Differential structure of Abelian functions

The space of Abelian functions of a principally polarized abelian variety (J,Θ) is studied as a module over the ring D of global holomorphic differential operators on J. We construct a D free resolution in case Θ is non-singular. As an application, in the case of dimensions 2 and 3, we construct a new linear basis of the space of abelian functions which are singular only on Θ in terms of logarithmic derivatives of the higher-dimensional σ-function.