Complete Discrete Valuation Field Research Articles

Stable lattices in p-adic representations III. Indecomposable reduction and holonomy

Let G be a group, V a finite dimensional vector space over a complete discrete valuation field K, and ρ:G⟶GLK(V) an irreducible representation with bounded image (in the metric topology). First, we show that there are many stable lattices with indecomposable reduction, in the sense that their simplicial convex hull in the Bruhat-Tits building of GLK(V) is the set S(ρ) of stable norms.Then we investigate the reductions more closely, in case K has finite residue field and the simplicial set S(ρ) has dimension 2. When the reduction type (defined below) is (2,1), we show that there exists a stable lattice with indecomposable reduction that has at most 2 composition series (uniserial or biserial). When the reduction type is (3), we show that the number of v∈S(ρ)0 with semisimple reduction is bounded above by a constant times |S(ρ)0|.To prove these last two statements, we prove and apply an index theorem (of Gauss-Bonnet type).

Ramification filtration and differential forms

Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$.

Ramification filtration and differential forms

Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$. Bibliography: 21 titles.

Duality for local fields and sheaves on the category of fields

Duality for complete discrete valuation fields with perfect residue field with coefficients in (possibly p-torsion) finite flat group schemes was obtained by Begueri, Bester and Kato. In this paper, we give another formulation and proof of this result. We use the category of fields and a Grothendieck topology on it. This simplifies the formulation and proof and reduces the duality to classical results on Galois cohomology. A key point is that the resulting site correctly captures extension groups between algebraic groups.

Kurihara Invariants and Elimination of Wild Ramification

The paper continues a series of papers devoted to study of the connection between two approaches to classification of complete discrete valuation fields with imperfect residue fields and, in particular, 2-dimensional local fields in the case of mixed characteristic. One of these approaches was introduced by Masato Kurihara (1987) in terms of the module of differentials. Another one is based on Epp’s theory of elimination of wild ramification. A lower bound for the degree of a constant field extension that makes a given field into an almost standard one is established. This bound is expressed in terms of the invariant introduced in Kurihara’s paper.

Ramification of wild automorphisms of Laurent series fields

Let K K be a complete discrete valuation field with residue class field k k , where both are of positive characteristic p p . Then the group of wild automorphisms of K K can be identified with the group under composition of formal power series over k k with no constant term and X X -coefficient 1 1 . Under the hypothesis that p > b 2 p > b^2 , we compute the first nontrivial coefficient of the p p th iterate of a power series over k k of the form f = X + ∑ i ≥ 1 a i X b + i f = X + \sum _{i \geq 1} a_iX^{b+i} . As a result, we obtain a necessary and sufficient condition for an automorphism to be “ b b -ramified”, having lower ramification numbers of the form i n ( f ) = b ( 1 + ⋯ + p n ) i_n(f) = b(1 + \cdots + p^n) . This is a vast generalization of Nordqvist’s 2017 theorem on 2 2 -ramified power series, as well as the analogous result for minimally ramified power series which proved to be useful for arithmetic dynamics in a 2013 paper of Lindahl on linearization discs in C p \mathbf {C}_p and a 2015 result of Lindahl–Rivera-Letelier on optimal cycles over nonarchimedean fields of positive residue characteristic. The success of our computation is also promising progress towards a generalization of Lindahl–Nordqvist’s 2018 theorem bounding the norm of periodic points of 2 2 -ramified power series.

On ramification index of composition of complete discrete valuation fields

For an extension L/K of discrete valuation fields, let $$e_{L/K}$$ , $${\mathfrak {O}}_{L}$$ denote the ramification index and valuation ring of L/K respectively. Let K be a complete discrete valuation field and $$L_1/K$$ , $$L_2/K$$ be finite linearly disjoint extensions over K. We show that if $${\mathfrak {O}}_{L_1L_2} ={\mathfrak {O}}_{L_1}{\mathfrak {O}}_{L_2}$$ or $$\mathrm {gcd}(e_{L_1/K}, e_{L_2/K}) =1$$ , and one of the residue fields $$l_1/k,$$ $$l_2/k$$ is separable, then $$e_{L_1L_2/L_1} =e_{L_2/K}.$$ The analogous results for the residue degrees are also true.

Faltings extension and Hodge-Tate filtration for abelian varieties over p-adic local fields with imperfect residue fields

AbstractLet K be a complete discrete valuation field of characteristic $0$ , with not necessarily perfect residue field of characteristic $p>0$ . We define a Faltings extension of $\mathcal {O}_K$ over $\mathbb {Z}_p$ , and we construct a Hodge-Tate filtration for abelian varieties over K by generalizing Fontaine’s construction [Fon82] where he treated the perfect residue field case.

Grothendieck’s pairing on Néron component groups: Galois descent from the semistable case

In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This, in particular, implies Grothendieck’s conjecture on the perfectness of his pairing between the Néron component groups of an abelian variety and its dual. The point is that our formulation is well suited to Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives.

Extending finite-subgroup schemes of semistable abelian varieties via log-abelian varieties

We show—for a semistable abelian variety A K over a complete discrete valuation field K —that every finite-subgroup scheme of A K extends to a log finite-flat group scheme over the valuation ring of K endowed with the canonical log structure. To achieve this, we first give a positive answer to a question of Nakayama, namely whether every weak log-abelian variety over an fs (fine and saturated) log scheme with its underlying scheme locally noetherian is a sheaf for the Kummer-flat topology. We also give several equivalent conditions defining isogenies of log-abelian varieties.

Расширения Инабы полных полей характеристики 0

В статье изучаются p--расширения полных дискретно нормированных полей смешанной характеристики, где p-характеристика поля вычетов рассматриваемого поля. Известно, что любое вполне разветвленное расширение Галуа степени p-с немаксимальным скачком ветвления может быть задано уравнением Артина-Шрайера; при этом ограничение сверху на скачок ветвления соответствует ограничению снизу на нормирование правой части уравнения. Задача построения расширений с заданной группой Галуа произвольного конечного порядка не решена. В работах Инабы рассматривались p-расширения полей характеристики p, заданные матричным уравнением $$X^{(p)}=AX$$, которое мы здесь называем уравнением Инабы. В этом уравнении $$X^{(p)}$$ обозначает матрицу, полученную возведением каждого элемента квадратной матрицы X в степень p, а - некоторая унипотентная матрица A над данным полем. Такое уравнение задает последовательность расширений полей, каждое из которых задано уравнением Артина-Шрайера. Было доказано, что любое уравнение Инабы задает расширение Галуа, и обратно, любое конечное p-расширение Галуа задается уравнением такого вида. В настоящей работе для полей смешанной характеристики доказано, что расширение, задаваемое уравнением Инабы, является расширением Галуа, если нормирования элементов матрицы удовлетворяют некоторым оценкам снизу, т.е. если скачки промежуточных расширений степени p достаточно малы. Данная конструкция может применяться при решении задачи погружения расширений полей. Уравнение Инабы задает последовательность расширений полей, полученную последовательным присоединением элементов диагоналей матрицы. Это означает, что, если расширение L/K задано уравнением Инабы, и матрица A выбрана так, что на диагоналях с большими номерами записаны нули, то можно получать расширения, содержащие L/K, заменяя нули другими элементами. В работе доказано, что любое нециклическое расширение степени $$p^2$$ с достаточно маленькими скачками можно погрузить в расширение с группой Галуа, изоморфнной группе унипотентных матриц $$3\times 3$$ над полем из p элементов. В конце статьи сформулирован ряд открытых вопросов, при исследовании которых, возможно, окажется полезной данная конструкция.

Comparison of classifications of 2-dimensional local fields, type II

The article contributes to the theory of elimination of wild ramification for 2-dimensional local fields. It continues the study of classification of complete discrete valuation fields introduced in the work of Masato Kurihara. The main object of study is a 2-dimensional local field K of mixed characteristic with a finite residue field of odd characteristic. If such a field is weakly unramified over its constant subfield k (the maximal usual local field inside it), i. e., if eK/k = 1, its structure is well known. It is also known that any 2-dimensional local field can be turned into a standard one by means of a finite extension of its constant subfield (Epp’s theorem). However, the estimate of the minimal degree of such extension is an open question in general. In Kurihara’s article the 2-dimensional are subdivided into 2 types as follows. Consider a non-trivial linear relation between differentials of the two local parameters of the field. The field belongs to Type I, if the valuation of the coefficient by the uniformizer is less than that by the second local parameter, and to Type II otherwise. This paper is devoted to the fields of Type II. For them we consider the invariant Δ, the difference between valuations of coefficients in the above mentioned linear relation (it is non-positive for the fields of Type II). The minimal degree of the required extension of k cannot be less than eK/k for trivial reasons. However, such extension of degree eK/k does not exist in general. In this article it is proved that it exists if the absolute value of Δ is sufficiently large. The corresponding estimate for Δ depends only on eK/k.

Coincidence of two Swan conductors of abelian characters

There are two ways to define the Swan conductor of an abelian character of the absolute Galois group of a complete discrete valuation field. We prove that these two Swan conductors coincide.

Reduced norms of division algebras over complete discrete valuation fields of local-global type

Let [Formula: see text] be a complete discrete valuation field whose residue field [Formula: see text] is a global field of positive characteristic [Formula: see text]. Let [Formula: see text] be a central division [Formula: see text]-algebra of [Formula: see text]-power degree. We prove that the subgroup of [Formula: see text] consisting of reduced norms of [Formula: see text] is exactly the kernel of the cup product map [Formula: see text], if either [Formula: see text] is tamely ramified or of period [Formula: see text]. This gives a [Formula: see text]-torsion counterpart of a recent theorem of Parimala, Preeti and Suresh, where the same result is proved for division algebras of prime-to-[Formula: see text] degree.

Compatible systems and ramification

We show that compatible systems of$\ell$-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on$\ell$-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

Purely Inseparable Extensions and Ramification Filtrations

In this article, we investigate the shift of Abbes and Saito's ramification filtrations of the absolute Galois group of a complete discrete valuation field of positive characteristic under a purely inseparable extension. We also study a functoriality property for characteristic forms.

On the K-theory of division algebras over local fields

Let $K$ be a complete discrete valuation field with finite residue field of characteristic $p$, and let $D$ be a central division algebra over $K$ of finite index $d$. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers $\ell$ different from $p$ and integers $j \geq 1$ , there exists a "reduced norm" isomorphism of $\ell$-adic $K$-groups $\operatorname{Nrd}_{D/K} \colon K_j(D,\mathbb{Z}_{\ell}) \to K_j(K,\mathbb{Z}_{\ell})$ such that $d \cdot \operatorname{Nrd}_{D/K}$ is equal to the norm homomorphism $N_{D/K}$. The purpose of this paper is to prove the analogous result for the $p$-adic $K$-groups. To do so, we employ the cyclotomic trace map to topological cyclic homology and show that there exists a "reduced trace" equivalence $\operatorname{Trd}_{A/S} \colon \operatorname{THH}(A\,|\,D,\mathbb{Z}_p) \to \operatorname{THH}(S\,|\,K,\mathbb{Z}_p)$ between two $p$-complete cyclotomic spectra associated with $D$ and $K$, respectively. Interestingly, we show that if $p$ divides $d$, then it is not possible to choose said equivalence such that, as maps of cyclotomic spectra, $d \cdot \operatorname{Trd}_{A/S}$ agrees with the trace $\operatorname{Tr}_{A/S}$, although this is possible as maps of spectra with $\mathbb{T}$-action.

REFINED SWAN CONDUCTORS OF ONE-DIMENSIONAL GALOIS REPRESENTATIONS

For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.

LOCALLY ANALYTIC VECTORS AND OVERCONVERGENT -MODULES

AbstractLet$p$be a prime, let$K$be a complete discrete valuation field of characteristic$0$with a perfect residue field of characteristic$p$, and let$G_{K}$be the Galois group. Let$\unicode[STIX]{x1D70B}$be a fixed uniformizer of$K$, let$K_{\infty }$be the extension by adjoining to$K$a system of compatible$p^{n}$th roots of$\unicode[STIX]{x1D70B}$for all$n$, and let$L$be the Galois closure of$K_{\infty }$. Using these field extensions, Caruso constructs the$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-modules, which classify$p$-adic Galois representations of$G_{K}$. In this paper, we study locally analytic vectors in some period rings with respect to the$p$-adic Lie group$\operatorname{Gal}(L/K)$, in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules, we can establish the overconvergence property of the$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-modules.

The local lifting problem for D4

For a prime p, a cyclic-by-p group G and a G-extension L|K of complete discrete valuation fields of characteristic p with algebraically closed residue field, the local lifting problem asks whether the extension L|K lifts to characteristic zero. In this paper, we characterize D4-extensions of fields of characteristic two, determine the ramification breaks of (suitable) D4- extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every D4-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that D4 is a local Oort group for the prime 2.