Sur le rang des variétés abéliennes sur un corps de fonctions

— This text is a survey on the question of the rank of an abelian variety A defined over a one variable function field K over a base field k. We discuss not only an upper bound for this rank, but also study the behavior of this bound after taking a finite and abelian extension L of K. We ask ourselves : what happens if this hypothesis is suppressed? In a particular case, we discuss the validity of the Lang-Néron theorem. This validity depends on additional hypotheses. At the end of the text, we discuss situations in which these hypotheses are satisfied.