Periods of Jacobi forms and Hecke operator

TL;DR

This paper investigates the space of period functions of Jacobi forms, providing an explicit description of Hecke operators acting on this space, with applications to Jacobi Eisenstein series and connections to mock Jacobi forms.

Contribution

It introduces a new explicit description of Hecke operators on the period functions of Jacobi forms using Jacobi integrals, extending previous work on modular and Maass forms.

Findings

Explicit description of Hecke operators on Jacobi period functions

Application to Jacobi Eisenstein series of weight 2 and index 1

Connections to mock Jacobi forms and Mordell integrals

Abstract

A Hecke action on the space of periods of cusp forms, which is compatible with that on the space of cusp forms, was first computed using continued fraction 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. As an application an elementary proof of the Eichler-Selberg trace formula was derived. Similar modification has been applied to period space of Maass cusp forms with spectral parameter $s$. In this paper we study the space of period functions of Jacobi forms by means of Jacobi integral and give an explicit description of Hecke operator acting on this space. a Jacobi Eisenstein series $E_{2,1}({\tau},z)$ of weight 2 and index 1 is discussed as an example. Periods of Jacobi integrals are already appeared as a disguised form in the work of Zwegers to study Mordell integral coming from Lerch sums and mock Jacobi forms are typical example of Jacobi integral.