Non-vanishing of Jacobi Poincar\'{e} series

TL;DR

This paper establishes conditions under which Jacobi Poincaré series of exponential type do not vanish, constructs bases for classical Jacobi forms, and proves related sum equalities, advancing understanding of their non-vanishing properties.

Contribution

It provides new non-vanishing criteria for Jacobi Poincaré series, constructs bases for classical Jacobi forms, and proves sum equalities related to Kloosterman sums.

Findings

Jacobi Poincaré series of exponential type do not vanish under certain conditions.

A basis of classical Jacobi forms is constructed from initial Poincaré series.

Equality of specific Kloosterman-type sums is established.

Abstract

We prove that under suitable conditions, the Jacobi Poincar\'{e} series of exponential type of integer weight and matrix index does not vanish identically. For classical Jacobi forms, we construct a basis consisting of the "first" few Poincar\'{e} series and also give conditions both dependent and independent of the weight, which ensures non-vanishing of classical Jacobi Poincar\'{e} series. Equality of certain Kloosterman-type sums is proved. Also, a result on the non-vanishing of Jacobi Poincar\'{e} series is obtained when an odd prime divides the index.