Hermitian Maass lift for General Level

TL;DR

This paper generalizes the Hermitian Maass lift to arbitrary levels for imaginary quadratic fields, establishing an isomorphism between hermitian Jacobi forms and plus forms, and constructing a new lift from elliptic modular forms.

Contribution

It extends Krieg's results to arbitrary levels and introduces a generalized Hermitian Maass lift for hermitian modular forms.

Findings

Established an isomorphism between hermitian Jacobi forms and plus forms at arbitrary level.

Proved the existence of a generalized Hermitian Maass lift from elliptic modular forms.

Connected the lift to recent work by Berger and Klosin and Ikeda's construction.

Abstract

For an imaginary quadratic field $K$ of discriminant $-D$, let $\chi = \chi_K$ be the associated quadratic character. We will show that the space of special hermitian Jacobi forms of level $N$ is isomorphic to the space of plus forms of level $DN$ and nebentypus $\chi$ (the hermitian analogue of Kohnen's plus space) for any integer $N$ prime to $D$. This generalizes the results of Krieg from $N = 1$ to arbitrary level. Combining this isomorphism with the recent work of Berger and Klosin and a modification of Ikeda's construction we prove the existence of a lift from the space of elliptic modular forms to the space of hermitian modular forms of level $N$ which can be viewed as a generalization of the classical hermitian \Maass lift to arbitrary level.