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Polyharmonic Maass forms for \(\text {PSL}(2,{\mathbb Z})\)

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Abstract

We discuss the space of polyharmonic Maass forms of even integer weight on \(\text {PSL}(2,\mathbb Z)\backslash \mathbb H\). We explain the role of the real-analytic Eisenstein series \(E_k(z,s)\) and the differential operator \(\frac{\partial }{\partial s}\) in this theory.

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Notes

  1. The partial differential equations literature assigns order 2m to \(\left( \Delta _k \right) ^m\) when treated in terms of the differential operators \(\frac{\partial }{\partial x}\) and \(\frac{\partial }{\partial y}\). A second conflict is that p-harmonic Maass form is used by Bruggeman [9] with a different meaning, to refer to a function annihilated by the p-Laplacian, in which p corresponds to the weight parameter k in our notation.

  2. Bruggeman [8, Sect. 1.2.4] uses a family variable s that equals \(s_{usual}-\frac{1}{2}\) with \(s=s_{usual}\) used here.

  3. With our scaling \(E_{k}(z, 0) = \frac{1}{2}G_k(z)\), where \(G_k(z)\) is the usual unnormalized holomorphic Eisenstein series in Serre [29].

  4. The terminology is used in Bruinier, Ono and Rhoades [11], who call this space \(H_k\), omitting the \(+\), see [11, Remark 6].

  5. Siegel’s definition of the Eisenstein series has an extra factor of 2 compared to (1.3).

  6. The constant can be found using Proposition 7.1 as \(s \rightarrow 0\) in \(\xi _2( {\widehat{E}}_2(z, s))=(\overline{s} +1) (\overline{s} {\widehat{E}}_0(z, - \overline{s})),\) using the simple pole of \({\widehat{E}}_0(z, s)\) at \(s=0\) with residue \(-\frac{1}{2}\).

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Acknowledgments

The authors thank N. Andersen, W. Duke and the reviewer for helpful comments. The first author thanks the Clay Foundation for support as a Clay Senior Fellow at ICERM, an NSF-supported institute, where some work on this paper was done.

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Authors and Affiliations

  1. Department of Mathematics, The University of Michigan, Ann Arbor, MI, 48109-1043, USA

    Jeffrey C. Lagarias

  2. Center for Communications Research, Princeton, NJ, 08540, USA

    Robert C. Rhoades

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  1. Jeffrey C. Lagarias
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  2. Robert C. Rhoades
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Correspondence to Robert C. Rhoades.

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Dedicated to the memory of Marvin Knopp.

Research of the first author was partially supported by NSF Grants DMS-1101373 and DMS-1401224. Research of the second author was partially supported by an NSF Mathematical Sciences Postdoctoral Fellowship.

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Lagarias, J.C., Rhoades, R.C. Polyharmonic Maass forms for \(\text {PSL}(2,{\mathbb Z})\) . Ramanujan J 41, 191–232 (2016). https://doi.org/10.1007/s11139-015-9729-1

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