On a canonical class of Green currents for the unit sections of abelian schemes

TL;DR

This paper constructs a canonical Green current for the zero-section of abelian schemes over complex varieties, generalizing classical elliptic curve functions and establishing key properties like distribution and reciprocity laws.

Contribution

It introduces a canonical Green current for abelian schemes that generalizes Siegel functions and proves their fundamental properties.

Findings

Existence of a Green current for the zero-section in abelian schemes.

Generalization of classical properties of Siegel functions.

Establishment of distribution relations, limit formulas, and reciprocity laws.

Abstract

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.