Group Structures of Elliptic Curves Over Finite Fields

Abstract

It is well known that if E is an elliptic curve over the finite field ⁠, then for some positive integers m,k. Let S(M,K) denote the set of pairs (m,k) with m≤M and k≤K for which there exists an elliptic curve over some prime finite field whose group of points is isomorphic to ⁠. Banks, Pappalardi, and Shparlinski recently conjectured that if ⁠, then a density zero proportion of the groups in question actually arises as the group of points on some elliptic curve over some prime finite field. On the other hand, if ⁠, they conjectured that a density 1 proportion of the groups in question arises as the group of points on some elliptic curve over some prime finite field. We prove that the first part of their conjecture holds in the full range ⁠, and we prove that the second part of their conjecture holds in the limited range K≥M4+ϵ. In the wider range K≥M2, we show that at least a positive density of the groups in question actually occurs.