staple
With cedram.org

Search the site

Table of contents for this issue | Next article
Victor Abrashkin
Modified proof of a local analogue of the Grothendieck conjecture
Journal de théorie des nombres de Bordeaux, 22 no. 1 (2010), p. 1-50, doi: 10.5802/jtnb.703
Article PDF | Reviews MR 2675872 | Zbl pre05831861

Résumé - 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 $p>2$ (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.

Bibliography

[1] V.A. Abrashkin, Ramification filtration of the Galois group of a local field. II. Proceeding of Steklov Math. Inst. 208 (1995), 18–69.  MR 1730256 |  Zbl 0884.11047
[2] V.A. Abrashkin, Ramification filtration of the Galois group of a local field. III. Izvestiya RAN, ser. math. 62 (1998), 3–48.  MR 1680900 |  Zbl 0918.11060
[3] V.A. Abrashkin, A local analogue of the Grothendieck conjecture. Int. J. of Math. 11 (2000), 3–43.  MR 1754618 |  Zbl 1073.12501
[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.  MR 1086886 |  Zbl 0753.14041
[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.  MR 1106901 |  Zbl 0743.11066
[6] K. Iwasawa, Local class field theory. Oxford University Press, 1986  MR 863740 |  Zbl 0604.12014
[7] Sh.Mochizuki, A version of the Grothendieck conjecture for $p$-adic local fields. Int. J. Math. 8 (1997), 499–506.  MR 1460898 |  Zbl 0894.11046
[8] J.-P.Serre, Lie algebras and Lie groups. Lectures given at Harvard University. New-York-Amsterdam, Bevjamin, 1965.  MR 218496 |  Zbl 0132.27803
[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.  MR 31944 |  Zbl 0071.03302
[10] J.-P. Wintenberger, Extensions abéliennes et groupes d’automorphismes de corps locaux, C. R. Acad. Sc. Paris, Série A 290 (1980), 201–203.  MR 564309 |  Zbl 0428.12012
[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.
Numdam |  MR 719763 |  Zbl 0516.12015