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

TL;DR

This paper develops a method to explicitly realize elements of the Tate-Shafarevich group of an elliptic curve using Kolyvagin classes, providing concrete examples that violate the Hasse principle.

Contribution

It introduces a computational approach to convert Kolyvagin cohomology classes into geometric models of genus one curves representing Tate-Shafarevich group elements.

Findings

Explicit equations for genus one curves representing non-trivial Tate-Shafarevich group elements.

Counterexamples to the Hasse principle for p-torsion elements with p ≤ 11.

Method to compute Kolyvagin classes as elements of E(L) and their geometric models.

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.