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

Keywords

  • Eichler–Shimura cohomology
  • Jacobi form
  • Real weight

ASJC Scopus subject areas

  • Mathematics(all)

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