A direct proof of Mono-Rolen-Stumpenhusen and new constructions via the Maass raising operators

Abstract

In this paper, we give a direct conceptual proof of the main result of Mono, Rolen, and Stumpenhusen, using differential operators. More precisely, we realize their functions ${\omega }_{k+1,D}$ as images of the quadratic form Poincaré series $f_{k,D}$ under the Maass raising operator. This perspective gives a natural explanation for the modularity and Laplace eigenvalue prop erties of ${\omega }_{k+1,D}$. We further extend these results by investigating the images of more general local Maass forms under the Maass raising and lowering operators.