Does the Set of Points of an Elliptic Curve Determine the Group?
Let E be an elliptic curve over a field k, given in Weierstrass form. As is well known, the set E(k)of points of E over k forms an abelian group in a natural way, the point at infinity being the zero element. It is often heard that the group structure on E(k) is “determined” by the fact that three points of E(k) add up to zero if they lie on a line. In this paper we investigate whether this statement is correct if taken literally. We find that it is not. In fact, we exhibit a field k and two elliptic curves E, E′ over k for which E(k) and E′ (k) are equal as subsets of the set of points of the projective plane over k, but have different group structures. Our main result states that this is a rare phenomenon: it occurs if and only if k has characteristic 2 and E(k) has order 5. We also encounter an elliptic curve E for which E(k) has a Z[i]-module structure for many fields k, even though E does not have complex multiplication by Z[i].
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