The modular differential equation for skew-holomorphic Jacobi forms

Shotaro Kimura

doi:10.1007/s11139-021-00547-z

The modular differential equation for skew-holomorphic Jacobi forms

Shotaro Kimura^*

^*Corresponding author for this work

School of Fundamental Science and Engineering

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

Abstract

In this paper, we study the modular differential equation for skew-holomorphic Jacobi forms, which are non-holomorphic modular forms. This differential equation is a second-order linear ordinary differential equation and similar to the case of elliptic modular forms, whose studies were initiated by Kaneko and Zagier. On the other hand, this equation differs from the case of holomorphic Jacobi forms introduced by Kiyuna in the types of differential equations and dependences on the index. We show the same properties as previous studies: the solution space of the differential equation is modular invariant and the differential equation is unique.

Original language	English
Pages (from-to)	1137-1146
Number of pages	10
Journal	Ramanujan Journal
Volume	59
Issue number	4
DOIs	https://doi.org/10.1007/s11139-021-00547-z
Publication status	Published - 2022 Dec

Keywords

Differential operator
Kaneko–Zagier differential equation
Modular differential equation
Skew-holomorphic Jacobi forms

ASJC Scopus subject areas

Algebra and Number Theory

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10.1007/s11139-021-00547-z

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title = "The modular differential equation for skew-holomorphic Jacobi forms",

abstract = "In this paper, we study the modular differential equation for skew-holomorphic Jacobi forms, which are non-holomorphic modular forms. This differential equation is a second-order linear ordinary differential equation and similar to the case of elliptic modular forms, whose studies were initiated by Kaneko and Zagier. On the other hand, this equation differs from the case of holomorphic Jacobi forms introduced by Kiyuna in the types of differential equations and dependences on the index. We show the same properties as previous studies: the solution space of the differential equation is modular invariant and the differential equation is unique.",

keywords = "Differential operator, Kaneko–Zagier differential equation, Modular differential equation, Skew-holomorphic Jacobi forms",

author = "Shotaro Kimura",

note = "Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.",

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AU - Kimura, Shotaro

PY - 2022/12

Y1 - 2022/12

N2 - In this paper, we study the modular differential equation for skew-holomorphic Jacobi forms, which are non-holomorphic modular forms. This differential equation is a second-order linear ordinary differential equation and similar to the case of elliptic modular forms, whose studies were initiated by Kaneko and Zagier. On the other hand, this equation differs from the case of holomorphic Jacobi forms introduced by Kiyuna in the types of differential equations and dependences on the index. We show the same properties as previous studies: the solution space of the differential equation is modular invariant and the differential equation is unique.

AB - In this paper, we study the modular differential equation for skew-holomorphic Jacobi forms, which are non-holomorphic modular forms. This differential equation is a second-order linear ordinary differential equation and similar to the case of elliptic modular forms, whose studies were initiated by Kaneko and Zagier. On the other hand, this equation differs from the case of holomorphic Jacobi forms introduced by Kiyuna in the types of differential equations and dependences on the index. We show the same properties as previous studies: the solution space of the differential equation is modular invariant and the differential equation is unique.

KW - Differential operator

KW - Kaneko–Zagier differential equation

KW - Modular differential equation

KW - Skew-holomorphic Jacobi forms

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The modular differential equation for skew-holomorphic Jacobi forms

Abstract

Keywords

ASJC Scopus subject areas

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