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Nonvanishing of the central critical value of the third symmetric power L -functions

Nonvanishing of the central critical value of the third symmetric power L -functions Abstract. In this paper we characterize the nonvanishing of the central critical value of the third symmetric power L-functions of irreducible cuspidal automorphic representations p of GL2 A in terms of the occurrence of p in the spectral decomposition of the tensor product of two automorphic theta representations on the cubic cover of GL2 A, which was constructed by Kazhdan and Patterson in [KP]. 1991 Mathematics Subject Classi®cation: 11F, 22E. 1 Introduction Let p be an irreducible cuspidal automorphic representation of GL2 A and n be a positive integer. One can de®ne, following Langlands, the n-th symmetric power L-function LsY pY Sym n . For the signi®cance of this family of automorphic L-functions, we refer to [Sh], for instance. The symmetric cube L-function, LsY pY Sym 3 , was ®rst studied by F. Shahidi in [Sh1] using the Langlands-Shahidi method to establish the analytic properties as conjectured by Langlands (meromorphic continuation and functional equation, . . .). In a recent work of D. Bump, D. Ginzburg and J. Ho¨stein [BGH], an integral representation was found for the symmetric cube L-functions, by which they can prove that the partial symmetric cube L-function is holomorphic for Res b 3 except http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Nonvanishing of the central critical value of the third symmetric power L -functions

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References (20)

Publisher
de Gruyter
Copyright
Copyright © 2001 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2001.001
Publisher site
See Article on Publisher Site

Abstract

Abstract. In this paper we characterize the nonvanishing of the central critical value of the third symmetric power L-functions of irreducible cuspidal automorphic representations p of GL2 A in terms of the occurrence of p in the spectral decomposition of the tensor product of two automorphic theta representations on the cubic cover of GL2 A, which was constructed by Kazhdan and Patterson in [KP]. 1991 Mathematics Subject Classi®cation: 11F, 22E. 1 Introduction Let p be an irreducible cuspidal automorphic representation of GL2 A and n be a positive integer. One can de®ne, following Langlands, the n-th symmetric power L-function LsY pY Sym n . For the signi®cance of this family of automorphic L-functions, we refer to [Sh], for instance. The symmetric cube L-function, LsY pY Sym 3 , was ®rst studied by F. Shahidi in [Sh1] using the Langlands-Shahidi method to establish the analytic properties as conjectured by Langlands (meromorphic continuation and functional equation, . . .). In a recent work of D. Bump, D. Ginzburg and J. Ho¨stein [BGH], an integral representation was found for the symmetric cube L-functions, by which they can prove that the partial symmetric cube L-function is holomorphic for Res b 3 except

Journal

Forum Mathematicumde Gruyter

Published: Jan 5, 2001

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