Computing the Cassels Pairing on Kolyvagin Classes in the Shafarevich-Tate Group

  • Kirsten Eisenträger
  • Dimitar Jetchev
  • Kristin Lauter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5209)


Kolyvagin has shown how to study the Shafarevich-Tate group of elliptic curves over imaginary quadratic fields via Kolyvagin classes constructed from Heegner points. One way to produce explicit non-trivial elements of the Shafarevich-Tate group is by proving that a locally trivial Kolyvagin class is globally non-trivial, which is difficult in practice. We provide a method for testing whether an explicit element of the Shafarevich-Tate group represented by a Kolyvagin class is globally non-trivial by determining whether the Cassels pairing between the class and another locally trivial Kolyvagin class is non-zero. Our algorithm explicitly computes Heegner points over ring class fields to produce the Kolyvagin classes and uses the efficiently computable cryptographic Tate pairing.


Elliptic Curve Elliptic Curf Cohomology Class Galois Representation Hyperelliptic Curve 
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© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Kirsten Eisenträger
    • 1
  • Dimitar Jetchev
    • 2
  • Kristin Lauter
    • 3
  1. 1.Department of MathematicsThe Pennsylvania State University
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeley
  3. 3.Microsoft ResearchRedmond

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