Shifted polyharmonic Maass forms for PSL(2,Z)

TL;DR

This paper investigates shifted polyharmonic Maass forms of weight k for PSL(2,Z), establishing their finite-dimensionality, exploring their structure, and analyzing the role of Eisenstein series and differential operators in their theory.

Contribution

It extends previous work on polyharmonic Maass forms by including a shift parameter or urther understanding of their structure and properties.

Findings

V_k^m() is finite-dimensional and its dimension is bounded.

The role of Eisenstein series E_k(z,s) with =s(s+k-1) is clarified.

The differential operator d/ds is shown to be significant in the theory.

Abstract

We study the vector space V_k^m(\lambda) of shifted polyharmonic Maass forms of weight k \in 2Z, depth m \geq 0, and shift \lambda \in C. This space is composed of real-analytic modular forms of weight k for PSL(2,Z) with moderate growth at the cusp which are annihilated by (\Delta_k - \lambda)^m, where \Delta_k is the weight k hyperbolic Laplacian. We treat the case \lambda \neq 0, complementing work of the second and third authors on polyharmonic Maass forms (with no shift). We show that V_k^m(\lambda) is finite-dimensional and bound its dimension. We explain the role of the real-analytic Eisenstein series E_k(z,s) with \lambda=s(s+k-1) and of the differential operator d/ds in this theory.