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.

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 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.

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 Product formula for p -adic epsilon factors

2014 ◽  
Vol 14 (2) ◽  
pp. 275-377 ◽  
Author(s):  
Tomoyuki Abe ◽  
Adriano Marmora
Keyword(s):  
Finite Field ◽  
Stationary Phase ◽  
Product Formula ◽  
Formula One ◽  
Etale Cohomology ◽  
Rigid Cohomology ◽  
Étale Cohomology ◽  
Analogous Formula ◽  
Epsilon Factors

AbstractLet $X$ be a smooth proper curve over a finite field of characteristic $p$. We prove a product formula for $p$-adic epsilon factors of arithmetic $\mathscr{D}$-modules on $X$. In particular we deduce the analogous formula for overconvergent $F$-isocrystals, which was conjectured previously. The $p$-adic product formula is a counterpart in rigid cohomology of the Deligne–Laumon formula for epsilon factors in $\ell$-adic étale cohomology (for $\ell \neq p$). One of the main tools in the proof of this $p$-adic formula is a theorem of regular stationary phase for arithmetic $\mathscr{D}$-modules that we prove by microlocal techniques.

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 FUJITA DECOMPOSITION AND MASSEY PRODUCT FOR FIBERED VARIETIES

2021 ◽  
pp. 1-29
Author(s):  
LUCA RIZZI ◽  
FRANCESCO ZUCCONI
Keyword(s):  
General Type ◽  
Structure Theorem ◽  
Smooth Curve ◽  
Local System ◽  
Base Change ◽  
Smooth Variety ◽  
Massey Product

Abstract Let $f\colon X\to B$ be a semistable fibration where X is a smooth variety of dimension $n\geq 2$ and B is a smooth curve. We give the structure theorem for the local system of the relative $1$ -forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of $f_*\omega _{X/B}$ . We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally, we give a criterion to have that X is not of Albanese general type if $B=\mathbb {P}^1$ .

scholarly journals On the action of the unitary group on the projective plane over a local field

1997 ◽  
Vol 62 (3) ◽  
pp. 371-397 ◽  
Author(s):  
Harm Voskuil
Keyword(s):  
Projective Plane ◽  
Local Field ◽  
Unitary Group ◽  
Residue Field ◽  
Equivariant Map ◽  
Rank One ◽  
Characteristic 2 ◽  
Set Of Points ◽  
Stable Points ◽  
Split Torus

AbstractLet G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ≠ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.

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.

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