Galois Deformation Theory for Norm Fields and its Arithmetic Applications.

Abstract

Let K be a finite extension of Q_p, and choose a uniformizer pi in K. Choose pi_{n+1} such that pi_1:=pi and pi_{n+1}^p=pi_n, and let K_infty denote the field extension of K obtained by adjoining pi_{n+1} for all n. We introduce a new technique using restriction to Gal(Kbar/K_infty) to study deformations and mod p reductions in p-adic Hodge theory. One of our main results in deformation theory is the existence of deformation rings for Gal(Kbar/K_infty)-representations "of height <= h" for any positive integer h, and we analyze their local structure. Using these Gal(Kbar/K_infty)-deformation rings, we give a different proof of Kisin's connected component analysis of flat deformation rings of a certain fixed Hodge type, which we used to prove the modularity of potentially Barsotti-Tate representations. This new proof works ``more uniformly'' for $p=2$, and does not use Zink's theory of windows and displays.

We also study the equi-characteristic analogue of crystalline representations in the sense of Genestier-Lafforgue and Hartl. We show the full faithfulness of a natural functor from semilinear algebra objects, so-called local shtukas, into representations of the absolute Galois group of a local field of characteristic p>0. We also obtain equi-characteristic deformation rings for Galois representations that come from local shtukas, and study their local structure.