  We show that on any abelian scheme over a complex quasi-projective smooth
variety, there is a Green current for the zero-section, which is axiomatically
determined up to $\partial$ and $\bar\partial$-exact differential forms. This
current generalizes the Siegel functions defined on elliptic curves. We prove
generalizations of classical properties of Siegel functions, like distribution
relations, limit formulae and reciprocity laws.
