Imperfect Residue Field Case Research Articles

On 𝑝-adic Simpson and Riemann–Hilbert correspondences in the imperfect residue field case

Let K K be a mixed characteristic complete discrete valuation field with residue field admitting a finite p p -basis, and let G K G_K be the Galois group. Inspired by Liu and Zhu’s construction of p p -adic Simpson and Riemann–Hilbert correspondences over rigid analytic varieties, we construct such correspondences for representations of G K G_K . As an application, we prove a Hodge–Tate (resp. de Rham) “rigidity” theorem for p p -adic representations of G K G_K , generalizing a result of Morita.

Strongly divisible lattices and crystalline cohomology in the imperfect residue field case

Strongly divisible lattices and crystalline cohomology in the imperfect residue field case

On differential modules associated to de Rham representations in the imperfect residue field case

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$, whose residue field may not be perfect, and $G_K$ the absolute Galois group of $K$. In the first part of this paper, we prove that Scholl's generalization of fields of norms over $K$ is compatible with Abbes-Saito's ramification theory. In the second part, we construct a functor $\mathbb{N}_{\mathrm{dR}}(V)$ associating a de Rham representation $V$ with a $(\varphi,\nabla)$-module in the sense of Kedlaya. Finally, we prove a compatibility between Kedlaya's differential Swan conductor of $\mathbb{N}_{\mathrm{dR}}(V)$ and Swan conductor of $V$, which generalizes Marmora's formula.

On Lie algebras arising from p-adic representations in the imperfect residue field case

Let K be a complete discrete valuation field of mixed characteristic (0,p) with residue field kK such that [kK:kKp]=pd<∞. Let GK be the absolute Galois group of K and ρ:GK→GLh(Qp) a p-adic representation. When kK is perfect, Shankar Sen described the Lie algebra of ρ(GK) in terms of so-called Sen's operator Θ for ρ. When kK may not be perfect, Olivier Brinon defined d+1 operators Θ0,…,Θd for ρ, which reduce to Sen's operator Θ in the case of d=0. In this paper, we describe the Lie algebra of ρ(GK) in terms of Brinon's operators Θ0,…,Θd, which is a generalization of Sen's result.

The p-adic monodromy theorem in the imperfect residue field case

Let K be a complete discrete valuation field of mixed characteristic (0,p) and G_K the absolute Galois group of K. In this paper, we will prove the p-adic monodromy theorem for p-adic representations of G_K without any assumption on the residue field of K, for example the finiteness of a p-basis of the residue field of K. The main point of the proof is a construction of (phi,G_K)-module Nrig^+(V) for a de Rham representation V, which is a generalization of Pierre Colmez' Nrig^+(V). In particular, our proof is essentially different from Kazuma Morita's proof in the case when the residue field admits a finite p-basis. We also give a few applications of the p-adic monodromy theorem, which are not mentioned in the literature. First, we prove a horizontal analogue of the p-adic monodromy theorem. Secondly, we prove an equivalence of categories between the category of horizontal de Rham representations of G_K and the category of de Rham representations of an absolute Galois group of the canonical subfield of K. Finally, we compute H^1 of some p-adic representations of G_K, which is a generalization of Osamu Hyodo's results.

A note on Sen’s theory in the imperfect residue field case

In Sen’s theory in the imperfect residue field case, Brinon defined a functor from the category of \({{\mathbb C}_p}\)-representations to the category of linear representations of a certain Lie algebra. We give a comparison theorem between the continuous Galois cohomology of \({{\mathbb C}_p}\)-representations and the Lie algebra cohomology of the associated representations. The key ingredients of the proof are Hyodo’s calculation of Galois cohomology and the effaceability of Lie algebra cohomology for solvable Lie algebras.

A ring of periods for Sen modules in the imperfect residue field case

We construct a ring B Sen of Colmez in the imperfect residue field case.

Galois Theory of [formula omitted] in the imperfect residue field case

We generalize a work of Iovita and Zaharescu on the Galois theory of B dR + to the imperfect residue field case. The proof is based on a structure theorem of Colmez's higher Kähler differentials.

Hodge-Tate and de Rham representations in the imperfect residue field case

Soit K un corps local p-adique de corps residuel k tel que [k: k P ] = p e < +∞ et soit V une representation p-adique de Gal(K/K). Nous utilisons la theorie des modules differentiels p-adiques pour montrer que V est une representation de Hodge-Tate (resp. de Rham) de Gal(K/K) si et seulement si V est une representation de Hodge-Tate (resp. de Rham) de Gal(k pf /k pf ) ou K pf /K est un certain corps local p-adique de corps residuel le plus petit corps parfait k pf contenant k.

On ramification theory in the imperfect residue field case

This paper is devoted to the ramification theory of complete discrete valuation fields such that the residue field has prime characteristic p and the cardinality of a p-base is 1. This class contains two-dimensional local and local-global fields. A new definition of ramification filtration for such fields is given. It turns out that Hasse-Herbrand type functions can be defined with all the usual properties. Thanks to this, a theory of upper ramification groups and the ramification theory of infinite extensions can be developed. The case of two-dimensional local fields of equal characteristic is studied in detail. A filtration on the second K-group of the field in question is introduced that is different from the one induced by the standard filtration on the multiplicative group. The reciprocity map of two-dimensional local class field theory is proved to identify this filtration with the ramification filtration.

On the Hodge-Tage decomposition in the imperfect residue field case.

On the Hodge-Tage decomposition in the imperfect residue field case.

Crystalline and semi-stable representations in the imperfect residue field case

Let $K$ be a $p$-adic local field with residue field $k$ such that $[k : k^p] = p^e \lt \infty$ and $V$ be a $p$-adic representation of $\mathrm{Gal}(\overline{K} / K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a potentially crystalline (resp. potentially semi-stable) representation of $\mathrm{Gal}(\overline{K} / K)$ if and only if $V$ is a potentially crystalline (resp. potentially semi-stable) representation of $\mathrm{Gal}(\overline{K^\mathrm{pf}} / K^\mathrm{pf})$ where $K^\mathrm{pf} / K$ is a certain $p$-adic local field whose residue field is the smallest perfect field $k^\mathrm{pf}$ containing $k$. As an application, we prove the $p$-adic monodromy theorem of Fontaine in the imperfect residue field case.