Abrashkin's Home Page

University of Durham --- Department of Mathematical Sciences

Courses

Galois Theory

Linear Algebra

Research

Preprints

Publications

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
Higher dimensional local fields
Finite flat group schemes
$p$-adic representations

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;

Galois modules arising from Faltings's strict modules, Preprint series of Durham University, 2003

An analogue of the field-of-norms functor and the Grothendieck Conjecture, Preprint series of Durham University, 2004

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
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.

Linear Algebra (Epiphany 2005)

Courses

Research

Algebraic number theory & Arithmetic algebraic geometry

Preprints

Publications

Linear Algebra (Epiphany 2005)