POTENTIALLY SEMI-STABLE DEFORMATION RINGS

摘要

Let K/Q_p be a finite extension and G_K = Gal(K/K) the Galois group of an algebraic closure K. Let F be a finite field of characteristic p, and V_F a finite dimensional F-vector space equipped with a continuous action of GK. The study of the deformation theory of Galois representations was initiated by Mazur [Ma], who showed that if V_F has no non-trivial endomorphisms, then it admits a universal deformation ring R_(V_F). After the work of Wiles [Wi] and the conjectures of Fontaine-Mazur [FM] it became clear that for arithmetic applications it was important to understand certain quotients of R_(V_F) corresponding to deformations satisfying certain conditions. For example Wiles uses deformations which arise from finite flat group schemes, and the corresponding quotient of R_(V_F) was constructed by Ramakrishna [Ra].