Some remarks on almost rational torsion points

John Boxall[1]; David Grant[2]

  • [1] Laboratoire de Mathématiques Nicolas Oresme, CNRS – UMR 6139 Université de Caen boulevard Maréchal Juin BP 5186, 14032 Caen cedex, France
  • [2] Department of Mathematics University of Colorado at Boulder Boulder, Colorado 80309-0395 USA

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 1, page 13-28
  • ISSN: 1246-7405

Abstract

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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 ⊆ 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 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.

How to cite

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Boxall, John, and Grant, David. "Some remarks on almost rational torsion points." Journal de Théorie des Nombres de Bordeaux 18.1 (2006): 13-28. <http://eudml.org/doc/249655>.

@article{Boxall2006,
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.},
affiliation = {Laboratoire de Mathématiques Nicolas Oresme, CNRS – UMR 6139 Université de Caen boulevard Maréchal Juin BP 5186, 14032 Caen cedex, France; Department of Mathematics University of Colorado at Boulder Boulder, Colorado 80309-0395 USA},
author = {Boxall, John, Grant, David},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Elliptic curves; torsion; almost rational; elliptic curves},
language = {eng},
number = {1},
pages = {13-28},
publisher = {Université Bordeaux 1},
title = {Some remarks on almost rational torsion points},
url = {http://eudml.org/doc/249655},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Boxall, John
AU - Grant, David
TI - Some remarks on almost rational torsion points
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 1
SP - 13
EP - 28
AB - 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.
LA - eng
KW - Elliptic curves; torsion; almost rational; elliptic curves
UR - http://eudml.org/doc/249655
ER -

References

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