Elliptic Curves with Isomorphic Groups of Points over Finite Field Extensions

Abstract

Consider a pair of ordinary elliptic curves $E$ and $E^{\prime }$ defined over the same finite field ${\mathbb{F}}_q$. Suppose they have the same number of ${\mathbb{F}}_q$-rational points, i.e. $|E({\mathbb{F}}_q)|=|E^{\prime }({\mathbb{F}}_q)|$. In this paper we characterise for which finite field extensions ${\mathbb{F}}_{q^k}$, $k\ge 1$ (if any) the corresponding groups of ${\mathbb{F}}_{q^k}$-rational points are isomorphic, i.e. $E({\mathbb{F}}_{q^k})\cong E^{\prime }({\mathbb{F}}_{q^k})$.

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