Pairings of harmonic Maass-Jacobi forms involving special values of partial $L$-functions

TL;DR

This paper explores the relationship between harmonic Maass-Jacobi forms and Jacobi integrals through cohomology theory, revealing a pairing linked to special values of partial L-functions and connecting it to the Petersson inner product, akin to Haberland's formula.

Contribution

It introduces a novel pairing between Jacobi integrals based on partial L-values, extending classical modular form results to Jacobi forms using cohomological methods.

Findings

Established a pairing between Jacobi integrals via partial L-values.

Connected the pairing to the Petersson inner product for skew-holomorphic Jacobi cusp forms.

Presented an analogue of Haberland's formula for Jacobi forms.

Abstract

In this paper, considering the Eichler-Shimura cohomology theory for Jacobi forms, we study connections between harmonic Maass-Jacobi forms and Jacobi integrals. As an application we study a pairing between two Jacobi integrals, which is defined by special values of partial $L$-functions of skew-holomorphic Jacobi cusp forms. We obtain connections between this pairing and the Petersson inner product for skew-holomorphic Jacobi cusp forms. This result can be considered as analogue of Haberland formula of elliptic modular forms for Jacobi forms.