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Ramification Filtration and Differential Forms (2022)
Journal Article
Abrashkin, V. (2023). Ramification Filtration and Differential Forms. Izvestiâ Rossijskoj akademii nauk. Seriâ matematičeskaâ Известия Российской академии наук. Серия математическая (Online), 87(3), 5-22. https://doi.org/10.4213/im9322

Let L be a complete discrete valuation field of prime characteristic p with finite residue field. Denote by Γ(v)L the ramification subgroups of ΓL=Gal(Lsep/L). We consider the category MΓLieL of finite Zp[ΓL]-modules H, satisfying some additional (Li... Read More about Ramification Filtration and Differential Forms.

Groups of automorphisms of local fields of period p^M and nilpotent class < p (2016)
Journal Article
Abrashkin, V. (2017). Groups of automorphisms of local fields of period p^M and nilpotent class < p. Annales de l'Institut Fourier, 67(2), 605-635. https://doi.org/10.5802/aif.3093

Suppose K is a finite field extension of Qp containing a pM-th primitive root of unity. For 1 6 s < p denote by K[s, M] the maximal p-extension of K with the Galois group of period pM and nilpotent class s. We apply the nilpotent Artin–Schreier theor... Read More about Groups of automorphisms of local fields of period p^M and nilpotent class < p.

Ramification estimate for Fontaine-Laffaille Galois modules (2015)
Journal Article
Abrashkin, V. (2015). Ramification estimate for Fontaine-Laffaille Galois modules. Journal of Algebra, 427, 319-328. https://doi.org/10.1016/j.jalgebra.2014.11.029

Suppose K is unramified over Qp and View the MathML source. Let H be a torsion ΓK-equivariant subquotient of crystalline Qp[ΓK]-module with HT weights from [0,p−2]. We give a new proof of Fontaine's conjecture about the triviality of action of some r... Read More about Ramification estimate for Fontaine-Laffaille Galois modules.

Group schemes of period p>2 (2010)
Journal Article
Abrashkin, V. (2010). Group schemes of period p>2. Proceedings of the London Mathematical Society, 101(1), 207-259. https://doi.org/10.1112/plms/pdp052

For a prime number p>2 we give a direct proof of Breuil's classification of finite flat group schemes killed by p over the valuation ring of a p-adic field with perfect residue field. As application we establesh a correspondence between finite flat g... Read More about Group schemes of period p>2.

Modified proof of a local analogue of the Grothendieck Conjecture (2010)
Journal Article
Abrashkin, V. (2010). Modified proof of a local analogue of the Grothendieck Conjecture. Journal de théorie des nombres de Bordeaux, 22(1), 1-50. https://doi.org/10.5802/jtnb.703

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 and the category of their Galois groups together with their ramification filtrati... Read More about Modified proof of a local analogue of the Grothendieck Conjecture.

Characteristic p analogue of modules with finite crystalline height (2009)
Journal Article
Abrashkin, V. (2009). Characteristic p analogue of modules with finite crystalline height. Pure and Applied Mathematics Quarterly, 5(1), 469-494. https://doi.org/10.4310/pamq.2009.v5.n1.a14

In the case of local fields of positive characateristic we introduce an analogue of Fontaine's concept of Galois modules with finite crystalline height h. If h=1 these modules appear as geometric points of Faltings's strict modules. We obtain upper e... Read More about Characteristic p analogue of modules with finite crystalline height.

An analogue of the field-of-norms functor and of the Grothendieck Conjecture (2007)
Journal Article
Abrashkin, V. (2007). An analogue of the field-of-norms functor and of the Grothendieck Conjecture. Journal of Algebraic Geometry, 16(4), 671-730. https://doi.org/10.1090/s1056-3911-07-00470-5

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... Read More about An analogue of the field-of-norms functor and of the Grothendieck Conjecture.

Galois modules arising from Faltings's strict modules (2006)
Journal Article
Abrashkin, V. (2006). Galois modules arising from Faltings's strict modules. Compositio Mathematica, 142(4), 867-888. https://doi.org/10.1112/s0010437x06002041

Suppose that $O=\Bbb F_q[\pi ]$ is a polynomial ring and $R$ is a commutative unitary $O$-algebra. The category of finite group schemes over $R$ with a strict action of $O$ was recently introduced by Faltings and appears as an equal characteristic an... Read More about Galois modules arising from Faltings's strict modules.

Towards explicit description of the ramification filtration in the 2-dimensional case (2004)
Journal Article
Abrashkin, V. (2004). Towards explicit description of the ramification filtration in the 2-dimensional case. Journal de théorie des nombres de Bordeaux, 16(2), 293-333

The principal result of this paper is an explicit description of the structure of ramification subgroups of the Galois group of 2-dimensional local field modulo its subgroup of commutators of order 3. This result plays a clue role in the author's pro... Read More about Towards explicit description of the ramification filtration in the 2-dimensional case.

An analogue of the Grothendieck conjecture for two-dimensional local fields of finite characteristic (2003)
Journal Article
Abrashkin, V. (2003). An analogue of the Grothendieck conjecture for two-dimensional local fields of finite characteristic. Proceedings of the Steklov Institute of Mathematics, 241, 2-34

In the case of a local field $K$ of finite characteristic $p>0$, a local analogue of the grothendieck Conjecture appears as a characterization of "analytic" automorphisms of the Galois group $\Gamma _K$ of $K$, i.e. those which are induced by a field... Read More about An analogue of the Grothendieck conjecture for two-dimensional local fields of finite characteristic.