A Converse Theorem for Jacobi Forms

AbstractLetf(qτ, qz)=∑n, rc(n, r)qnτqrzbe a power series whose coefficients satisfy a particular periodicity condition depending on the integerrmodulo 2m. We first associate tof(qτ, qz) a 2m-vector-valued functionΛ(f, s) via a generalized Mellin transform. Then we show that the functionΛ(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 indexminvariant 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