A Converse Theorem for Jacobi Forms Original Research Article

Letf(qτ, qz)=∑n, r c(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, image)times sign, left closedimage2. This is the direct analogue of Heckeʹs converse theorem for elliptic cusp forms in the context of Jacobi cusp forms on SL(2, image)times sign, left closedimage2.