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 $\mathbb{Q}$ 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 the p-torsion part of the Tate-Shafarevich group of $E$, for $p \leq 11$, and hence are counterexamples to the Hasse principle.