  In this paper we establish the relationships between theta functions of
arbitrary order and their derivatives. We generalize our previous work
math.AG/0310085 and prove that for any n>1 the map sending an abelian variety
to the set of Gauss images of its points of order 2n is an embedding into an
appropriate Grassmannian (note that for n=1, i.e. points of order 2, we only
get generic injectivity). We further discuss the generalizations of Jacobi's
derivative formula for any dimension and any order.
  The series of embedding of the level covers of moduli spaces of principally
polarized abelian varieties into Grassmannians that we define can be considered
as the odd counterpart of the classical embeddings into projective spaces, by
taking the values of even theta constants, considered by Igusa and Mumford.
