Kohnen's limit process for real-analytic Siegel modular forms

TL;DR

This paper explores a limit process for Siegel modular forms that yields Jacobi forms, introduces harmonic skew-Maass-Jacobi forms, and confirms the existence of a space linking these forms, answering Kohnen's question.

Contribution

The paper defines harmonic skew-Maass-Jacobi forms and harmonic Siegel-Maass forms, establishing a connection between them and confirming Kohnen's limit process produces skew-holomorphic Jacobi forms.

Findings

Improved Fourier coefficient bounds for harmonic Siegel-Maass forms

Established a link between harmonic skew-Maass-Jacobi forms and harmonic Siegel-Maass forms

Confirmed Kohnen's question about the limit process producing skew-holomorphic Jacobi forms

Abstract

Kohnen introduced a limit process for Siegel modular forms that produces Jacobi forms. He asked if there is a space of real-analytic Siegel modular forms such that skew-holomorphic Jacobi forms arise via this limit process. In this paper, we initiate the study of harmonic skew-Maass-Jacobi forms and harmonic Siegel-Maass forms. We improve a result of Maass on the Fourier coefficients of harmonic Siegel-Maass forms, which allows us to establish a connection to harmonic skew-Maass-Jacobi forms. In particular, we answer Kohnen's question in the affirmative.