The Eichler–Shimura cohomology theorem for Jacobi forms

Dohoon Choi, Subong Lim

Research output: Contribution to journalArticle

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 languageEnglish
Pages (from-to)271-288
Number of pages18
JournalMonatshefte fur Mathematik
Volume182
Issue number2
DOIs
Publication statusPublished - 2017 Feb 1
Externally publishedYes

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

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 journalArticle

@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 -