Automorphism Groups of Algebraic Curves in Positive Characteristic

Explore the intricate relationship between automorphism groups and algebraic curves in positive characteristic through this 52-minute mathematical lecture. Delve into the fundamental concepts of algebraic curves defined over algebraically closed fields of positive characteristic p, examining their function fields and K-automorphism groups. Learn about classical results including Schnid's theorem on the finiteness of automorphism groups for curves of genus at least two, and discover how every finite group can be realized as an automorphism group of some algebraic curve. Investigate the crucial role of invariants such as genus and p-rank in determining the structure and size of automorphism groups, with particular attention to Nakajima's influential work on p-Sylow subgroups and ordinary curves. Examine three major open problems in the field: the maximum size of d-groups of automorphisms when d≠p, the sharpness of Nakajima's bound for ordinary curves, and the existence of optimal functions relating automorphism group size to p-rank zero curves. Compare results in positive characteristic with the classical Hurwitz bound from complex algebraic geometry, and analyze improvements by Stichtenoth and Henn including the quartic bound and exceptional curves. Gain insights into recent contributions addressing these fundamental questions and their applications to determining isomorphism classes of algebraic curves over finite fields, bridging number theory, algebraic geometry, and coding theory through Goppa's AG codes.