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

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 $$\pi: E\times F\to X$$.

MSC:

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