On the rank of abelian varieties over function fields

Abstract

Let $\cac$ be a smooth projective curve defined over a number field $k$, $A/k(\cac)$ an abelian variety and $(\tau ,B)$ the $k(\cac)/k$-trace of $A$. We estimate how the rank of $A(k(\cac))/τB(k)$ varies when we take a finite cover $π:\cac'\to\cac$ defined over $k$ geometrically abelian.

final version, to appear Manuscripta Mathematica