Automorphism groups of rational elliptic surfaces with section and constant j map

2013-10-12
I will present the second leg of the classification project for the automorphism groups of rational elliptic surfaces (RES) with section which concerns those RES with constant J-Map. In the first leg of this study, it was shown that the group Aut(B) of regular automorphisms (biholomorphic maps) of a relatively minimal RES B over the field C is the semi-direct product MW(B) o Autσ(B) of its Mordell-Weil group MW(B) (the group of sections) and the subgroup Autσ(B) of the automorphisms preserving the zero section of the surface. MW(B) has been classified by Oguiso and Shioda with respect to the configuration of singular fibers on B. Autσ(B) was classified for RES with non-constant J-Map in the first leg of this study. In this talk I will discuss the results in the constant J-Map case. RES with constant J-Map have richer automorphism groups. While Autσ(B) has size at most 24 in the non-constant J-Map case, it can have size 144 or can even be infinite depending on the configuration of singular fibers on B if the J-Map is constant. One reason for having more symmetry in that second case is the existence of automorphisms which act as complex multiplication of order 3,4 or 6 on every smooth elliptic curve fiber of the surface.
Citation Formats
T. Karayayla, “Automorphism groups of rational elliptic surfaces with section and constant j map,” presented at the Fall Eastern Section Meeting of the AMS, October 12-13, 2013, Philadelphia, USA, 2013, Accessed: 00, 2021. [Online]. Available: http://www.ams.org/amsmtgs/2209_abstracts/1093-14-28.pdf.