Classification of Automorphism Groups of Rational Elliptic Surfaces

In this paper, we give a classification of (regular) automorphism groups of relatively minimal rational elliptic surfaces with section over the field which have non-constant J-maps. The automorphism group of such a surface B is the semi-direct product of its Mordell–Weil group and the subgroup of the automorphisms preserving the zero section σ of the rational elliptic surface B. The configuration of singular fibers on the surface determines the Mordell–Weil group as has been shown by Oguiso and Shioda (1991), and also depends on the singular fibers. The classification of automorphism groups in this paper gives the group in terms of the configuration of singular fibers on the surface. In general, is a finite group of order less than or equal to 24 which is a extension of either , , (the Dihedral group of order 2n) or (the Alternating group of order 12). The configuration of singular fibers does not determine the group uniquely; however we list explicitly all the possible groups and the configurations of singular fibers for which each group occurs.