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

  • H. W. LenstraJr.
  • J. Pila
Part of the Mathematics and Its Applications book series (MAIA, volume 325)


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].


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


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

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • H. W. LenstraJr.
    • 1
  • J. Pila
    • 2
  1. 1.Department of Mathematics # 3840University of California at BerkeleyBerkeleyUSA
  2. 2.Department of MathematicsThe University of Melbourne ParkvilleVictoriaAustralia

Personalised recommendations