 A quantitative primitive divisor result for points on elliptic curves
Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 609-634.

Let $E/K$ be an elliptic curve defined over a number field, and let $P\in E\left(K\right)$ be a point of infinite order. It is natural to ask how many integers $n\ge 1$ fail to occur as the order of $P$ modulo a prime of $K$. For $K=ℚ$, $E$ a quadratic twist of ${y}^{2}={x}^{3}-x$, and $P\in E\left(ℚ\right)$ as above, we show that there is at most one such $n\ge 3$.

Soient $E/K$ une courbe elliptique définie sur un corps de nombres et $P\in E\left(K\right)$ un point d’ordre infini. Il est naturel de se demander combien de nombres entiers $n\ge 1$ n’apparaissent pas comme ordre du point $P$ modulo un idéal premier de $K$. Dans le cas où $K=ℚ$, $E$ une tordue quadratique de ${y}^{2}={x}^{3}-x$ et $P\in E\left(ℚ\right)$ comme ci-dessus, nous démontrons qu’il existe au plus un tel $n\ge 3$.

Published online:
DOI: 10.5802/jtnb.691
Classification: 11G05,  11B39
Patrick Ingram 1

1 Department of Mathematics University of Toronto Toronto, Canada Current address: Department of Pure Mathematics University of Waterloo Waterloo, Canada
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Patrick Ingram. A quantitative primitive divisor result for points on elliptic curves. Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 609-634. doi : 10.5802/jtnb.691. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.691/

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