The Eichler cohomology theorem for Jacobi forms

TL;DR

This paper extends Eichler cohomology theory to Jacobi forms, establishing an isomorphism between cohomology groups and spaces of Jacobi forms, and analyzing the properties of cocycles and their relation to L-functions.

Contribution

It introduces a new cohomology framework for Jacobi forms and proves an Eichler cohomology theorem for arbitrary real weights, generalizing previous results.

Findings

Cohomology groups for Jacobi groups are defined and analyzed.

Every cocycle in the theory is parabolic.

An isomorphism between cohomology groups and Jacobi forms is established in certain cases.

Abstract

Let $\Gamma$ be a finitely generated Fuchsian group of the first kind which has at least one parabolic class. Eichler introduced a cohomology theory for Fuchsian groups, called as "Eichler cohomology theory", and established the $\CC$-linear isomorphism from the direct sum of two spaces of cusp forms on $\Gamma$ with the same integral weight to the Eichler cohomology group of $\Gamma$. After the results of Eichler, the Eichler cohomology theory was generalized in various ways. For example, these results were generalized by Knopp to the cases with arbitrary real weights. In this paper, we extend the Eichler cohomology theory to the context of Jacobi forms. We define the cohomology groups of Jacobi groups which are analogues of Eichler cohomology groups and prove an Eichler cohomology theorem for Jacobi forms of arbitrary real weights. Furthermore, we prove that every cocycle is parabolic and that for some special cases we have an isomorphism between the cohomology group and the space of Jacobi forms in terms of the critical values of partial $L$-functions of Jacobi cusp forms.