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We study the collection of group structures that can be realized as a groupof rational points on an elliptic curve over a finite field (such groups arewell known to be of rank at most two). We also study various subsets of thiscollection which correspond to curves over prime fields or to curves with aprescribed torsion. Some of our results are rigorous and are based on recentadvances in analytic number theory, some are conditional under certain widelybelieved conjectures, and others are purely heuristic in nature.

On group structures realized by elliptic curves over arbitrary finite fields