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Victor Abrashkin

Publications and source records attributed to Victor Abrashkin.

14 recordsLinked to original sources
arXiv2025

Ramification filtration via deformations, II

Let $\mathcal K$ be a field of formal Laurent series with coefficients in a finite field of characteristic $p$. For $M\ge 1$, let $\mathcal G_{<p,M}$ be the maximal quotient of the Galois group of $\mathcal K$ of period $p^M$ and nilpotent class $<p$ and $\{\mathcal G_{<p,M}^{(v)}\}_{v\geqslant 0}$ -- the ramification subgroups in upper numbering. Let $\mathcal G_{<p,M}=G(\mathcal L)$ be the identification of nilpotent Artin-Schreier theory: here $G(\mathcal L)$ is the group obtained from a suitable profinite Lie $\mathbb{Z}/p^M$-algebra $\mathcal L$ via the Campbell-Hausdorff composition law. We develop new techniques to obtain a ``geometrical'' construction of the ideals $\mathcal L^{(v)}$ such that $G(\mathcal L^{(v)})=\mathcal G_{<p,M}^{(v)}$. Given $v_0\geqslant 1$, we construct a decreasing central filtration $\mathcal L(w)$, $1\leqslant w\leqslant p$, on $\mathcal L$, an epimorphism of Lie $\mathbb{Z}/p^M$-algebras $\bar{\mathcal V}:\bar{\mathcal L}^{†}\to \bar{\mathcal L}:=\mathcal L/\mathcal L(p)$, and a unipotent action $Ω$ of $\mathbb{Z} /p^M$ on $\bar{\mathcal L}^{†}$, which induces the identity action on $\bar{\mathcal L}$. Suppose $dΩ=B^{†}$, where $B^{†}\in\operatorname{Diff}\bar{\mathcal L}^{†}$, and $\bar{\mathcal L}^{†[v_0]}$ is the ideal of $\bar{\mathcal L}^{†}$ generated by the elements of $B^{†}(\bar{\mathcal L}^{†})$. Our main result states that the ramification ideal $\mathcal L^{(v_0)}$ appears as the preimage of the ideal in $\bar{\mathcal L}$ generated by $\bar{\mathcal V}B^{†}(\bar{\mathcal L}^{†[v_0]})$. In the last section we apply this to the explicit construction of generators of $\bar{\mathcal L}^{(v_0)}$. The paper justifies a geometrical origin of ramification subgroups of $Γ_K$ and can be used for further developing of non-abelian local class field theory.

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arXiv2022

Ramification filtration and differential forms

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

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arXiv2021

Galois groups of p-extensions of higher local fields

Suppose $\mathcal K$ is $N$-dimensional local field of characteristic $p$, $\mathcal G =\mathop{Gal}(\mathcal K_{sep}/\mathcal K)$, $\mathcal G_{<p}$ is the maximal quotient of $\mathcal G$ of period $p$ and nilpotent class $<p$ and $\mathcal K_{<p}\subset \mathcal K_{sep}$ is such that $\mathop{Gal}(\mathcal K_{<p}/\mathcal K)=\mathcal G_{<p}$. We use nilpotent Artin-Schreier theory to identify $\mathcal G_{<p}$ with the group $G(\mathcal L)$ obtained from a profinite Lie $\mathbb F_p$-algebra $\mathcal L$ via the Campbell-Hausdorff composition law. The canonical $\mathcal P$-topology on $\mathcal K$ is used to define a dense Lie subalgebra $\mathcal L^{\mathcal P}$ in $\mathcal L$. The algebra $\mathcal L^{\mathcal P}$ can be provided with a system of $\mathcal P$-topological generators and its $\mathcal P$-open subalgebras correspond to all $N$-dimensional extensions of $\mathcal K$ in $\mathcal K_{<p}$. These results are applied to higher local fields $K$ of characteristic 0 containing primitive $p$-th root of unity. If $Γ=\mathop{Gal}(K_{alg}/K)$ we introduce similarly the quotient $Γ_{<p}=G(L)$, a dense $\mathbb F_p$-Lie algebra $L^{\mathcal P}\subset L$, and describe the structure of $L^{\mathcal P}$ in terms of generators and relations. The general result is illustrated by explicit presentation of $Γ_{<p}$ modulo third commutators.

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arXiv2021

Ramification filtration via deformations

Let $\mathcal K$ be a field of formal Laurent series with coefficients in a finite field of characteristic $p$, $\mathcal G_{<p}$ -- the maximal quotient of $\operatorname{Gal} (\mathcal K_{sep}/\mathcal K)$ of period $p$ and nilpotent class $<p$ and $\{\mathcal G_{<p}^{(v)}\}_{v\geqslant 0}$ -- its filtration by ramification subgroups in the upper numbering. Let $\mathcal G_{<p}=G(\mathcal L)$ be the identification of nilpotent Artin-Schreier theory: here $G(\mathcal L)$ is the group obtained from a suitable profinite Lie $\mathbb{F}_p$-algebra $\mathcal L$ via the Campbell-Hausdorff composition law. We develop a new technique to describe the ideals $\mathcal L^{(v)}$ such that $G(\mathcal L^{(v)})=\mathcal G_{<p}^{(v)}$ and to find their generators. Given $v_0\geqslant 1$ we construct epimorphism of Lie algebras $\barη^{†}:\mathcal L\longrightarrow \bar{\mathcal L}^{†}$ and an action $Ω_U$ of the formal group of order $p$, $α=_p=\operatorname{Spec}\,\mathbb{F}_p[U]$, $U^p=0$, on $\bar{\mathcal L}^{†}$. Suppose $dΩ_U=B^{†}U$, where $B^{†}\in\operatorname{Diff}\bar{\mathcal L}^{†}$, and $\bar{\mathcal L}^{†}[v_0]$ is the ideal of $\bar{\mathcal L}^{†}$ generated by the elements of $B^{†}(\bar{\mathcal L}^{†})$. The main result of the paper states that $\mathcal L^{(v_0)}=(\barη^{†})^{-1}\bar{\mathcal L}^{†}[v_0]$. In the last sections we relate this result to the explicit construction of generators of $\mathcal L^{(v_0)}$ obtained earlier by the author, develop its more efficient version and apply it to the recovering of the whole ramification filtration of $\mathcal G_{<p}$ from the set of its jumps.

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arXiv2017

Automorphisms of local fields of period $p$ and nilpotent class $<p$

Suppose $K$ is a finite field extension of $\mathbb{Q} _p$ containing a primitive $p$-th root of unity. Let $Γ_{<p}$ be the Galois group of a maximal $p$-extension of $K$ with the Galois group of period $p$ and nilpotent class $<p$. In the paper we describe the ramification filtration $\{Γ_{<p}^{(v)}\}_{v\geqslant 0}$ and relate it to an explicit form of the Demushkin relation for $Γ_{<p}$. The results are given in terms of Lie algebras attached to involved groups by the classical equivalence of the categories of $p$-groups and Lie algebras of nilpotent class $<p$.

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arXiv2014

Ramification estimate for Fontaine-Laffaille Galois modules

Suppose $K$ is unramified over $\mathbb Q _p$ and $Γ_K=\operatorname{Gal}(\bar K/K)$. Let $H$ be a torsion $Γ_K$-equivariant subquotient of crystalline $\mathbb Q _p[Γ_K]$-module with HT weights from $[0,p-2]$. We give a new proof of Fontaine's conjecture about the triviality of action of some ramification subgroups $Γ_K^{(v)}$ on $H$. The earlier author's proof from [1] contains a gap and proves this conjecture only for some subgroups of index $p$ in $Γ_K^{(v)}$.

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arXiv2012

Group schemes of period 2

We give an explicit construction of the antiequivalence of the category of finite flat commutative group schemes of period 2 defined over the valuation ring of a 2-adic field with algebraically closed residue field. This result extends the earlier author's approach to group schemes of period p>2 from Proceedings of LMS, 101, 2010, 207-259.

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arXiv2012

Projective varieties with bad reduction at 3 only

Suppose F=W(k)[1/p] where W(k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p>2. We construct integral theory of p-adic semi-stable representations of the absolute Galois group of F with Hodge-Tate weights from [0,p). This modification of Breuil's theory results in the following application in the spirit of Shafarevich's Conjecture. If Y is a projective algebraic variety over the field of rational numbers with good reduction modulo all primes different from 3 and semi-stable reduction modulo 3 then for the Hodge numbers of the complexification Y_C of Y, it holds h^2(Y_C)=h^{1,1}(Y_C).

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arXiv2010

The field-of-norms functor and the Hilbert symbol for higher local fields

The field-of-norms functor is applied to deduce explicit reciprocity formulae for the Hilbert symbol in the mixed characteristic case from the explicit formula for the Witt symbol in characteristic p > 2 in the context of higher local fields. Is is shown that a "very special case" of this construction gives Vostokov's explicit formula

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arXiv2009

Group schemes of period p>2

For a prime number p>2, we give a direct proof of Breuil's classification of killed by p finite flat group schemes over the valuation ring of a p-adic field with perfect residue field. As application we prove that the Galois modules of geometric points of such group schemes and of their characteristic p analogues coming from Faltings's strict modules can be identified via the Fontaine-Wintenberger field-of-norms functor.

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arXiv2009

Modified proof of a local analogue of the Grothendieck conjecture

A local analogue of the Grothendieck Conjecture is an equivalence of the category of complete discrete valuation fields $K$ with finite residue fields of characteristic $p\ne 0$ and the category of absolute Galois groups of fields $K$ together with their ramification filtrations. The case of characteristic 0 fields $K$ was considered by Mochizuki several years ago. Then the author proved it by different method if $p>2$ (but $\operatorname{char}K=0$ or $p$). This paper represents a modified approach: it covers the case $p=2$, contains considerable technical simplifications and replaces the Galois group of $K$ by its maximal pro-$p$-quotient. Special attention is paid to the procedure of recovering field isomorphisms coming from isomorphisms of Galois groups, which are compatible with the corresponding ramification filtrations.

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arXiv2005

An analogue of the field-of-norms functor and the Grothendieck Conjecture

The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied to deduce the mixed characteristic case of a local analogue of the Grothendieck Conjecture for these fields from its characteristic p case, which was proved earlier by the author.

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arXiv2004

Galois modules arising from Faltings's strict modules

Suppose O is a complete discrete valuation ring of positive characteristic with perfect residue field. The category of finite flat strict modules was introduced recently by Faltings and appears as an equal characteristic analogue of the classical category of finite flat group schemes. In this paper we obtain a classification of these modules and apply it to prove analogues of properties, which were known earlier for group schemes.

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