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. © Springer-Verlag Berlin Heidelberg 2008.
CITATION STYLE
Eisenträger, K., Jetchev, D., & Lauter, K. (2008). Computing the cassels pairing on Kolyvagin classes in the shafarevich-tate group. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 5209 LNCS, pp. 113–125). https://doi.org/10.1007/978-3-540-85538-5_8
Mendeley helps you to discover research relevant for your work.