On the Tate-Shafarevich groups of certain elliptic curves
Let \(p\) be a prime number congruent to \(1\pmod 4\), \(p= \pi\cdot\overline\pi\) in \(\mathbb Z[i]\), where \(\pi= 1(\text{mod}(1+ i)^3)\) and \(q:= p^f\). Consider the elliptic curve \[ E\mid t(t- 1)y^2= x(x- 1)(x- t) \] defined over the global field of characteristic \(p\), \(\mathbb F_q(t)\). In this case it is known [\textit{J. Tate}, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Dix Exposés Cohomologie Schémas, Advanced Studies Pure Math. 3, 189--214 (1968) = Sém. Bourbaki 1965/66, Exp. No. 306, 415--440 (1966; Zbl 0199.55604)] that the corresponding Tate-Shafarevich group \(\text Ш(\mathbb F_q(t), E)\) is finite. For every odd prime \(\ell\neq p\), the \(\ell^\infty\)-torsion of \(\text Ш\,W\) is identified, via étale cohomology, to a group of points of an elliptic curve with complex multiplication by \(\mathbb Z[i]\) (the kernel of multiplication by \([(\pi/\overline\pi)^f- 1]\) on \(\mathbb Z[i]\otimes \mathbb Z_\ell\)), from which follows that \(\text Ш(\mathbb F_q(t), E)[\ell^\infty]\) depends only on the valuation of \([(\pi/\overline\pi)^f- 1]\) at the places of \(\mathbb Q(i)\) lying above \(\ell\). By using this identification, the Cassels-Tate pairing has been also calculated.
scientific article; zbMATH DE number 5759075
scientific article