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Shifted polyharmonic Maass forms for \(\text{PSL} (2,{\mathbb Z})\). (English) Zbl 1440.11078
Summary: 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.

11F55 Other groups and their modular and automorphic forms (several variables)
11F37 Forms of half-integer weight; nonholomorphic modular forms
11F12 Automorphic forms, one variable
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