Cohomological relation between Jacobi forms and skew-holomorphic Jacobi forms

TL;DR

This paper establishes a cohomological framework linking holomorphic and skew-holomorphic Jacobi forms, revealing a new isomorphism that enhances understanding of their structural relationship in modular form theory.

Contribution

It introduces a novel isomorphism connecting spaces of Jacobi and skew-holomorphic Jacobi cusp forms to Eichler cohomology, advancing the theoretical understanding of their interplay.

Findings

Established an isomorphism via group cohomology

Connected spaces of Jacobi and skew-holomorphic forms

Enhanced understanding of modular form liftings

Abstract

Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between holomorphic Jacobi forms and skew-holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on $\Gamma^J$ and the space of skew-holomorphic Jacobi cusp forms on $\Gamma^J$ with the same half-integral weight to the Eichler cohomology group of $\Gamma^J$ with a coefficient module coming from polynomials.