scholarly journals On ramification filtrations and p-adic differential equations, II: mixed characteristic case

2011 ◽  
Vol 148 (2) ◽  
pp. 415-463 ◽  
Author(s):  
Liang Xiao
Keyword(s):  
Differential Equations ◽  
Residue Field ◽  
Mixed Characteristic ◽  
Group Schemes ◽  
Characteristic Case ◽  
Valuation Field

AbstractLet K be a complete discrete valuation field of mixed characteristic (0,p), with possibly imperfect residue field. We prove a Hasse–Arf theorem for the arithmetic ramification filtrations on GK, except possibly in the absolutely unramified and non-logarithmic case, or the p=2 and logarithmic case. As an application, we obtain a Hasse–Arf theorem for filtrations on finite flat group schemes over 𝒪K.

scholarly journals Facteurs epsilon p-adiques

2008 ◽  
Vol 144 (2) ◽  
pp. 439-483 ◽  
Author(s):  
Adriano Marmora
Keyword(s):  
Residue Field ◽  
Smooth Curve ◽  
Local System ◽  
Product Formula ◽  
Mixed Characteristic ◽  
Characteristic Case ◽  
Rank One ◽  
Epsilon Factors ◽  
Valuation Field ◽  
Field Of Norms

AbstractWe develop and study the epsilon factor of a ‘local system’ of p-adic coefficients over the spectrum of a complete discrete valuation field K with finite residue field of characteristic p>0. In the equal characteristic case, we define the epsilon factor of an overconvergent F-isocrystal over Spec(K), using the p-adic monodromy theorem. We conjecture a global formula, the p-adic product formula, analogous to Deligne’s formula for étale ℓ-adic sheaves proved by Laumon, which explains the importance of this local invariant. Namely, for an overconvergent F-isocrystal over an open subset of a projective smooth curve X, the constant of the functional equation of the L-series is expressed as a product of the local epsilon factors at the points of X. We prove the conjecture for rank-one overconvergent F-isocrystals and for finite unit-root overconvergent F-isocrystals. In the mixed characteristic case, we study the behavior of the epsilon factor by deformation to the field of norms.

ON GALOIS EQUIVARIANCE OF HOMOMORPHISMS BETWEEN TORSION CRYSTALLINE REPRESENTATIONS

2016 ◽  
Vol 229 ◽  
pp. 169-214
Author(s):  
YOSHIYASU OZEKI
Keyword(s):  
Galois Group ◽  
Residue Field ◽  
Absolute Galois Group ◽  
Mixed Characteristic ◽  
Crystalline Representations ◽  
Valuation Field ◽  
The Absolute

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field. Let $(\unicode[STIX]{x1D70B}_{n})_{n\geqslant 0}$ be a system of $p$-power roots of a uniformizer $\unicode[STIX]{x1D70B}=\unicode[STIX]{x1D70B}_{0}$ of $K$ with $\unicode[STIX]{x1D70B}_{n+1}^{p}=\unicode[STIX]{x1D70B}_{n}$, and define $G_{s}$ (resp. $G_{\infty }$) the absolute Galois group of $K(\unicode[STIX]{x1D70B}_{s})$ (resp. $K_{\infty }:=\bigcup _{n\geqslant 0}K(\unicode[STIX]{x1D70B}_{n})$). In this paper, we study $G_{s}$-equivariantness properties of $G_{\infty }$-equivariant homomorphisms between torsion crystalline representations.

scholarly journals On nearby cycles and 𝒟-modules of log schemes in characteristic p>0

2010 ◽  
Vol 146 (6) ◽  
pp. 1552-1616 ◽  
Author(s):  
Takeshi Tsuji
Keyword(s):  
Residue Field ◽  
Characteristic P ◽  
Mixed Characteristic ◽  
Crystalline Cohomology ◽  
Ring Of Integers ◽  
Nearby Cycles ◽  
Valuation Field ◽  
Weight Spectral Sequence ◽  
Log Schemes ◽  
Stable Scheme

AbstractLet K be a complete discrete valuation field of mixed characteristic (0,p) with a perfect residue field k. For a semi-stable scheme over the ring of integers OK of K or, more generally, for a log smooth scheme of semi-stable type over k, we define nearby cycles as a single 𝒟-module endowed with a monodromy ∂logt, whose cohomology should give the log crystalline cohomology. We also explicitly describe the monodromy filtration of the 𝒟-module with respect to the endomorphism ∂logt, and construct a weight spectral sequence for the cohomology of the nearby cycles.

scholarly journals RECIPROCITY SHEAVES AND THEIR RAMIFICATION FILTRATIONS

2021 ◽  
pp. 1-74
Author(s):  
Kay Rülling ◽  
Shuji Saito
Keyword(s):  
Residue Field ◽  
Group Scheme ◽  
Canonical Extension ◽  
General Notion ◽  
Base Field ◽  
Ground Field ◽  
Group Schemes ◽  
Geometric Type ◽  
Categorical Framework ◽  
Valuation Field

Abstract We define a motivic conductor for any presheaf with transfers F using the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. If F is a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on $F(L)$ , where L is any henselian discrete valuation field of geometric type over the perfect ground field. We show that if F is a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; if F assigns to X the group of finite characters on the abelianised étale fundamental group of X, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and if F assigns to X the group of integrable rank $1$ connections (in characteristic $0$ ), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type with perfect residue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.

scholarly journals NOTES ON EXTREMAL AND TAME VALUED FIELDS

2016 ◽  
Vol 81 (2) ◽  
pp. 400-416
Author(s):  
SYLVY ANSCOMBE ◽  
FRANZ-VIKTOR KUHLMANN
Keyword(s):  
Approximation Property ◽  
Elementary Theory ◽  
Residue Field ◽  
Axiom System ◽  
Complete Characterization ◽  
Valued Fields ◽  
Mixed Characteristic ◽  
Valued Field ◽  
Residue Fields

AbstractWe extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.

scholarly journals Splitting properties of the reduction of semi-abelian varieties

2016 ◽  
Vol 12 (08) ◽  
pp. 2241-2264
Author(s):  
Alan Hertgen
Keyword(s):  
Abelian Variety ◽  
Abelian Varieties ◽  
Residue Field ◽  
Characteristic Exponent ◽  
Jacobian Variety ◽  
Ring Of Integers ◽  
Identity Component ◽  
Genus Formula ◽  
Valuation Field

Let [Formula: see text] be a complete discrete valuation field. Let [Formula: see text] be its ring of integers. Let [Formula: see text] be its residue field which we assume to be algebraically closed of characteristic exponent [Formula: see text]. Let [Formula: see text] be a semi-abelian variety. Let [Formula: see text] be its Néron model. The special fiber [Formula: see text] is an extension of the identity component [Formula: see text] by the group of components [Formula: see text]. We say that [Formula: see text] has split reduction if this extension is split. Whereas [Formula: see text] has always split reduction if [Formula: see text] we prove that it is no longer the case if [Formula: see text] even if [Formula: see text] is tamely ramified. If [Formula: see text] is the Jacobian variety of a smooth proper and geometrically connected curve [Formula: see text] of genus [Formula: see text], we prove that for any tamely ramified extension [Formula: see text] of degree greater than a constant, depending on [Formula: see text] only, [Formula: see text] has split reduction. This answers some questions of Liu and Lorenzini.

scholarly journals Stabilité de l’holonomie sur les variétés quasi-projectives

2011 ◽  
Vol 147 (6) ◽  
pp. 1772-1792 ◽  
Author(s):  
Daniel Caro
Keyword(s):  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Mixed Characteristic ◽  
Complete Discrete Valuation Ring ◽  
The Stability

AbstractLet 𝒱 be a mixed characteristic complete discrete valuation ring with perfect residue field k. We solve Berthelot’s conjectures on the stability of the holonomicity over smooth projective formal 𝒱-schemes. Then we build a category of F-complexes of arithmetic 𝒟-modules over quasi-projective k-varieties with bounded and holonomic cohomology. We obtain its stability under Grothendieck’s six operations.

scholarly journals -neighborhoods and comparison theorems

2015 ◽  
Vol 151 (10) ◽  
pp. 1945-1964 ◽  
Author(s):  
Piotr Achinger
Keyword(s):  
Valuation Ring ◽  
Open Neighborhood ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Universal Cover ◽  
Comparison Theorems ◽  
Main Ingredient ◽  
Generic Fiber ◽  
Mixed Characteristic ◽  
Original Approach

A technical ingredient in Faltings’ original approach to$p$-adic comparison theorems involves the construction of$K({\it\pi},1)$-neighborhoods for a smooth scheme$X$over a mixed characteristic discrete valuation ring with a perfect residue field: every point$x\in X$has an open neighborhood$U$whose generic fiber is a$K({\it\pi},1)$scheme (a notion analogous to having a contractible universal cover). We show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in$p$-adic Hodge theory. The main ingredient of the proof is a variant of a trick of Nagata used in his proof of the Noether normalization lemma.

scholarly journals On the ramification of étale cohomology groups

2019 ◽  
Vol 2019 (749) ◽  
pp. 295-304 ◽  
Author(s):  
Isabel Leal
Keyword(s):  
Positive Characteristic ◽  
Residue Field ◽  
Cohomology Groups ◽  
Generic Fiber ◽  
Codimension One ◽  
Etale Cohomology ◽  
Rank One ◽  
Étale Cohomology ◽  
Valuation Field ◽  
Dimension One

Abstract Let K be a complete discrete valuation field whose residue field is perfect and of positive characteristic, let X be a connected, proper scheme over \mathcal{O}_{K} , and let U be the complement in X of a divisor with simple normal crossings. Assume that the pair (X,U) is strictly semi-stable over \mathcal{O}_{K} of relative dimension one and K is of equal characteristic. We prove that, for any smooth \ell -adic sheaf \mathcal{G} on U of rank one, at most tamely ramified on the generic fiber, if the ramification of \mathcal{G} is bounded by t+ for the logarithmic upper ramification groups of Abbes–Saito at points of codimension one of X, then the ramification of the étale cohomology groups with compact support of \mathcal{G} is bounded by t+ in the same sense.

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