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/{mathbb 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))^{mathrm {Gal}(L/{mathbb Q})}, where p is prime and L is a dihedral extension of {mathbb Q} of degree 2p, to a geometric model as a genus one curve embedded in mathbb { P} ^{p-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 , for p le 11, and hence are counterexamples to the Hasse principle.