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Traces of CM values and cycle integrals of polyharmonic Maass forms

  • Toshiki MatsusakaEmail author
Research
  • 97 Downloads

Abstract

Lagarias and Rhoades generalized harmonic Maass forms by considering forms which are annihilated by a number of iterations of the action of the \(\xi \)-operator. In our previous work, we considered polyharmonic weak Maass forms by allowing the exponential growth at cusps, and constructed a basis of the space of such forms. This paper focuses on the case of half-integral weight. We construct a basis as an analogue of our work, and give arithmetic formulas for the Fourier coefficients in terms of traces of CM values and cycle integrals of polyharmonic weak Maass forms. These results put the known results into a common framework.

Keywords

Polyharmonic Maass forms Harmonic Modular forms Fourier coefficients 

Mathematics Subject Classification

Primary 11F37 Secondary 11F12 

Notes

Author's contributions

Acknowledgements

The author would like to thank Masanobu Kaneko, Soon-Yi Kang, Chang Heon Kim, and Markus Schwagenscheidt for their helpful comments. He also thanks the referees for their comments on a preliminary version of this paper. 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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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan

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