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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.