A converse theorem for Jacobi forms

Abstract

Let f(qτ, qz) = ∑n, rc(n, r) qnτ qrz be a power series whose coefficients satisfy a particular periodicity condition depending on the integer r modulo 2m. We first associate to f(qτ,qz) a 2m-vector-valued function Λ(f,s) via a generalized Mellin transform. Then we show that the function Λ(f, s) is entire, bounded on vertical strips and satisfies certain matrix functional equation if, and only if, f(qτ, qz) is the Fourier expansion of a Jacobi cusp form of index m invariant under the group SL(2, ℤ) ⋉ ℤ2. This is the direct analogue of Hecke's converse theorem for elliptic cusp forms in the context of Jacobi cusp forms on SL(2, ℤ) ⋉ ℤ2. © 1996 Academic Press, Inc.