# Series expansion of the period function and representations of Hecke operators

Dohoon Choi, Subong Lim, Tobias Mühlenbruch, Wissam Raji

Research output: Contribution to journalArticle

### 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 301-340 40 Journal of Number Theory 171 https://doi.org/10.1016/j.jnt.2016.07.020 Published - 2017 Feb 1 Yes

Period Function
Hecke Operators
Cusp Form
Series Expansion
Number theory
Equivariant
Isomorphism
Polynomial

### Keywords

• Eichler–Shimura cohomology
• Hecke operators
• Period functions

### ASJC Scopus subject areas

• Algebra and Number Theory

### Cite this

Series expansion of the period function and representations of Hecke operators. / Choi, Dohoon; Lim, Subong; Mühlenbruch, Tobias; Raji, Wissam.

In: Journal of Number Theory, Vol. 171, 01.02.2017, p. 301-340.

Research output: Contribution to journalArticle

Choi, Dohoon ; Lim, Subong ; Mühlenbruch, Tobias ; Raji, Wissam. / Series expansion of the period function and representations of Hecke operators. In: Journal of Number Theory. 2017 ; Vol. 171. pp. 301-340.
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