A parametrized set of explicit elements of $\Sha(E/\Q)[3]$

TL;DR

This paper constructs explicit examples of elliptic curves and plane cubics over  with elements of order 3 in their Tate-Shafarevich group, illustrating failures of the local-global principle.

Contribution

It provides explicit equations for homogeneous spaces related to a degree 3 rational isogeny and constructs examples of plane cubics with local points but no global points.

Findings

Explicit equations for homogeneous spaces of degree 3 isogenies.

Construction of elliptic curves with order 3 elements in .

Examples of plane cubics with local points but no global points.

Abstract

In this paper, we give explicit equations for homogeneous spaces corresponding to a rational isogeny of degree $3$. An explicit set of elliptic curves with elements of order $3$ in their Tate-Shafarevich group is constructed. Combining this gives explicit examples of plane cubics over $\Q$ that have a point everywhere locally, but not globally.