Polyharmonic weak Maass forms of higher depth for SL_2(Z)

TL;DR

This paper introduces polyharmonic weak Maass forms of higher depth for SL_2(Z), relaxing growth conditions and constructing a basis, thereby generalizing previous work and enabling preimages under the xi-operator.

Contribution

It generalizes the concept of polyharmonic Maass forms by relaxing growth conditions and constructs a basis for these forms, extending Lagarias-Rhoades' framework.

Findings

Constructed a basis for polyharmonic weak Maass forms

Generalized Lagarias-Rhoades' basis construction

Obtained preimages under the xi-operator

Abstract

The space of polyharmonic Maass forms was introduced by Lagarias-Rhoades, recently. They constructed its basis from the Taylor coefficients of the real analytic Eisenstein series. In this paper, we introduce polyharmonic weak Maass forms, that is, we relax the moderate growth condition at cusp, and we construct a basis as a generalization of Lagarias-Rhoades' works. As a corollary, we can obtain a preimage of an arbitrary polyharmonic weak Maass form under the xi-operator.