The Eichler–Shimura cohomology theorem for Jacobi forms

Dohoon Choi; Subong Lim

doi:10.1007/s00605-016-0940-y

The Eichler–Shimura cohomology theorem for Jacobi forms

Dohoon Choi, Subong Lim

Research output: Contribution to journal › Article

1 Citation (Scopus)

Abstract

Let Γ be a subgroup of finite index in SL (2 , Z). Eichler defined the first cohomology group of Γ with coefficients in a certain module of polynomials. Eichler and Shimura established that this group is isomorphic to the direct sum of two spaces of cusp forms on Γ with the same integral weight. These results were generalized by Knopp to cusp forms of real weights. In this paper, we define the first parabolic cohomology groups of Jacobi groups Γ ⁽ ¹ ^, ^j ⁾ and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if j= 1 and the weights of Jacobi cusp forms are in 12Z-Z, then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.

Original language	English
Pages (from-to)	271-288
Number of pages	18
Journal	Monatshefte fur Mathematik
Volume	182
Issue number	2
DOIs	https://doi.org/10.1007/s00605-016-0940-y
Publication status	Published - 2017 Feb 1
Externally published	Yes

Fingerprint

Jacobi Forms

Cusp Form

Cohomology

Theorem

Cohomology of Groups

Isomorphic

L-function

Direct Sum

Jacobi

Skew

Isomorphism

Subgroup

Partial

Module

Polynomial

Coefficient

Keywords

Eichler–Shimura cohomology
Jacobi form
Real weight

ASJC Scopus subject areas

Mathematics(all)

Cite this

Choi, D., & Lim, S. (2017). The Eichler–Shimura cohomology theorem for Jacobi forms. Monatshefte fur Mathematik, 182(2), 271-288. https://doi.org/10.1007/s00605-016-0940-y

The Eichler–Shimura cohomology theorem for Jacobi forms. / Choi, Dohoon; Lim, Subong.

In: Monatshefte fur Mathematik, Vol. 182, No. 2, 01.02.2017, p. 271-288.

Research output: Contribution to journal › Article

Choi, D & Lim, S 2017, 'The Eichler–Shimura cohomology theorem for Jacobi forms', Monatshefte fur Mathematik, vol. 182, no. 2, pp. 271-288. https://doi.org/10.1007/s00605-016-0940-y

Choi D, Lim S. The Eichler–Shimura cohomology theorem for Jacobi forms. Monatshefte fur Mathematik. 2017 Feb 1;182(2):271-288. https://doi.org/10.1007/s00605-016-0940-y

Choi, Dohoon ; Lim, Subong. / The Eichler–Shimura cohomology theorem for Jacobi forms. In: Monatshefte fur Mathematik. 2017 ; Vol. 182, No. 2. pp. 271-288.

@article{b592e3c26c934ee4b9e3ce324ad29868,

title = "The Eichler–Shimura cohomology theorem for Jacobi forms",

abstract = "Let Γ be a subgroup of finite index in SL (2 , Z). Eichler defined the first cohomology group of Γ with coefficients in a certain module of polynomials. Eichler and Shimura established that this group is isomorphic to the direct sum of two spaces of cusp forms on Γ with the same integral weight. These results were generalized by Knopp to cusp forms of real weights. In this paper, we define the first parabolic cohomology groups of Jacobi groups Γ ( 1 , j ) and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if j= 1 and the weights of Jacobi cusp forms are in 12Z-Z, then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.",

keywords = "Eichler–Shimura cohomology, Jacobi form, Real weight",

author = "Dohoon Choi and Subong Lim",

year = "2017",

month = "2",

day = "1",

doi = "10.1007/s00605-016-0940-y",

language = "English",

volume = "182",

pages = "271--288",

journal = "Monatshefte fur Mathematik",

issn = "0026-9255",

publisher = "Springer Wien",

number = "2",

}

TY - JOUR

T1 - The Eichler–Shimura cohomology theorem for Jacobi forms

AU - Choi, Dohoon

AU - Lim, Subong

PY - 2017/2/1

Y1 - 2017/2/1

N2 - Let Γ be a subgroup of finite index in SL (2 , Z). Eichler defined the first cohomology group of Γ with coefficients in a certain module of polynomials. Eichler and Shimura established that this group is isomorphic to the direct sum of two spaces of cusp forms on Γ with the same integral weight. These results were generalized by Knopp to cusp forms of real weights. In this paper, we define the first parabolic cohomology groups of Jacobi groups Γ ( 1 , j ) and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if j= 1 and the weights of Jacobi cusp forms are in 12Z-Z, then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.

AB - Let Γ be a subgroup of finite index in SL (2 , Z). Eichler defined the first cohomology group of Γ with coefficients in a certain module of polynomials. Eichler and Shimura established that this group is isomorphic to the direct sum of two spaces of cusp forms on Γ with the same integral weight. These results were generalized by Knopp to cusp forms of real weights. In this paper, we define the first parabolic cohomology groups of Jacobi groups Γ ( 1 , j ) and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if j= 1 and the weights of Jacobi cusp forms are in 12Z-Z, then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.

KW - Eichler–Shimura cohomology

KW - Jacobi form

KW - Real weight

UR - http://www.scopus.com/inward/record.url?scp=84975140868&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975140868&partnerID=8YFLogxK

U2 - 10.1007/s00605-016-0940-y

DO - 10.1007/s00605-016-0940-y

M3 - Article

AN - SCOPUS:84975140868

VL - 182

SP - 271

EP - 288

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 2

ER -

Access to Document

10.1007/s00605-016-0940-y