\(L\)-functions for Jacobi forms of arbitrary degree
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 \(f(Z,W)\).
scientific article; zbMATH DE number 1416660
scientific article