Bounds for curves in abelian varieties

A uniform bound of intersection multiplicities of curves and divisors on abelian varieties is proved by algebraic geometric methods. It extends and improves a result obtained by A. Buium with a different method based on Kolchin's differential algebra. The problem is modeled after the "abc-Conjecture'' of Masser-Oesterle for abelian varieties over the function field of a curve. As an application a finiteness theorem will be proved for maps from a curve into an abelian variety omitting an ample divisor.