The Eichler–Shimura cohomology theorem for Jacobi forms

Abstract

Let $\Gamma$ be a subgroup of finite index in $\mathrm {SL}(2,\mathbb {Z})$. Eichler defined the first cohomology group of $\Gamma$ 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 $\Gamma$ 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 $\Gamma ^{(1,j)}$ 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 $\frac{1}{2}\mathbb {Z}-\mathbb {Z}$, then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.