Sur le rang des variétés abéliennes sur un corps de fonctions
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.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/pmb.7
Classification : 11G10
Mots clés : Abelian varieties, Tate’s conjecture, Selmer groups.
@article{PMB_2014___2_31_0,
     author = {Pacheco, Am{\'\i}lcar},
     title = {Sur le rang des vari\'et\'es ab\'eliennes sur un corps de fonctions},
     journal = {Publications Math\'ematiques de Besan\c{c}on - Alg\`ebre et Th\'eorie des Nombres},
     pages = {31--46},
     publisher = {Presses universitaires de Franche-Comt\'e},
     number = {2},
     year = {2014},
     doi = {10.5802/pmb.7},
     zbl = {1366.11082},
     language = {fr},
     url = {http://archive.numdam.org/articles/10.5802/pmb.7/}
}
Pacheco, Amílcar. Sur le rang des variétés abéliennes sur un corps de fonctions. Publications Mathématiques de Besançon - Algèbre et Théorie des Nombres, no. 2 (2014), pp. 31-46. doi : 10.5802/pmb.7. http://archive.numdam.org/articles/10.5802/pmb.7/

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