The classification of automorphism groups of rational elliptic surfaces with section

In this paper, we give a classification of (regular) automorphism groups of relatively minimal rational elliptic surfaces with section over the field C which have non-constant J-maps. The automorphism group Aut(B) of such a surface B is the semi-direct product of its Mordell–Weil group MW(B) and the subgroup Autσ(B) 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 Autσ(B) also depends on the singular fibers. The classification of automorphism groups in this paper gives the group Autσ(B) in terms of the configuration of singular fibers on the surface. In general, Autσ(B) is a finite group of order less than or equal to 24 which is a Z/2Z extension of either Z/nZ, Z/2Z×Z/2Z, Dn (the Dihedral group of order 2n) or A4 (the Alternating group of order 12). The configuration of singular fibers does not determine the group Autσ(B) uniquely; however we list explicitly all the possible groups Autσ(B) and the configurations of singular fibers for which each group occurs.