Elliptic \(K3\) surfaces admitting a Shioda-Inose structure (Q2839764)

Let \(X\) be a \(K3\) surface with Picard number 17. Then \(X\) admits a Shioda-Inose structure if and only if \(\mathrm{Pic}(X)\) is isomorphic with \(E_8\oplus E_8\oplus \langle 2d\rangle\).NEWLINENEWLINEFor fixed \(d\) the surfaces of this type form a three-dimensional family. Explicit examples of such three-dimensional families seem only to be known for \(d\in \{1,2\}\). In both cases a general member of the family admits a Jacobian elliptic fibration, such that its Mordell-Weil group is finite and of even order and such that the Shioda-Inose structure is induced by the translation by a section of order two.NEWLINENEWLINEIn this paper, the author shows that if an elliptic \(K3\) surface with Picard number 17 and finite Mordell-Weil grou admits a Shioda-Inose structure induced by translation by a section of order two, then \(d\in \{1,2,3,5,7\}\). Moreover, the author constructs explicit examples for each of these values for \(d\).