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John Boxall; David Grant
Some remarks on almost rational torsion points
Journal de théorie des nombres de Bordeaux, 18 no. 1 (2006), p. 13-28, doi: 10.5802/jtnb.531
Article PDF | Reviews MR 2245873 | Zbl 05070445
Keywords: Elliptic curves, torsion, almost rational.

Résumé - Abstract

For a commutative algebraic group $G$ over a perfect field $k$, Ribet defined the set of almost rational torsion points $G^{\operatorname{ar}}_{\operatorname{tors},k}$ 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 $g$, $T^{\operatorname{ar}}_{\operatorname{tors},k}\subseteq T[U_{d,g}]$. 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 $k$, $G^{\operatorname{ar}}_{\operatorname{tors},k}$ 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 $A^{\operatorname{ar}}_{\operatorname{tors},k}$ is infinite.

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