Small rational points on elliptic curves over number fields.

Petsche, Clayton

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Explicit local heights.

Everest, Graham (1999)

The New York Journal of Mathematics [electronic only]

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A quantitative primitive divisor result for points on elliptic curves

Patrick Ingram (2009)

Journal de Théorie des Nombres de Bordeaux

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Let $E / K$ be an elliptic curve defined over a number field, and let $P \in E (K)$ be a point of infinite order. It is natural to ask how many integers $n \geq 1$ 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 $P \in E (ℚ)$ as above, we show that there is at most one such $n \geq 3$ .