Sur le rang des variétés abéliennes sur un corps de fonctions

Pacheco, Amílcar

Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 2 (2014), pp. 31-46.

Ce texte est un survey concernant la question du rang d’une variété abélienne $A$ sur un corps de fonctions $K$ en une variable sur un corps de base $k$. Il s’agit non seulement de discuter une borne supérieure pour ce rang, mais aussi d’étudier le comportement de cette borne si on prend une extension abélienne finie $L$ de $K$. On se demande aussi : que se passe-t-il quand on enlève cette dernière hypothèse ? Dans un cas particulier, on discute de la validité d’un analogue du théorème de Lang-Néron. Pour cela, il nous faudra des hypothèses additionnelles. À la fin du texte, nous discutons des situations où ces hypothèses sont vérifiées.

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.

Classification : 11G10
Mots clés : Abelian varieties, Tate’s conjecture, Selmer groups.

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