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 Trudy Mat. Inst. Steklova, 2003, Volume 241, Pages 8–42 (Mi tm386)

An Analogue of the Grothendieck Conjecture for Two-Dimensional Local Fields of Finite Characteristic

V. A. Abrashkinab

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Durham

Abstract: 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. the automorphisms of the topological group $\Gamma _K$ induced by conjugation by the automorphisms of the algebraic closure $\overline K$ of $K$ that leave the field $K$ invariant. Earlier, it was proved by the author that necessary and sufficient conditions for such a characterization in the case of one-dimensional local fields $K$ of characteristic $p\geq 3$ are the compatibility of these fields with the ramification filtration of the Galois group $\Gamma _K$. In the present paper, it is shown that, in the case of multidimensional local fields, the compatibility with the ramification filtration supplemented with certain natural topological conditions is still sufficient for the characterization of analytic automorphisms of $\Gamma _K$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 241, 2–34

Bibliographic databases:
UDC: 512.7

Citation: V. A. Abrashkin, “An Analogue of the Grothendieck Conjecture for Two-Dimensional Local Fields of Finite Characteristic”, Number theory, algebra, and algebraic geometry, Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich, Trudy Mat. Inst. Steklova, 241, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 8–42; Proc. Steklov Inst. Math., 241 (2003), 2–34

Citation in format AMSBIB
\Bibitem{Abr03} \by V.~A.~Abrashkin \paper An Analogue of the Grothendieck Conjecture for Two-Dimensional Local Fields of Finite Characteristic \inbook Number theory, algebra, and algebraic geometry \bookinfo Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich \serial Trudy Mat. Inst. Steklova \yr 2003 \vol 241 \pages 8--42 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm386} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2024042} \zmath{https://zbmath.org/?q=an:1125.11355} \transl \jour Proc. Steklov Inst. Math. \yr 2003 \vol 241 \pages 2--34 

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This publication is cited in the following articles:
1. I. B. Zhukov, “On ramification theory in the imperfect residue field case”, Sb. Math., 194:12 (2003), 1747–1774
2. Abrashkin V., “An analogue of the field–of–norms functor and of the Grothendieck conjecture”, Journal of Algebraic Geometry, 16:4 (2007), 671–730
3. St. Petersburg Math. J., 26:5 (2015), 695–740
4. Xiao L., Zhukov I., “Ramification of Higher Local Fields, Approaches and Questions”, Valuation Theory in Interaction, EMS Ser. Congr. Rep., eds. Campillo A., Kuhlmann F., Teissier B., Eur. Math. Soc., 2014, 600–656
5. Vostokov S.V., Afanas'eva S.S., Bondarko M.V., Volkov V.V., Demchenko O.V., Ikonnikova E.V., Zhukov I.B., Nekrasov I.I., Pital P.N., “Explicit Constructions and the Arithmetic of Local Number Fields”, Vestnik St. Petersburg Univ. Math., 50:3 (2017), 242–264
6. V. A. Abrashkin, “Ramification filtration via deformations”, Sb. Math., 212:2 (2021), 135–169
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