Visualizing elements of order three in the Shafarevich-Tate group
Let \(E\) be an elliptic curve defined over \(\mathbb Q\) with a rational point \(T\) of order \(2\). Then there exists an isogeny \(\phi: E \rightarrow E'\) of degree \(2\) with \(T\) in its kernel. The Selmer group attached to \(\phi\) consists of curves of genus \(1\) (called torsors or homogeneous spaces) that have \(\mathbb Q_p\)-rational points over every completion \(\mathbb Q_p\). These torsors represent non-trivial elements of the \(2\)-part of the Tate-Shafarevich group of \(E\) if they do not contain a rational point. The first examples of such torsors were discovered by \textit{H. Reichardt} [J. Reine Angew. Math. 184, 12--18 (1942; Zbl 0026.29701)] and \textit{C.-E. Lind} [Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins, Diss. Uppsala (1940; Zbl 0025.24802)]. Reichardt considered torsors such as \(z^2 = x^4 + 17y^4\); his results show that, over \(k = \mathbb Q(\sqrt{-2})\), every torsor in the \(2\)-Selmer group has a \(k\)-rational point, so the \(2\)-part of the Tate-Shafarevich group of \(E/k\) is trivial; then he used Galois theory to show that the torsor in question does not have a \(\mathbb Q\)-rational point. In modern language, this implies that the \(2\)-part of the Tate-Shafarevich group of \(E/\mathbb Q\) becomes trivial over \(k\): in analogy with class groups in number field we speak of capitulation. Now assume that \(E/\mathbb Q\) is an arbitrary elliptic curve embedded in an abelian variety \(J\) defined over \(\mathbb Q\), so there is an exact sequence \(0 \rightarrow E \rightarrow J \rightarrow B \rightarrow 0\) of abelian varieties. Let \(G\) denote the absolute Galois group of \(\mathbb Q\). An element \(\sigma \in H^1(G,E)\) is said to be visible in \(J\) if it is in the kernel of the induced map \(H^1(G,E) \rightarrow H^1(G,J)\). Since \(E(k)\) can be viewed as an abelian variety of dimension \(2\) defined over \(\mathbb Q\), the capitulation of Tate-Shafarevich groups in extensions of number fields is a special case of visibility. Numerical experiments [\textit{J. E. Cremona} and \textit{B. Mazur}, Visualizing elements in the Shafarevich-Tate group; Exp. Math. 9, 13--28 (2001; Zbl 0972.11049)] have shown that a surprisingly large number of elliptic curves with small conductor have Tate-Shafarevich groups that become visible in \(2\)-dimensional abelian varieties of a very special type. In this paper, the author explains a small part of this observation by proving that if \(E\) is an elliptic curve defined over a number field \(K\), and if \(\sigma\) is an element of order \(3\) in the Tate-Shafarevich group of \(E/K\), then there exists an abelian surface \(J/K\) containing \(E\) such that \(\sigma\) is visible in \(J\).
scientific article; zbMATH DE number 1531032