Computational Algebra and Number Theory pp 111-118 | Cite as

# Does the Set of Points of an Elliptic Curve Determine the Group?

## Abstract

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*].

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]R. Hartshorne,
*Algebraic geometry*, New York: Springer—Verlag, 1977.zbMATHGoogle Scholar - [2]D. S. Kubert,
*Universal bounds on the torsion of elliptic curves*, Proc. London Math. Soc.**33**(1976), 193–237.Google Scholar - [3]S. Lang,
*Fundamentals of diophantine geometry*, New York: Springer—Verlag, 1983.zbMATHGoogle Scholar - [4]K. Oguiso and T. Shioda,
*The Mordell—Weil lattice of a rational elliptic surface*, Comment. Math. Univ. St. Paul.**40**(1991), 83–99.Google Scholar - [5]T. Shioda,
*On the Mordell—Weil lattices*, Comment. Math. Univ. St. Paul.**39**(1990), 211–240.Google Scholar - [6]J. H. Silverman,
*The arithmetic of elliptic curves*, New York: Springer—Verlag, 1986.zbMATHCrossRefGoogle Scholar