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

Volume 185 / 2018

Nickolas Andersen, Jeffrey C. Lagarias, Robert C. Rhoades Acta Arithmetica 185 (2018), 39-79 MSC: 11F55, 11F37, 11F12. DOI: 10.4064/aa170905-7-3 Published online: 15 June 2018


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.


  • Nickolas AndersenDepartment of Mathematics
    Los Angeles, CA 90095, U.S.A.
  • Jeffrey C. LagariasDepartment of Mathematics
    University of Michigan
    Ann Arbor, MI 48109-1043, U.S.A.
  • Robert C. RhoadesCenter for Communications Research
    Princeton, NJ 08540, U.S.A.
    Susquehanna International Group
    401 City Ave.
    Bala Cynwyd, PA 19004, U.S.A.

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