Periods of Jacobi forms and the Hecke operator

A Hecke action on the space of periods of cusp forms, that is compatible with that on the space of cusp forms, was first computed using continued fraction  [20] and an explicit algebraic formula of Hecke operators acting on the space of period functions of modular forms was derived by studying the rational period functions [9]. As an application an elementary proof of the Eichler–Selberg trace formula was derived  [27]. A similar modification has been applied to the space of period functions of Maass cusp forms with spectral parameter s [22,23,21]. In this paper we study the space of period functions of Jacobi forms by means of the Jacobi integral and give an explicit description of the action of Hecke operators on this space. A Jacobi–Eisenstein series E 2 , 1 ( τ , z ) of weight 2 and index 1 is discussed as an example. Periods of Jacobi integrals already appeared in a disguised form in the work of Zwegers in his study of the Mordell integral coming from Lerch sums  [28], and mock Jacobi forms are typical examples of the Jacobi integral  [10].