The Kohnen plus space for Hilbert-Siegel modular forms

TL;DR

This paper extends the concept of Kohnen plus space to Hilbert-Siegel modular forms, establishing an isomorphism with a related space, generalizing previous results for various modular forms.

Contribution

It introduces a new analogue of the Kohnen plus space for Hilbert-Siegel modular forms, expanding the theoretical framework and previous generalizations.

Findings

Established the structure of the Kohnen plus space for Hilbert-Siegel modular forms

Proved an isomorphism with a space of related modular forms

Generalized previous results to a broader class of modular forms

Abstract

The Kohnen plus space, roughly speaking, is a space consisting of modular forms of half integral weight with some property in Fourier coefficients. For example, the $n$-th coefficient of a normal modular form of weight $k+1/2$ in the plus space is $0$ unless $(-1)^kn$ is congruent to some square modulo $4$. The concept of plus space was initially introduced by Kohnen in 1980. Eichler and Zagier showed that the plus space is isomorphic to the space of Jacobi forms in the one variable case. Later, Ibukiyama generalized these results to the cases for Siegel modular forms in 1992. Also, Hiraga and Ikeda generalized these results to the cases for Hilbert modular forms in 2013. In this paper, we continue to consider the case of Hilbert-Siegel modular forms. An analogue of the previous results will be given.