Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues

TL;DR

This paper investigates differential operators on harmonic weak Maass forms, revealing that forms dual to certain newforms with vanishing Hecke eigenvalues have algebraic coefficients, and characterizes the image of a specific differential operator.

Contribution

It introduces new insights into the algebraic nature of coefficients of harmonic weak Maass forms related to special newforms and characterizes the image of a differential operator using regularized inner products.

Findings

Harmonic weak Maass forms dual to newforms with vanishing Hecke eigenvalues have algebraic coefficients.

The operator $D^{k-1}$'s image is characterized via regularized inner products.

Certain harmonic weak Maass forms exhibit algebraic coefficients despite the general expectation of transcendentality.

Abstract

For integers $k\geq 2$, we study two differential operators on harmonic weak Maass forms of weight $2-k$. The operator $\xi_{2-k}$ (resp. $D^{k-1}$) defines a map to the space of weight $k$ cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are "dual" under $\xi_{2-k}$ to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of $D^{k-1}$.