### 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 |
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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 |

### Keywords

- Eichler–Shimura cohomology
- Jacobi form
- Real weight

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Monatshefte fur Mathematik*,

*182*(2), 271-288. https://doi.org/10.1007/s00605-016-0940-y