Elliptic curves with isomorphic groups of points over finite field extensions

Consider a pair of ordinary elliptic curves E and E' defined over the same finite field F-q. Suppose they have the same number of F-q-rational points, i.e. vertical bar E(F-q)vertical bar = vertical bar E'(F-q)vertical bar. In this paper we characterise for which finite field extensions F(q)k k >= 1 (if any) the corresponding groups of F(q)k-rational points are isomorphic, i.e. E(F(q)k) congruent to (F(q)k). (C) 2017 The Author(s). Published by Elsevier Inc.