On automorphic \(L\)-functions for the Jacobi group of degree one and a relation with \(L\)-functions for Jacobi forms
type: Item
instance of: scholarly article
title: On automorphic \(L\)-functions for the Jacobi group of degree one and a relation with \(L\)-functions for Jacobi forms
zbMATH Open document ID: 0905.11026
DOI: 10.1007/BF02678190
author: Jost Homrighausen
author: Rolf Berndt
published in: Manuscripta Mathematica
publication date: 1999-01-05T00:00:00Z
full work available at URL: https://eudml.org/doc/156261
review text: The paper under review discusses automorphic \(L\)-functions attached to certain automorphic representations of the Jacobi group of degree one. Following the two well-known methods in the \(GL(2)\)-theory, the authors investigate the local \(L\)-factor given by the greatest common divisor of certain zeta integrals \(\zeta_p (W,s)\), \(W\) running over the Whittaker model, and compare this factor with the \(L\)-factor given by the zeta integral \(\zeta_p (W^0,s)\) of a normalized spherical Whittaker function \(W^0\). In contrast to the \(GL(2)\)-theory, these two methods lead in general to different local \(L\)-factors and the second one even depends on the character specifying the Whittaker model. The authors give an explanation of the above situation in the special case of an automorphic representation attached to a Hecke eigenform \(f\) of index one. They show (under a certain condition on \(f)\) that the first method leads to the local \(L\)-factor attached to \(f\) by \textit{V. Gritsenko} [Math. USSR, Sb. 47, 237-268 (1984); translation from Mat. Sb., Nov. Ser. 119(161), 248-277 (1982; Zbl 0507.10017); and Mathematica Gottingensis, SFB Göttingen, Heft 12] and \textit{T. Arakawa} [Comment. Math. Univ. St. Pauli 43, 181-216 (1994; Zbl 0837.11027)] using the Hecke eigenvalues of \(f\), whereas the zeta integral \(\zeta_p(W^0,s)\) is equal to the \(L\)-factor of the modular form of half-integral weight associated to \(f\) as defined by \textit{G. Shimura} [Ann. Math., II. Ser. 97, 440-481 (1973; Zbl 0266.10022)]. Moreover it is shown that a certain linear combination of these latter \(L\)-functions with local factors \(\zeta_p (W^0,s)\) is equal to the Mellin transform of \(f\) as defined by \textit{R. Berndt} [Manuscr. Math. 84, 101-112 (1994; Zbl 0827.11028)].
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F50
Mathematics Subject Classification ID: 33C15
zbMATH DE Number: 1019709
zbMATH Keywords: automorphic \(L\)-functions
zbMATH Keywords: local \(L\)-factor
zbMATH Keywords: automorphic representations of the Jacobi group
zbMATH Keywords: zeta integrals
zbMATH Keywords: Whittaker function
zbMATH Keywords: Whittaker model
zbMATH Keywords: Mellin transform
reviewed by: Aloys Krieg
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1019709