Harmonic Maass forms associated with CM newforms

TL;DR

This paper constructs harmonic Maass forms linked to CM newforms using regularized theta lifts, demonstrating algebraic Fourier coefficients and rationality properties, thus addressing a question in the field.

Contribution

It introduces a new method using regularized theta lifts to study harmonic Maass forms associated with CM newforms, revealing their algebraic and rationality properties.

Findings

Fourier coefficients of the harmonic Maass forms are algebraic.

Rationality properties of coefficients of forms linked to CM newforms are established.

Addresses a question posed by Bruinier, Ono, and Rhoades.

Abstract

In this paper, we use a regularized theta lifting to construct harmonic Maass forms corresponding to binary theta functions of weight $k \ge 2$ under the $\xi$-operator. As a result, we show that their holomorphic parts have algebraic Fourier coefficients, with compatible Galois action. As an application, we prove rationality properties of coefficients of harmonic Maaass forms corresponding to CM newforms, answering a question of Bruinier, Ono and Rhoades.