Miranda-persson’s Problem on Extremal Elliptic K3 Surfaces

In one of their early works, Miranda and Persson have classified all possible configurations of singular fibers for semistable extremal elliptic fibrations on K3 surfaces. They also obtained the Mordell-Weil groups in terms of the singular fibers except for 17 cases where the determination and the uniqueness of the groups were not settled. In this paper, we settle these problems completely. We also show that for all cases with ‘larger’ Mordell-Weil groups, this group, together with the singular fibre type, determines uniquely the fibration structure of the K3 surface (up to based fibre-space isomorphisms).