On the Tate-Shafarevich group of certain elliptic curves
Let \(K\subset \mathbb C\) be an imaginary quadratic field with prime discriminant \(-p< -3\), with ring of integers \(\mathfrak O\) and class group \(\text{Cl}_K\) of order \(h\). Let \(F\) be the field generated by the \(j\)-invariant of \(\mathfrak O\) of degree \(h\) such that \(H=FK\) is the Hilbert class field of \(K\), and let \(A/F\) be a \(\mathbb Q\)-curve with \(j\)-invariant \(j(\mathfrak O).\) Let \(B= \text{Res}_{F/\mathbb Q}A,\) \(\;T^{+}= \text{End}_{\mathbb Q}(B)\otimes \mathbb Q\). The objective of this paper is to compute the Tate-Shafarevich group of \(B/\mathbb Q\) in certain situations. For \(p\equiv 3\pmod 8\) let \(A=A(p)^{-3},\) where \(A(p)\) is the canonical curve of the discriminant ideal \((-p^3)\). The authors construct a non-zero algebraic integer which gives rise to an integral ideal \(\left({\mathfrak f}\right) \) of \(\text{End}_{\mathbb Q}(B)\). Numerous tables for \(p\) and \(\mathfrak f\) are provided in the final section. A conjecture made here is that the order ideal of the Tate-Shafarevich group is \({\mathfrak f}^2\). This conjecture relates to the Buhler-Gross conjecture [\textit{J. P. Buhler} and \textit{B. H. Gross}, Invent. Math. 79, 11--29 (1985; Zbl 0584.14027)]. The authors show that \(A(F)\) is finite. Furthermore, if the Birch-Swinnerton-Dyer conjecture holds for \(A/F\) (or for \(B/\mathbb Q\)) then the order of its Tate-Shafarevich group is \(N_{T^{+}/\mathbb Q} (\mathfrak f)^{2}.\) The proof of this result uses the \(L\)-series \(L(s,A/F),\) which is presented as a product of Hecke \(L\)-series. Finally, several questions involving a certain cohomology class posed by \textit{B. H. Gross} [Compos. Math. 45, 155--164 (1982; Zbl 0541.14010)] are solved.
scientific article; zbMATH DE number 1309084