## Shifted polyharmonic Maass forms for ${\rm PSL} (2,{\mathbb Z})$

### Volume 185 / 2018

#### Abstract

We study the vector space $V_k^m(\lambda)$ of shifted polyharmonic Maass forms of weight $k\in 2\mathbb Z$, depth $m\geq 0$, and shift $\lambda\in \mathbb C$. This space is composed of real-analytic modular forms of weight $k$ for $\operatorname{PSL}(2,\mathbb Z)$ with moderate growth at the cusp which are annihilated by $(\varDelta_k - \lambda)^m$, where $\varDelta_k$ 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 $\frac{\partial}{\partial s}$ in this theory.