The Cassels-Tate pairing and the Platonic solids.
The paper presents some descent calculations for families of elliptic curves whose \(m\)-torsion splits as \(\mu_m\times\mathbb{Z}/m\mathbb{Z}\) for \(m=3,4\) or 5. These computations provide both theoretical results and a large amount of numerical data and examples. In particular, by considering these curves, the author shows that the Tate-Shafarevich group of an elliptic curve over a number field may contain arbitrarily many elements of order \(m\). A significant part of the article is devoted to the Cassels-Tate pairing, extending the methods by \textit{W. McCallum} [Invent. Math. 93, No. 3, 637--666 (1988; Zbl 0661.14033)] and \textit{C. DeLorme Beaver} [J. Number Theory 82, No. 1, 25--46 (2000; Zbl 0983.11029)]. In the particular case \(K=\mathbb{Q}\), the Cassels-Tate pairing is used to describe the Selmer group of the considered elliptic curves as the kernel of a skew-symmetric matrix, and estimates for the rank of the curves are derived. This explicit description allows the author to perform a number of computations and exhibit examples of elliptic curves with interesting Tate-Shafarevich groups and compile tables of numerical data.
scientific article; zbMATH DE number 1902870
scientific article