Abrashkin's Home Page

University of Durham --- Department of Mathematical Sciences

Research

Algebraic number theory & Arithmetic algebraic geometry

I work in these general areas. Here are some more specific areas of interest

Ramification filtration of the Galois group of a local field (one of possible applications to a local analogue of the Grothendieck Conjecture was
outlined at my lecture at the Newton Institute)
Higher dimensional local fields
Finite flat group schemes
$p$-adic representations

More information (including potential PhD students) is available about the research in the department in arithmetic algebraic geometry, algebraic number theory and algebraic K-theory. The people involved together run the arithmetic study group.

Preprints

Report on the Ramification Filtration of the Galois Group of a Local Field, Proceedings of the Research Conference on "Number Theory and Arithmetical Geometry: Arithmetical applications of Modular Forms" (San Feliu de Guixols, Spain, 24-29 October, 1997), Inst. fur Exp. Math. Universitat Essen, preprint No.14 (1998), pp.47-53;

Towards arithmetical description of the Galois group of a local field , Talk given at Oberwolfach meeting ``Pro-$p$-extensions of Global fields and pro-$p$-groups'', May 21st-May 27th, 2006

Publications

Note: The papers listed here are not all linked to retrievable files. And those that are are not necessarily the final versions and so they should be treated as "for information only".

Determination of the imaginary quadratic fields of class number 2 with even discriminant by the Heegner method, Matem. Zametki Akad. Nauk SSSR, 15 (1974) pp. 241-246.
Good reduction of 2-dimensional abelian varieties, Izv. Akad. Nauk SSSR, ser. matem., 40 (1976), pp.262-272; Engl. transl. in Math. USSR Izv. 10 (1977)
2-divisible groups over $\Bbb Z$, Matem. Zametki Akad. Nauk SSSR, 19 (1976), pp.717-726; Engl. transl. in Math. USSR Izv. 10 (1977)
Moduli of relevements of $p$-divisible groups and non-supersingular abelian varieties, Uspekhi Matem. Nauk, 31 (1976), pp.199-200; Engl. transl. in Russ. Math. Surveys 31 (1976)
$p$-divisible groups over $\Bbb Z$, Izv. Akad. Nauk SSSR, ser. matem., 41 (1977), pp.435-459; Engl. transl. in Math. USSR Izv. 11 (1977)
Group schemes of period $p$, Izv. Akad. Nauk SSSR, ser. matem., 46 (1982), pp.262-272; Engl. transl. in Math. USSR Izv. 28 (1982)
Group schemes of period $p$ over the ring of Witt vectors, Dokl. Akad. Nauk SSSR, 283 (1985), pp.1289-1294; Engl. transl. in Soviet Math. Dokl. 32 (1985)
Honda systems of group schemes of period $p$, Izv. Akad. Nauk SSSR, ser. matem., 51 (1987), pp.451-484; Engl. transl. in Math. USSR Izv. 130 (1987)
Galois modules of period $p$ group schemes over the ring of Witt vectors, Izv. Akad. Nauk SSSR, ser. matem., 51 (1987), pp.691-736; Engl. transl. in Math. USSR Izv. 31 (1988)
Modular crystalline representations, Uspekhi matem. nauk, 43(1988), pp.167-168; English transl. in Russ. Math. Surveys 43(1988)
Finite group schemes of a period $p$ over a discrete valuation ring, Uspekhi matem. nauk 43(1988), pp.195-196; English transl. in Russ. Math. Surveys 43(1988)
Modular crystalline representations and a generalisation of the Shafarevich conjecture, Algebraic Geometry, Yaroslavl.Gos.Ped. Inst.,Yaroslavl, 1988, Manuscript N6418-B88, deposited at VINITI, 1988, 204-229 (Russian); Ref.Zh.Mat. 1988, 12A 425.
Group schemes over a discrete valuation ring with small ramification, Algebra i analiz, 1(1989), pp.57-97; English transl. in Leningrad Math.J., 1(1990)
Modification of the Fontaine-Laffaille functor, Izv. Akad. Nauk SSSR, ser. matem., 53(1989), pp.451-496; English transl. in Math. USSR Izv. 34(1990)
Modular representations of the Galois group of a local field and a generalisation of the Shafarevich conjecture, Izv. Akad. Nauk SSSR, ser. matem., 53(1989), pp.1135-1182; English transl. in Math. USSR Izv. 35(1990)
Ramification in etale cohomology, Inv. math., 101(1990), pp.631-640
A ramification filtration of the Galois group of a local field, Trudy St. Petersb. Mat. Obshch. 3(1995), pp.47-127; English transl., Proc. St. Petersburg Math. Soc. III, Amer. Math. Soc. Transl., Ser.2, 166, Amer. Math. Soc., Providence 1996, pp.35-100;
Ramification filtration of the Galois group of a local field. II Trudy Mat. Inst. Steklov 208(1995), pp.18-69; English transl. in Proc. of Steklov Math. Inst., 208(1995);
The field of norms functor and the Bruckner-Vostokov formula, Math. Annalen, 308 (1997), pp. 5-19
Explicit formulae for the Hilbert symbol of the formal group over Witt vectors, Izvestia RAN, ser. math., 61(1997), pp.3-56; English transl., Izvestiya: Mathematics 61:3, pp.463-515;
The ramification filtration of the Galois group of a local field III, Izvestiya RAN, ser. matem., 62 (1998), pp.3-48; English transl., Izvestiya: Mathematics 62:5, pp.857-900;
One group-theoretical property of the ramification filtration, Izvestiya RAN, ser. matem., 62(1998), pp.3-26; English transl., Izvestiya: Mathematics 62:6;
The image of the Galois group for some crystalline representations, Izvestiya RAN, ser. matem., 63(1999); English transl., Izvestiya: Mathematics 63:1, pp.1-36;
A local analogue of the Grothendieck conjecture , Int. J. of Math., 11(2000), pp. 3-43;
Ramification theory for higher dimensional local fields, Contemp. Math., 300 (2002), pp.1-16
Characteristic $p$ case of the Grothendieck conjecture for 2-dimensional local fields, Proc. of Steklov Institute, 241(2003), pp.1-35
Towards explicit description of the ramification filtration in the 2-dimensional case, Journal de Theorie des Nombres de Bordeaux, 16(2004), pp.1-41
Galois modules arising from Faltings's strict modules, Compositio Mathematica, 142(2006), pp.867-888
An analogue of the field-of-norms functor and the Grothendieck Conjecture, ps-file pdf-file, Journal of Algebraic Geometry, 16(2007), pp.671-730
Characteristic p analogue of modules with finite crystalline height , Pure and Applied Mathematics Quarterly, volume 5, Number 1 (January 2009) (Special issue: In Honor of Jean-Pierre Serre, Part 2 of 2), 469-494
Modified proof of a local analogue of the Grothendieck Conjecture , Journal Theorie des Nombres de Bordeaux 22 (2010), 1-50
Group schemes of period p>2 , Proc. London Math. Soc. (3) 101 (2010) 207-259
Projective varieties with bad semi-stable reduction at 3 only Documenta Math. 18 (2013), 547-619
Group schemes of period 2 Manuscripta Math, 143 (2014), no. 3-4, 317-353
The field-of-norms functor and the Hilbert symbol for higher local fields (with Ruth Jenni) JTNB, tome 24, n.1 (2012), p.1-39
A Semi-stable Case of the Shafarevich Conjecture Proceedings of Durham Symposia, London Mathematical Society lecture note series, v.414, pp.1-31
Galois groups of local fields, Lie algebras and ramification Proceedings of HIM (Bonn), London Mathematical Society lecture note series, v.420
Automorphisms of local fields of period p and nilpotent class less than p, I International Journal of Mathematics, v.28(06): 1750043
Automorphisms of local fields of period p and nilpotent class less than p, II International Journal of Mathematics, v.28(10):1750066
Ramification estimate for Fontaine-Laffaille Galois modules Journal of Algebra, v.427 (2015), pp.319-328
Automorphisms of local fields of period p^M and nilpotent class less than p Annales de l'Institute Fourier, v.67(2): 605-635
Ramification filtration via deformations Sbornik: Mathematics, 2021, v.212, no.2
Galois groups of p-extensions of higher local fields (to appear in JTNB)
Ramification filtration and differential forms Izvestiya Math, ser. matem., 87 (2023), no.3, 421-438
Ramification filtration via deformations,II (placed onto NT arXive)
If you have Netscape you should be able to view the linked articles on screen by following the link. If this does not work, try adding the lines:

application/x-dvi; xdvi %s
application/postscript; ghostview %s
(or similar) to your .mailcap file.