Elliptic Curves with Isomorphic Groups of Points over Finite Field Extensions
Clemens Heuberger, Michela Mazzoli

TL;DR
This paper characterizes the conditions under which two ordinary elliptic curves over a finite field have isomorphic groups of points over various finite field extensions, given they have the same number of points over the base field.
Contribution
It provides a complete characterization of when elliptic curves with equal base field point counts have isomorphic groups over extension fields.
Findings
Identifies specific extension degrees where groups are isomorphic
Provides criteria based on elliptic curve properties
Enhances understanding of elliptic curve group structures
Abstract
Consider a pair of ordinary elliptic curves and defined over the same finite field . Suppose they have the same number of -rational points, i.e. . In this paper we characterise for which finite field extensions , (if any) the corresponding groups of -rational points are isomorphic, i.e. .
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