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Shun Ohkubo - One of the best experts on this subject based on the ideXlab platform.
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On differential modules associated to de Rham representations in the imperfect Residue Field case
Algebra & Number Theory, 2015Co-Authors: Shun OhkuboAbstract: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.Comment: 50pages; v4: minor correction
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on lie algebras arising from p adic representations in the imperfect Residue Field case
Journal of Algebra, 2014Co-Authors: Shun OhkuboAbstract:Abstract Let K be a complete discrete valuation Field of mixed characteristic ( 0 , p ) with Residue Field k K such that [ k K : k K p ] = p d ∞ . Let G K be the absolute Galois group of K and ρ : G K → GL h ( Q p ) a p -adic representation. When k K is perfect, Shankar Sen described the Lie algebra of ρ ( G K ) in terms of so-called Sen's operator Θ for ρ . When k K may not be perfect, Olivier Brinon defined d + 1 operators Θ 0 , … , Θ d 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 ( G K ) in terms of Brinon's operators Θ 0 , … , Θ d , which is a generalization of Sen's result.
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The p-adic monodromy theorem in the imperfect Residue Field case
Algebra & Number Theory, 2013Co-Authors: Shun OhkuboAbstract:報告番号: 甲28380 ; 学位授与年月日: 2012-03-22 ; 学位の種別: 課程博士 ; 学位の種類: 博士(数理科学) ; 学位記番号: 博数理第388号 ; 研究科・専攻: 数理科学研究科数理科学専攻
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on lie algebras arising from p adic representations in the imperfect Residue Field case
arXiv: Number Theory, 2013Co-Authors: Shun OhkuboAbstract:Let $K$ be a complete discrete valuation Field of mixed characteristic $(0,p)$ with Residue Field $k_K$ such that $[k_K:k_K^p]=p^d<\infty$. Let $G_K$ be the absolute Galois group of $K$ and $\rho:G_K\to GL_h(\Q_p)$ a $p$-adic representation. When $k_K$ is perfect, Shankar Sen described the Lie algebra of $\rho(G_K)$ in terms of so-called Sen's operator $\Theta$ for $\rho$. When $k_K$ may not be perfect, Olivier Brinon defined $d+1$ operators $\Theta_0,...,\Theta_d$ for $\rho$, which coincides with Sen's operator $\Theta$ in the case of $d=0$. In this paper, we describe the Lie algebra of $\rho(G_K)$ in terms of Brinon's operators $\Theta_0,...,\Theta_d$, which is a generalization of Sen's result.
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the p adic monodromy theorem in the imperfect Residue Field case
arXiv: Number Theory, 2012Co-Authors: Shun OhkuboAbstract: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.
Sudesh K. Khanduja - One of the best experts on this subject based on the ideXlab platform.
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On a query of Adrian Wadsworth
1999Co-Authors: Sudesh K. Khanduja, Usha GargAbstract:Lei v 0 be a valuation of a Field K 0 with Residue Field k 0 of characteristic #2. Suppose that K is a function Field of a conic over K 0 and v is a prolongation of v 0 to K. whose Residue Field k is not algebraic over k 0 . It is well known (cf [Residue Fields of valued function Fields of conies, Proc. Edinb. math. Sac 36 (1993), 469-78]) that either k is a simple transcendental extension of a finite extension of k 0 ,or k is a regular function Field of a conic over k 0 . The paper contains the answer to a natural question posed by Wadsworth which states that if v 1 , v 1 are any two prolongations of v 0 to K with Residue Fields k 1 k 2 non-algebraic over k 0 such that neither k 1 , or k 2 is a simple transcendental extension of a finite extension of k 0 , is it true that k 1 , is t 0 -isomorphic to k T .
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On value groups and Residue Fields of some valued function Fields
1994Co-Authors: Sudesh K. KhandujaAbstract:Let K = K0(x,y) be a function Field of transcendence degree one over a Field K0 with x,y satisfying y2 = F(x), F(x) being any polynomial over K0. Let v0 be a valuation of K0 having a Residue Field K0 and v be a prolongation of v0 to K with Residue Field k. In the present paper, it is proved that if G0⊆G are the value groups of v0 and v, then either G/G0 is a torsion group or there exists an (explicitly constructive) subgroup G1 of G containing G0 with [G1:G0]
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On value groups and Residue Fields of some valued function Fields
Proceedings of the Edinburgh Mathematical Society, 1994Co-Authors: Sudesh K. KhandujaAbstract:Let K = K0(x, y) be a function Field of transcendence degree one over a Field K0 with x, y satisfying y2 = F(x), F(x) being any polynomial over K0. Let υ0 be a valuation of K0 having a Residue Field k0 and υ be a prolongation of υ to K with Residue Field k. In the present paper, it is proved that if G0⊆G are the value groups of υ0 and υ, then either G/G0 is a torsion group or there exists an (explicitly constructible) subgroup G1 of G containing G0 with [G1:G0]<∞ together with an element γ of G such that G is the direct sum of G1 and the cyclic group ℤγ. As regards the Residue Fields, a method of explicitly determining k has been described in case k/k0 is a non-algebraic extension and char k0≠2. The description leads to an inequality relating the genus of K/K0 with that of k/k0: this inequality is slightly stronger than the one implied by the well-known genus inequality (cf. [Manuscripta Math.65 (1989), 357–376’, [Manuscripta Math.58 (1987), 179–214]).
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Residue Fields of valued function Fields of conics
Proceedings of the Edinburgh Mathematical Society, 1993Co-Authors: Sudesh K. Khanduja, Usha GargAbstract:Suppose that K is a function Field of a conic over a subField K0. Let v0 be a valuation of K0 with Residue Field k0 of characteristic ≠2. Let v be an extension of v0 to K having Residue Field k. It has been proved that either k is an algebraic extension of k0 or k is a regular function Field of a conic over a finite extension of k0. This result can also be deduced from the genus inequality of Matignon (cf. [On valued function Fields I, Manuscripta Math. 65 (1989), 357–376]) which has been proved using results about vector space defect and methods of rigid analytic geometry. The proof given here is more or less self-contained requiring only elementary valuation theory.
Uri Onn - One of the best experts on this subject based on the ideXlab platform.
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Representations of automorphism groups of finite o-modules of rank two
Advances in Mathematics, 2008Co-Authors: Uri OnnAbstract:Abstract Let o be a complete discrete valuation domain with finite Residue Field. In this paper we describe the irreducible representations of the groups Aut ( M ) for any finite o -module M of rank two. The main emphasis is on the interaction between the different groups and their representations. An induction scheme is developed in order to study the whole family of these groups coherently. The results obtained depend on the ring o in a very weak manner, mainly through the degree of the Residue Field. In particular, a uniform description of the irreducible representations of GL 2 ( o / p l ) is obtained, where p is the maximal ideal of o .
Karim Johannes Becher - One of the best experts on this subject based on the ideXlab platform.
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a ruled Residue theorem for function Fields of conics
arXiv: Commutative Algebra, 2019Co-Authors: Parul Gupta, Karim Johannes BecherAbstract:The ruled Residue theorem characterises Residue Field extensions for valuations on a rational function Field. It is extended here to algebraic function Fields of genus zero.
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A ruled Residue theorem for function Fields of conics
Journal of Pure and Applied Algebra, 1Co-Authors: Parul Gupta, Karim Johannes BecherAbstract:Abstract The ruled Residue theorem characterises Residue Field extensions for valuations on a rational function Field. Under the assumption that the characteristic of the Residue Field is different from 2 this theorem is extended here to function Fields of conics. The main result is that there is at most one extension of a valuation on the base Field to the function Field of a conic for which the Residue Field extension is transcendental but not ruled. Furthermore the situation when this valuation is present is characterised.
Kazuma Morita - One of the best experts on this subject based on the ideXlab platform.
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Crystalline and semi-stable representations in the imperfect Residue Field case
Asian Journal of Mathematics, 2014Co-Authors: Kazuma MoritaAbstract:Let K be a p-adic local Field with Residue Field k such that [k : k] = p < ∞ and V be a p-adic representation of Gal(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 Gal(K/K) if and only if V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(Kpf/K) where K/K is a certain p-adic local Field whose Residue Field is the smallest perfect Field k containing k. As an application, we prove the p-adic monodromy theorem of Fontaine in the imperfect Residue Field case.
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Hodge-Tate and de Rham representations in the imperfect Residue Field case
Annales scientifiques de l'École normale supérieure, 2010Co-Authors: Kazuma MoritaAbstract: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.
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Galois cohomology of a $p$-adic Field via $(\Phi,\Gamma)$-modules in the imperfect Residue Field case
arXiv: Number Theory, 2005Co-Authors: Kazuma MoritaAbstract:For a $p$-adic local Field $K$ with perfect Residue Field, L. Herr constructed a complex which computes the Galois cohomology of a $p$-torsion representation of the absolute Galois group of $K$ by using the theory of $(\Phi,\Gamma)$-modules. We shall generalize his work to the imperfect Residue Field (the Residue Field has a finite $p$-basis) case.