### Abstract

The period polynomial of a cusp form of an integral weight plays an important role in the number theory. In this paper, we study the period function of a cusp form of real weight. We obtain a series expansion of the period function of a cusp form of real weight for SL(2,Z) by using the binomial expansion. Furthermore, we study two kinds of Hecke operators acting on cusp forms and period functions, respectively. With these Hecke operators we show that there is a Hecke-equivariant isomorphism between the space of cusp forms and the space of period functions. As an application, we obtain a formula for a certain L-value of a Hecke eigenform by using the series expansion of its period function.

Original language | English |
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Pages (from-to) | 301-340 |

Number of pages | 40 |

Journal | Journal of Number Theory |

Volume | 171 |

DOIs | |

Publication status | Published - 2017 Feb 1 |

Externally published | Yes |

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### Keywords

- Eichler–Shimura cohomology
- Hecke operators
- Period functions

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

*171*, 301-340. https://doi.org/10.1016/j.jnt.2016.07.020