On Jacobian fibrations on the Kummer surfaces of the product of non-isogenous elliptic curves. (English) Zbl 0703.14024

Oguiso, Keiji

J. Math. Soc. Japan 41, No. 4, 651-680 (1989).

Let \(E\) and \(F\) be non-isogenous elliptic curves over \(\mathbb C\), and let \(X\) be the Kummer surface of \(E\times F\). The number \(J(X)\) of Jacobian fibrations of \(X\), modulo the action of \(\operatorname{Aut}(X)\) is finite (H. Sterk). Because only 11 types of singular fibers can occur, the set \(J(X)\) is divided into 11 classes. An explicit complete set of representatives for these classes, together with the associated Mordell-Weil groups, is given. The description only depends on whether or not \(E\) or \(F\) admit non-trivial automorphisms. The proof depends on the study of the 24 nodal curves of the natural map \(\pi: E\times F\to X\).

Reviewer: Frédéric Campana (Vandœuvre-les-Nancy)

MSC:

14J28	\(K3\) surfaces and Enriques surfaces
14G05	Rational points
14H52	Elliptic curves
14J50	Automorphisms of surfaces and higher-dimensional varieties

Keywords:

elliptic curves; Kummer surface; Jacobian fibrations; Mordell-Weil groups; automorphisms