An Eichler-Zagier map for Jacobi cusp forms on H×C(g,1)

We define an Eichler-Zagier map ZM on the space of Jacobi cusp forms of matrix index M and discuss its mapping properties. If the order of M is congruent to 1 mod 8 then we show that ZM maps certain Jacobi Poincaré series to half-integral weight Poincaré series and we construct a subspace of Jacobi cusp forms on which the map ZM is injective. By using the above results we relate the Fourier coefficients of certain Jacobi cusp forms to central values of the twisted modular L-functions and we improve certain non-vanishing results for Jacobi Poincaré series established by S. Das.