On rational points of algebraic curves of genus one (Q1891756)

In this very interesting article the author gives an excursion on unpublished papers of J. Tate. One of these unpublished results is the following theorem: Let \(K\) be a local field, \(E/K\) a Tate curve [for the definitions see \textit{J. H. Silverman}, ``The arithmetic of elliptic curves'', Grad. Texts Math. 106 (1986; Zbl 0585.14026), Appendix C, \S 14], and \(C\) a torsor (homogeneous space) of \(E\). Then there exists a minimal extension \(K_C\) of \(K\) associated with \(C\), such that \(C\) has a rational point over \(K_C\); moreover \(K_C\) is cyclic over \(K\). The author extends the above theorem to the case where \(T\) is the twist of a Tate curve, and applies his results to the global case. As a consequence of this he obtains the following theorem: If the equation \(w^2=30 -44 z^2 +(512/15)z^4\) has a solution over \(\mathbb{Q} (\sqrt d)\), \(d\) a squarefree integer, then \(d=15 (3u-1)\) with an integer \(u\) not divisible by 8.