Abstract
The space of polyharmonic Maass forms was introduced by Lagarias–Rhoades, recently. They constructed its basis from the Taylor coefficients of the real analytic Eisenstein series. In this paper, we introduce polyharmonic weak Maass forms, that is, we relax the moderate growth condition at cusp, and we construct a basis as a generalization of Lagarias–Rhoades’ works. As a corollary, we can obtain a preimage of an arbitrary polyharmonic weak Maass form under the \(\xi \)-operator.
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Acknowledgements
I would like to express my gratitude to Scott Ahlgren for letting me know polyharmonic Maass forms. Many thanks also to Daeyeol Jeon, Soon-Yi Kang, and Chang Heon Kim for their helpful comments on Theorem 1.4, and to Kathrin Bringmann and Jonas Kaszian for providing me a detailed history of polyharmonic Maass forms. Furthermore, it is a pleasure to thank Shunsuke Yamana for inviting me to MPIM, and also thank Soon-Yi Kang and Masanobu Kaneko for giving me the precious opportunity to discuss in NIMS. I am very grateful to the referee for careful reading of this paper and fruitful suggestions.
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This work is supported by Research Fellow (DC) of Japan Society for the Promotion of Science, Grant-in-Aid for JSPS Fellows 18J20590 and JSPS Overseas Challenge Program for Young Researchers. .
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Matsusaka, T. Polyharmonic weak Maass forms of higher depth for \(\mathrm {SL}_2({\mathbb {Z}})\). Ramanujan J 51, 19–42 (2020). https://doi.org/10.1007/s11139-018-0057-0
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DOI: https://doi.org/10.1007/s11139-018-0057-0