Hecke structures of weakly holomorphic modular forms and their algebraic properties

Dohoon Choi; Subong Lim

doi:10.1016/j.jnt.2017.08.029

Hecke structures of weakly holomorphic modular forms and their algebraic properties

Dohoon Choi, Subong Lim

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

Let S_k ^!(Γ₁(N)) be the space of weakly holomorphic cusp forms of weight k on Γ₁(N) with an even integer k>2 and M_k ^!(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 S_k ^!(Γ₁(N))/D^k−1(M_2−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.

Original language	English
Pages (from-to)	428-450
Number of pages	23
Journal	Journal of Number Theory
Volume	184
DOIs	https://doi.org/10.1016/j.jnt.2017.08.029
Publication status	Published - 2018 Mar
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2017 Elsevier Inc.

Keywords

Eichler–Shimura cohomology
Hecke operator
Weakly holomorphic modular form

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jnt.2017.08.029

Cite this

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title = "Hecke structures of weakly holomorphic modular forms and their algebraic properties",

abstract = "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.",

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author = "Dohoon Choi and Subong Lim",

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year = "2018",

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doi = "10.1016/j.jnt.2017.08.029",

language = "English",

volume = "184",

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

AU - Choi, Dohoon

AU - Lim, Subong

PY - 2018/3

Y1 - 2018/3

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

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U2 - 10.1016/j.jnt.2017.08.029

DO - 10.1016/j.jnt.2017.08.029

M3 - Article

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SN - 0022-314X

VL - 184

SP - 428

EP - 450

JO - Journal of Number Theory

JF - Journal of Number Theory

ER -

Hecke structures of weakly holomorphic modular forms and their algebraic properties

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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