Automorphisms of curves and the lifting conjecture

Abstract

It is an open question whether or not one can always lift Galois extensions of smooth algebraic curves in characteristic p to Galois extensions of smooth relative curves in characteristic 0. In this thesis we study some of the available techniques and partial solutions to this problem. Our studies include the techniques of Oort, Sekiguchi and Suwa where the lifting problem is approached via a connection with lifting group schemes. We then move to the topic of singular liftings and for this we study the approach of Garuti. Thereafter, we move to the wild smooth setting again where we study the crucial local − global principle, and apply it by illustrating how Green and Matignon solved the p2-lifting problem.