  The space of abelian functions of a principally polarized abelian variety J
is studied as a module over the ring D of global holomorphic differential
operators on J. We construct a D-free resolution in case the theta divisor is
non-singular. As an application, in the case of dimension 2 and 3, we construct
a new linear basis of the space of abelian functions in terms of logarithmic
derivatives of the higher dimensional sigma function.
