The heat operator and mock Jacobi forms

Abstract

We study a necessary and sufficient condition for Jacobi integrals of weight -r + j/2, r is an element of Z = 0, and index M((j)) on H x C(j) to have a dual Jacobi form of weight r + j/2 + 2 and index M((j)). Such a meromorphic Jacobi integral with a dual Jacobi form is called a mock Jacobi form whose concept was first introduced by Zagier in Seminaire Bourbaki, 60eme annee, 2006-2007, N(o) 986. In fact, we show the map L(M(j))(r+ 1) from the space of mock Jacobi forms to that of Jacobi forms is surjective by constructing the corresponding inverse image via Eichler integral of vector valued modular forms which are coming from the theta decomposition of Jacobi forms. We discuss Lerch sums as a typical example.