On integral kernels for Dirichlet series associated to Jacobi forms

Abstract

Every Jacobi cusp form of weight k and index m over SL 2 ( Z ) ⋉ Z 2 is in correspondence with 2 m Dirichlet series constructed with its Fourier coefficients. The standard way to get from one to the other is by a variation of the Mellin transform. In this paper, we introduce a set of integral kernels which yield the 2 m Dirichlet series via the Petersson inner product. We show that those kernels are Jacobi cusp forms and express them in terms of Jacobi Poincaré series. As an application, we give a new proof of the analytic continuation and functional equations satisfied by the Dirichlet series mentioned above.