The Classification of Automorphism Groups of Rational Elliptic Surfaces with Section

In this dissertation, 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 Autsigma( B) of the automorphisms preserving the zero section sigma 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, and Autsigma( B) also depends on the singular fibers. The classification of automorphism groups in this dissertation gives the group Autsigma( B) in terms of the configuration of singular fibers on the surface. In general, Autsigma(B) is a finite group of order less than or equal to 24 which is a Z/2Z extension of either Z/nZ,Z/2ZxZ /2Z , Dn (the Dihedral group of order 2 n) or A4 (the Alternating group of order 12). The configuration of singular fibers does not determine the group Autsigma(B) uniquely; however we list explicitly all the possible groups Autsigma( B) and the configurations of singular fibers for which each group occurs.