Let K be a finite extension of Qp , and choose a uniformizer pi ∈ K . Choose pin+1 : ppn such that ppn+1 = pin, and put Kinfinity:=⋃ nK (pin+1). We introduce a new technique using restriction to Gal( K/Kinfinity ) 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( K/Kinfinity )-representations "of height ≤ h" for any positive integer h, and we analyze their local structure. Using these Gal( K/Kinfinity )-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.
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