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Lie extensions and automorphism groups of local fields. (Extensions de Lie et groupes d’automorphismes de corps locaux.) (French) Zbl 0649.12012

Let \(K\) be a local field of residue characteristic \(p>0\). An infinite Galois extension \(L\) of \(K\) is called a Lie extension if the residue field extension is finite and if the Galois group \(G=\mathrm{Gal}(L:K)\) is a \(p\)-adic Lie group. Let \(X=X_ K(L)\) be the norm field of the Lie extension \(L:K\); \(X\) is a local field of characteristic \(p\) and \(\operatorname{Aut}(X)\) contains \(G\) as a compact subgroup. \(\operatorname{Aut}(X)\) and therefore \(G\) has a natural filtration \(G_x\), \(x\ge -1.\)
J.-P. Wintenberger [C. R. Acad. Sci., Paris, Sér. A 288, 477–479 (1979; Zbl 0401.12016)] conjectured that a pair \((X,G)\) consisting of a local field \(X\) of characteristic \(p\) and a compact subgroup \(G\) of \(\operatorname{Aut}(X)\) comes from a Lie extension in the above way, if and only if \(\liminf_{x\to \infty}(x/[G:G_x])>0\). The necessity of the condition is known; the author shows that it is also sufficient under the additional hypothesis that \(G\) is solvable.
The other main result of the paper is a characterization of those \((X,G)\) which are Lie extensions in the case that the residue field of \(X\) is quasi-finite and \(G\) is a semisimple \(p\)-adic Lie group.

MSC:

11S15 Ramification and extension theory
11S20 Galois theory
22E99 Lie groups

Citations:

Zbl 0401.12016
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References:

[1] J.-M. Fontaine : Un résultat de Sen sur les automorphismes de corps locaux, Séminaire Delange-Pisot-Poitou , Paris (1969-1970) exposé n^\circ . 6. · Zbl 0287.12023
[2] J.-M. Fontaine Corps de séries formelles et extensions galoisiennes de corps locaux, Séminaire de théorie des nombres de Grenoble, (1971-1972) pp. 28-38.
[3] J.-M. Fontaine : Groupes de ramification et représentations d’Artin , Ann. Sci. E.N.S., 4e série, t. 4, fasc. 3 (1971) 337-392. · Zbl 0232.12006 · doi:10.24033/asens.1214
[4] J.-M. Fontaine et J.-P. Wintenberger : Le corps des normes de certaines extensions algébriques de corps locaux , C.R. Acad. Sc., t. 288, série A (1979) 367-370. · Zbl 0475.12020
[5] J.-M. Fontaine et J.-P. Wintenberger : Extensions algébriques et corps des normes des extensions APF des corps locaux , C.R. Acad. Sci., t. 288, série A (1979) 441-444. · Zbl 0403.12018
[6] M Harris : p-adic représentations arising from descent on abelian varieties , Comp. Math. 39, fasc. 2 (1979) 177-245. · Zbl 0417.14034
[7] F. Laubie : Groupes de Lie p-adiques et ramification, Séminaire de théorie des nombres de Bordeaux (1982-1983) exposé n^\circ . 31. · Zbl 0534.12008
[8] J. Martinet : Characters theory and Artin L-functions in Algebraic Number Fields , A. Frölich (ed.) London New York, San Francisco, Academic Press (1977) pp. 1-87. · Zbl 0359.12015
[9] S. Sen : On automorphismes of local fields , Ann. of Math. 90 (1969) 33-46. · Zbl 0199.36301 · doi:10.2307/1970680
[10] S. Sen : Ramification in p-adic Lie extensions , Inventiones Math. 17 (1972) 44-50. · Zbl 0242.12012 · doi:10.1007/BF01390022
[11] J.-P. Serre : Corps Locaux , 2 e edn., Paris, Hermann (1968). · Zbl 0137.02601
[12] J.-P. Wintenberger : Automorphismes et extensions galoisiennes de corps locaux , Thèse de 3e cycle, Publication de l’Université Scientifique et Médicale de Grenoble, 1978.
[13] J.-P. Wintenberger : Extensions de Lie et groupes d’automorphismes des corps locaux de caractéristique p, C.R. Acad. Sc., t. 288, série A (1979) 477-479. · Zbl 0401.12016
[14] J.-P. Wintenberger : Extensions et groupes d’automorphismes de corps locaux , C.R. Acad. Sc., t. 290, série A (1980) 201-203. · Zbl 0428.12012
[15] J.-P. Wintenberger : Le corps des normes de certaines extensions infinies de corps locaux; applications , Ann. Sci. E.N.S., 4e série, t. 16 (1983) 59-89. · Zbl 0516.12015 · doi:10.24033/asens.1440
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