Mock modular forms whose shadows are Eisenstein series of integral weight.

*(English)*Zbl 07214390As Bruinier and Funke showed in their well-known paper defining harmonic Maass forms, the infinite-to-one operator \(\xi_{2-k}\) is a surjective map from harmonic Maass forms of weight \(2-k\) to modular forms of weight \(k\), called their shadows following Zagier. One of the fundamental problems in the theory is to find explicit preimages for given modular forms. One of the basic examples of this is to take a preimage of a Poincaré series of exponential type, which classically span the space of cusp forms. This naturally yields the important Maass-Poincaré series, fundamental building blocks in the theory of harmonic Maass forms, and is a key to many applications such as asymptotics and exact formulas, as well as existence-style proofs.

This paper focuses on a problem related to the aforementioned Poincaré series, namely Eisenstein series. Importantly, it studies the completely general case of any integral weight \(k\geq1\), any level, and with a character. The small weight cases are notoriously difficult for many studies, especially the unique case of \(k=1\). Harmonic Maass form theory often involves a duality between weight \(k\) and \(2-k\), and the closer one gets to the center of this symmetry, \(k=1\), the more difficult problems tend to be.

The authors work out such preimages in these difficult and very general situations. They give explicit formulas for the Fourier coefficients of these harmonic Maass form lifts. This is a technical achievement which will be useful for many authors working in the field. Moreover, the authors describe nice applications of these new lifts, for example to Hecke’s Eisenstein series for imaginary quadratic fields and theta series of weight 1.

This paper focuses on a problem related to the aforementioned Poincaré series, namely Eisenstein series. Importantly, it studies the completely general case of any integral weight \(k\geq1\), any level, and with a character. The small weight cases are notoriously difficult for many studies, especially the unique case of \(k=1\). Harmonic Maass form theory often involves a duality between weight \(k\) and \(2-k\), and the closer one gets to the center of this symmetry, \(k=1\), the more difficult problems tend to be.

The authors work out such preimages in these difficult and very general situations. They give explicit formulas for the Fourier coefficients of these harmonic Maass form lifts. This is a technical achievement which will be useful for many authors working in the field. Moreover, the authors describe nice applications of these new lifts, for example to Hecke’s Eisenstein series for imaginary quadratic fields and theta series of weight 1.

Reviewer: Larry Rolen (Dublin)