Modified proof of a local analogue of the Grothendieck conjecture

Victor Abrashkin[1]

  • [1] Math Dept of Durham University Sci Laboratories, South Road DH7 7QR Durham, UK

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 1, page 1-50
  • ISSN: 1246-7405

Abstract

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A local analogue of the Grothendieck Conjecture is an equivalence between the category of complete discrete valuation fields K with finite residue fields of characteristic p ≠ 0 and the category of absolute Galois groups of fields K together with their ramification filtrations. The case of characteristic 0 fields K was studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case p > 2 (but with no restrictions on the characteristic of K ). In this paper we suggest 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.

How to cite

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Abrashkin, Victor. "Modified proof of a local analogue of the Grothendieck conjecture." Journal de Théorie des Nombres de Bordeaux 22.1 (2010): 1-50. <http://eudml.org/doc/116396>.

@article{Abrashkin2010,
abstract = {A local analogue of the Grothendieck Conjecture is an equivalence between 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 studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case $p&gt;2$ (but with no restrictions on the characteristic of $K$). In this paper we suggest 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.},
affiliation = {Math Dept of Durham University Sci Laboratories, South Road DH7 7QR Durham, UK},
author = {Abrashkin, Victor},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {upper ramification groups; local Grothendieck conjecture; maximal -extension; Magnus algebra},
language = {eng},
number = {1},
pages = {1-50},
publisher = {Université Bordeaux 1},
title = {Modified proof of a local analogue of the Grothendieck conjecture},
url = {http://eudml.org/doc/116396},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Abrashkin, Victor
TI - Modified proof of a local analogue of the Grothendieck conjecture
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 1
SP - 1
EP - 50
AB - A local analogue of the Grothendieck Conjecture is an equivalence between 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 studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case $p&gt;2$ (but with no restrictions on the characteristic of $K$). In this paper we suggest 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.
LA - eng
KW - upper ramification groups; local Grothendieck conjecture; maximal -extension; Magnus algebra
UR - http://eudml.org/doc/116396
ER -

References

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  1. V.A. Abrashkin, Ramification filtration of the Galois group of a local field. II. Proceeding of Steklov Math. Inst. 208 (1995), 18–69. Zbl0884.11047MR1730256
  2. V.A. Abrashkin, Ramification filtration of the Galois group of a local field. III. Izvestiya RAN, ser. math. 62 (1998), 3–48. Zbl0918.11060MR1680900
  3. V.A. Abrashkin, A local analogue of the Grothendieck conjecture. Int. J. of Math. 11 (2000), 3–43. Zbl1073.12501MR1754618
  4. P. Berthelot, W. Messing, Théorie de Deudonné Cristalline III: Théorèmes d’Équivalence et de Pleine Fidélité. The Grotendieck Festschrift. A Collection of Articles Written in Honor of 60th Birthday of Alexander Grothendieck, volume 1, eds P.Cartier etc. Birkhauser, 1990, 173–247. Zbl0753.14041MR1086886
  5. J.-M. Fontaine, Representations p -adiques des corps locaux (1-ere partie). The Grothendieck Festschrift. A Collection of Articles Written in Honor of the 60th Birthday of Alexander Grothendieck, volume II, eds. P.Cartier etc. Birkhauser, 1990, 249–309. Zbl0743.11066MR1106901
  6. K. Iwasawa, Local class field theory. Oxford University Press, 1986 Zbl0604.12014MR863740
  7. Sh.Mochizuki, A version of the Grothendieck conjecture for p -adic local fields. Int. J. Math. 8 (1997), 499–506. Zbl0894.11046MR1460898
  8. J.-P.Serre, Lie algebras and Lie groups. Lectures given at Harvard University. New-York-Amsterdam, Bevjamin, 1965. Zbl0132.27803MR218496
  9. I.R. Shafarevich. A general reciprocity law (In Russian). Mat. Sbornik 26 (1950), 113–146; Engl. transl. in Amer. Math. Soc. Transl. Ser. 2, volume 2 (1956), 59–72. Zbl0071.03302MR31944
  10. J.-P. Wintenberger, Extensions abéliennes et groupes d’automorphismes de corps locaux, C. R. Acad. Sc. Paris, Série A 290 (1980), 201–203. Zbl0428.12012MR564309
  11. J.-P. Wintenberger, Le corps des normes de certaines extensions infinies des corps locaux; application. Ann. Sci. Ec. Norm. Super., IV. Ser 16 (1983), 59–89. Zbl0516.12015MR719763

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