Some remarks on almost rational torsion points

Boxall, John; Grant, David

Journal de théorie des nombres de Bordeaux, Tome 18 (2006) no. 1, pp. 13-28.

Lorsque $G$ désigne un groupe algébrique sur un corps parfait $k$, Ribet a défini l’ensemble des points de torsion presque rationnels $G_{tors,k}^{ar}$ de $G$ sur $k$. Si $d$, $g$ désignent des entiers positifs, nous montrons qu’il existe un entier $U_{d,g}$ tel que, pour tout tore $T$ de dimension au plus $d$ sur un corps de nombres de degré au plus $g$, on ait $T_{tors,k}^{ar}\subseteq T\left[U_{d,g}\right]$. Nous montrons le résultat analogue pour les variétés abéliennes à multiplication complexe puis, sous une hypothèse supplémentaire, pour les courbes elliptiques sans multiplication complexe. Enfin, nous montrons que, à l’exception d’un ensemble fini explicite de variétés semi-abéliennes $G$ sur un corps fini, $G_{tors,k}^{ar}$ est infini et nous utilisons ce résultat pour montrer que pour toute variété abélienne sur un corps $p$-adique $k$, il existe une extension finie de $k$ sur laquelle $A_{tors,k}^{ar}$ est infini.

For a commutative algebraic group $G$ over a perfect field $k$, Ribet defined the set of almost rational torsion points $G_{tors,k}^{ar}$ of $G$ over $k$. For positive integers $d$, $g,$ we show there is an integer $U_{d,g}$ such that for all tori $T$ of dimension at most $d$ over number fields of degree at most $g$, $T_{tors,k}^{ar}\subseteq T\left[U_{d,g}\right]$. We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit finite set of semi-abelian varieties $G$ over a finite field $k$, $G_{tors,k}^{ar}$ is infinite, and use this to show for any abelian variety $A$ over a $p$-adic field $k$, there is a finite extension of $k$ over which $A_{tors,k}^{ar}$ is infinite.

Mots clés : Elliptic curves, torsion, almost rational.

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