Explicit realization of elements of the Tate-Shafarevich group constructed from Kolyvagin classes

Abstract: We consider the Kolyvagin cohomology classes associated to an elliptic curve E defined over Q from a computational point of view. We explain how to go from a model of a class as an element of (E(L)/pE(L))Gal(L/Q), where p is prime and L is a dihedral extension of Q of degree 2p, to a geometric model as a genus one curve embedded in Pp−1. We adapt the existing methods to compute Heegner points to our situation, and explicitly compute them as elements of E(L). Finally, we compute explicit equations for several genus one curves that represent non-trivial elements of the p-torsion part of the Tate-Shafarevich group of E, for p≤11, and hence are counterexamples to the Hasse principle.

number theory

Audience: researchers in the topic

CRM-CICMA Québec Vermont Seminar Series

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