\(L\)-functions for Jacobi forms of arbitrary degree
type: Item
instance of: scholarly article
title: \(L\)-functions for Jacobi forms of arbitrary degree
zbMATH Open document ID: 0948.11021
DOI: 10.1007/BF02942550
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2000-03-19T00:00:00Z
review text: The author introduces a finite number of \(L\)-functions for every Jacobi cusp form \(f(Z,W)\) defined on \({\mathcal H}\times{{\mathbb C}^{(j,n)}}\), where \({\mathcal H}\) denotes the Siegel upper half space of degree \(n\) and \({\mathbb C}^{(j,n)}\) is the set of complex matrices with \(j\) rows and \(n\) columns. These \(L\)-functions are defined by infinite series constructed with the Fourier coefficients of \(f(Z,W)\) and a parameter \(s\in{\mathbb C}^n\). These new objects generalize the \(L\)-functions defined by the author [J. Number Theory 61, 181-193 (1996; Zbl 0869.11044)] \((n=j=1)\) to arbitrary positive integers \(n\) and \(j\). The author establishes the holomorphic continuation of the \(L\)-functions to the whole space \({\mathbb C}^n\) and exhibits a matrix functional equation satisfied by the row vectors formed with them. The proof is modeled on Hecke's original argument for elliptic modular forms, and it is based on integral representations for \(L\)-functions. Then the results are derived by using an argument similar to one for the meromorphic continuation of the Selberg-Eisenstein series and the functional equation relating \(f(-Z^{-1},WZ^{-1})\) and \(f(Z,W)\).
Mathematics Subject Classification ID: 11F50
Mathematics Subject Classification ID: 11F66
zbMATH DE Number: 1416660
zbMATH Keywords: Jacobi forms for arbitrary degree
zbMATH Keywords: \(L\)-functions of Jacobi forms
zbMATH Keywords: integral representations
zbMATH Keywords: Selberg Eisenstein series
author: Yves Martin
reviewed by: Taro Horie
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1416660