation, there is a recent work of J. Mahnkopf [27] [28] which provides us with such an interpretation for certain Rankin-Selberg type integrals. In particular, modulo a nonvanishing assumption on local archimedean Rankin-Selberg product L-functions for forms on GL n × GL n−1 , he defines a pair of periods, which seem to be in accordance with those of Deligne [9] and Shimura [43]. This work of Mahnkopf is quite remarkable and requires the use of both Rankin-Selberg and Langlands-Shahidi methods in studying the analytic (and arithmetic) properties of L-functions. His work therefore brings in the theory of Eisenstein series to play an important role. In §6 we briefly review this work of Mahnkopf.This article is an attempt to test the philosophy-to study the special values of L-functions while using functoriality-by means of recent cases of functoriality established for symmetric powers of automorphic forms on GL 2 [17] [21]. While a proof of the precise formulae in the conjectures of Deligne [9] still seem to be out of reach, we expect to be able to prove explicit connections between the special values of symmetric power L-functions twisted by Dirichlet characters and those of the original symmetric power L-functions using this work of Mahnkopf. These relations are formulated in this paper as Conjecture 7.1 which also follows from the more general conjectures of Blasius [3] and Panchiskin [33], although the heuristics underlying our conjecture are quite different.A standard assumption made in the study of special values of L-functions is that the representations (to which are attached the L-functions) are cohomological. This is the case in Mahnkopf's work. A global representation being cohomological is entirely determined by the archimedean components. For representations which are symmetric power lifts of a cusp form on GL 2 we have the following fact. Consider a holomorphic cusp form on the upper half plane of weight k. This corresponds to a cuspidal automorphic representation, which is cohomological if k ≥ 2, and any symmetric power lift, if cuspidal, is essentially cohomological. See Theorem 5...

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Section: 2mentioning

confidence: 99%

Functoriality and Special Values of L-Functions

Raghuram

Shahidi

Eisenstein Series and Applications

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“…First, by the construction of cuspidal automorphic representations and related cohomology classes for arithmetic subgroups of GL n over an algebraic number field ( [80], resp. [21]), there are nonvanishing results for the cohomology in degrees which cannot be detected by any modular symbol of type L, L a Levi subgroup of G. Second, modular symbols of a different type can contribute to cohomology in the same cohomological degree.…”

Section: Cornersmentioning

confidence: 99%

Geometric cycles, arithmetic groups and their cohomology

Schwermer

2010

Bull. Amer. Math. Soc.

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Abstract. It is the aim of this article to give a reasonably detailed account of a specific bundle of geometric investigations and results pertaining to arithmetic groups, the geometry of the corresponding locally symmetric space X/Γ attached to a given arithmetic subgroup Γ ⊂ G of a reductive algebraic group G and its cohomology groups H * (X/Γ, C). We focus on constructing totally geodesic cycles in X/Γ which originate with reductive subgroups H ⊂ G. In many cases, it can be shown that these cycles, to be called geometric cycles, yield non-vanishing (co)homology classes. Since the cohomology of an arithmetic group Γ is strongly related to the automorphic spectrum of Γ, this geometric construction of non-vanishing classes leads to results concerning, for example, the existence of specific automorphic forms.

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“…The space V G is a (g, K ; G(A f ))-module and admits a decomposition V G = {P}∈C V G ({P}) as a finite direct sum of (g, K ; G(A f ))-modules as proved by Langlands (cf. [2]). This decomposition descends to the submodule A E , that is…”

Section: Decomposition Along the Cuspidal Supportmentioning

confidence: 99%

“…Next, let Q ∈ {P} and π be an irreducible representation of L Q (A), unitary modulo the center, and such that (1) the central character χ π : A Q (A) → C × is trivial on A Q (Q), (2)π occurs in the cuspidal summand • L 2 (L Q (Q)\L Q (A)) χ π and (3) the infinitesimal character of π matches the infinitesimal character of the dual representationĚ. These conditions entail some compatibility requirements (cf.…”

Section: Decomposition Along the Cuspidal Supportmentioning

confidence: 99%

Eisenstein cohomology for congruence subgroups of SO(n, 2)

Gotsbacher

2009

Math. Z.

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The automorphic cohomology of a connected reductive algebraic group defined over Q decomposes as a direct algebraic sum of cuspidal and Eisenstein cohomology. The present paper investigates the Eisenstein cohomology for congruence subgroups of a rational form G of Q-rank 2 of SO(n, 2) in the generic case. The main results provide a description of the internal structure of the summands in the Eisenstein cohomology corresponding to the associate classes of maximal parabolic Q-subgroups.

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On the Cuspidal Cohomology of S -Arithmetic Subgroups of Reductive Groups over Number Fields

Cited by 31 publications

References 6 publications

Functoriality and Special Values of L-Functions

Functoriality and Special Values of L-Functions

Geometric cycles, arithmetic groups and their cohomology

Eisenstein cohomology for congruence subgroups of SO(n, 2)

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