Rankin's method and Jacobi forms of several variables

TL;DR

This paper extends Rankin's method and the computation of inner products involving Rankin-Cohen brackets from classical modular and Jacobi forms to Jacobi forms of several variables over higher-dimensional spaces.

Contribution

It generalizes the inner product formula for Rankin-Cohen brackets to Jacobi forms in multiple variables, broadening the scope of Rankin's method.

Findings

Derived a new inner product formula for multi-variable Jacobi forms.

Extended Rankin's method to higher-dimensional Jacobi forms.

Provided a framework for analyzing special values of convolution L-functions.

Abstract

Following Rankin's method, D. Zagier computed the $n$-th Rankin-Cohen bracket of a modular form $g$ of weight $k_1$ with the Eisenstein series of weight $k_2$ and then computed the inner product of this Rankin-Cohen bracket with a cusp form $f$ of weight $k = k_1+k_2+2n$ and showed that this inner product gives, upto a constant, the special value of the Rankin-Selberg convolution of $f$ and $g$. This result was generalized to Jacobi forms of degree 1 by Y. Choie and W. Kohnen. In this paper, we generalize this result to Jacobi forms defined over ${\mathcal H} \times {\mathbb C}^{(g, 1)}$.