### 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 | |

Publication status | Published - 2017 Feb 1 |

Externally published | Yes |

### Fingerprint

### Keywords

- Eichler–Shimura cohomology
- Jacobi form
- Real weight

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*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.

Research output: Contribution to journal › Article

*Monatshefte fur Mathematik*, vol. 182, no. 2, pp. 271-288. https://doi.org/10.1007/s00605-016-0940-y

}

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 -