Galois representations in the Tate-Shafarevich group of an elliptic curve (Q1919330)

This letter is a footnote to a paper of Kramer in which he proved the existence of a semistable elliptic curve \(E\) over \(\mathbb{Q}\) with arbitrarily large \(\text{{СХ}}(E/\mathbb{Q})_2\), where \(\text{Ш}(E/\mathbb{Q})\) denotes the Tate-Shafarevich group of \(E/\mathbb{Q}\) and \(\text{Ш}(E/\mathbb{Q})_2\) the subgroup of elements with order dividing \(2\). The author remarks that Kramer's construction also gives Tate-Shafarevich groups which are arbitrarily large as Galois modules. More precisely he proved: let \(K\) be a finite Galois extension of \(\mathbb{Q}\), \(n\) a positive integer. Then there exists a semistable elliptic curve \(E\) over \(\mathbb{Q}\) such that the natural representation of \(\text{Gal}(K/\mathbb{Q})\) on \(\text{Ш}(E/\mathbb{Q})_2\) contains a subrepresentation isomorphic to the direct sum of \(n\) copies of the regular representation of \(\text{Gal}(K/\mathbb{Q})\) over \(\mathbb{F}_2\).