Selmer groups for elliptic curves in ${\mathbb{Z}}_l^d$-extensions of function fields of characteristic $p$

Bandini, Andrea; Longhi, Ignazio

Annales de l'Institut Fourier, Tome 59 (2009) no. 6, pp. 2301-2327.

Soit $F$ un corps de fonctions de caractéristique $p>0$, $ℱ/F$ une ${\mathbb{Z}}_l^d$-extension (pour un nombre premier $l\ne p$) et $E/F$ une courbe elliptique non-isotrivale. Nous étudions le comportement des $r$-parties des groupes de Selmer pour les sous-extensions de $ℱ$ par des variantes du Théorème de contrôle de Mazur. Conséquemment, nous démontrons que la limite des groupes de Selmer est un module finiment co-engendré (parfois de cotorsion) sur l’algèbre d’Iwasawa de $ℱ/F$.

Let $F$ be a function field of characteristic $p>0$, $ℱ/F$ a ${\mathbb{Z}}_l^d$-extension (for some prime $l\ne p$) and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of the $r$-parts of the Selmer groups ($r$ any prime) in the subextensions of $ℱ$ via appropriate versions of Mazur’s Control Theorem. As a consequence we prove that the limit of the Selmer groups is a cofinitely generated (in some cases cotorsion) module over the Iwasawa algebra of $ℱ/F$.

Classification : 11G05, 11R23
Keywords: Selmer groups, elliptic curves, function fields, Iwasawa theory
Mot clés : groupes de Selmer, courbes elliptiques, corps de fonctions, théorie d’Iwasawa

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