On Extremal Elliptic K3 Surfaces

In [MP2], R.Miranda and U.Persson have classified possible semi-stable fibrations. The determination of all semi-stable fibrations has been done in [MP3] and [ATZ]. In this paper, we first classify all possible configurations of unsemi-stable fibrations (cf. Theorem 2.4). Then we calculate the possible Mordell-Weil Groups for Case(A) (cf.Theorem 3.1), i.e., the case where each singular fibre of f is not of tpye In. Finally, by using the method in [ATZ], we will precisely determine which cases in Table 1 are actually realizable (cf.Theorem 0.4).