The Eichler-Shimura cohomology theorem for Jacobi forms

Let Gamma be a subgroup of finite index in SL(2, Z). Eichler defined the first cohomology group of 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 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 and prove that these are isomorphic to the spaces of (skew-holomorphic) Jacobi cusp forms of real weights. We also show that if and the weights of Jacobi cusp forms are in , then these isomorphisms can be written in terms of special values of partial L-functions of Jacobi cusp forms.