### Abstract

Let S_{k} ^{!}(Γ_{1}(N)) be the space of weakly holomorphic cusp forms of weight k on Γ_{1}(N) with an even integer k>2 and M_{k} ^{!}(Γ_{1}(N)) be the space of weakly holomorphic modular forms of weight k on Γ_{1}(N). Further, let z denote a complex variable and D:=[Formula presented][Formula presented]. In this paper, we construct a basis of the space S_{k} ^{!}(Γ_{1}(N))/D^{k−1}(M_{2−k} ^{!}(Γ_{1}(N))) consisting of Hecke eigenforms by using the Eichler–Shimura cohomology theory. Further, we study algebraicity of CM values of weakly holomorphic modular forms in the basis. This applies to an analogue of the Chowla–Selberg formula for a mock modular form whose shadow is the Ramanujan delta function.

Original language | English |
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Pages (from-to) | 428-450 |

Number of pages | 23 |

Journal | Journal of Number Theory |

Volume | 184 |

DOIs | |

Publication status | Published - 2018 Mar 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Eichler–Shimura cohomology
- Hecke operator
- Weakly holomorphic modular form

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Hecke structures of weakly holomorphic modular forms and their algebraic properties.** / Choi, Dohoon; Lim, Subong.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 184, pp. 428-450. https://doi.org/10.1016/j.jnt.2017.08.029

}

TY - JOUR

T1 - Hecke structures of weakly holomorphic modular forms and their algebraic properties

AU - Choi, Dohoon

AU - Lim, Subong

PY - 2018/3/1

Y1 - 2018/3/1

N2 - Let Sk !(Γ1(N)) be the space of weakly holomorphic cusp forms of weight k on Γ1(N) with an even integer k>2 and Mk !(Γ1(N)) be the space of weakly holomorphic modular forms of weight k on Γ1(N). Further, let z denote a complex variable and D:=[Formula presented][Formula presented]. In this paper, we construct a basis of the space Sk !(Γ1(N))/Dk−1(M2−k !(Γ1(N))) consisting of Hecke eigenforms by using the Eichler–Shimura cohomology theory. Further, we study algebraicity of CM values of weakly holomorphic modular forms in the basis. This applies to an analogue of the Chowla–Selberg formula for a mock modular form whose shadow is the Ramanujan delta function.

AB - Let Sk !(Γ1(N)) be the space of weakly holomorphic cusp forms of weight k on Γ1(N) with an even integer k>2 and Mk !(Γ1(N)) be the space of weakly holomorphic modular forms of weight k on Γ1(N). Further, let z denote a complex variable and D:=[Formula presented][Formula presented]. In this paper, we construct a basis of the space Sk !(Γ1(N))/Dk−1(M2−k !(Γ1(N))) consisting of Hecke eigenforms by using the Eichler–Shimura cohomology theory. Further, we study algebraicity of CM values of weakly holomorphic modular forms in the basis. This applies to an analogue of the Chowla–Selberg formula for a mock modular form whose shadow is the Ramanujan delta function.

KW - Eichler–Shimura cohomology

KW - Hecke operator

KW - Weakly holomorphic modular form

UR - http://www.scopus.com/inward/record.url?scp=85030465149&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85030465149&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2017.08.029

DO - 10.1016/j.jnt.2017.08.029

M3 - Article

AN - SCOPUS:85030465149

VL - 184

SP - 428

EP - 450

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -