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Ramification groups in Artin-Schreier-Witt extensions. (English) Zbl 1207.11109

Summary: Let \(K\) be a local field of characteristic \(p>0\). The aim of this paper is to describe the ramification groups for the pro-\(p\) abelian extensions over \(K\) with regards to the Artin-Schreier-Witt theory. We carry out this investigation entirely in the usual framework of local class field theory. This leads to a certain non-degenerate pairing that we shall define in detail, generalizing in this way the Schmid formula to Witt vectors of length \(n\). Along the way, we recover a result of Brylinski but with a different proof which is more explicit and requires less technical machinery. A first attempt is finally made to extend these computations to the case where the perfect field of \(K\) is merely perfect.

MSC:

11S15 Ramification and extension theory
11S31 Class field theory; \(p\)-adic formal groups
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[1] E. Artin, O.Schreier, Eine Kennzeichnung der reell algeschlossenen Körper. Abh. Math. Sem. Hamburg 5 (1927), 225-231. · JFM 53.0144.01
[2] J.-L. Brylinski, Théorie du corps de classes de Kato et revêtements abéliens de surfaces. Ann. Inst. Fourier, 33.3, Grenoble (1983), 23-38. · Zbl 0524.12008
[3] I.B. Fesenko, S.V. Vostokov, Local Fields and Their Extensions. Translation of Mathematical Monographs 121, Amer. Math. Soc. (1993). · Zbl 0781.11042
[4] M. Garuti, Linear sytems attached to cyclic inertia. (Berkeley, CA, 1999), 377-386, Proc. Sympos. Pure Math., 70, Amer. Math. Soc., Providence, RI (2002). · Zbl 1072.14017
[5] H. Hasse, Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper. J. Reine Angew. Math. 172 (1934), 37-54. · Zbl 0010.00501
[6] M. Hazewinkel, Abelian extensions of local fields, Ph.D. thesis. Universiteit Nijmegen, Holland (1969). · Zbl 0193.35001
[7] E. Maus, Die Gruppentheoretische Struktur der Verzweigungsgruppenreihen. J. Reine. Angew. Math. 230 (1968), 1-28. · Zbl 0165.35703
[8] L. Ribes, P. Zalesskii, Profinite groups. A series of Modern Surveys in Mathematics, Volume 40, Springer (2000). · Zbl 0949.20017
[9] P. Roquette, Class Field Theory in characteristic \(p\). Its origin and development. K. Miyake (ed.), Class Field Theory- Its Centenary and Prospect, Advanced Studies In Pure Mathematics, vol. 30, Tokyo (2000), 549-631. · Zbl 1068.11073
[10] H.L. Schmid, Über das Reziprozitätsgezsetz in relativ-zyklischen algebraischen Funktionkörpern mit endlichem Konstantenkörper. Math. Z. 40 (1936), no. 1, 94-109. · Zbl 0011.14604
[11] H.L. Schmid, Zyklischen algebraische Funktionkörper vom Grade \(p^n\) über endlichem Konstantenkörper der Charakteristik \(p\). J. Reine Angew. Math. 175, (1936), 108-123. · Zbl 0014.00402
[12] H.L. Schmid, Zur Arithmetik der zyklischen \(p\)-Körper. J. Reine Angew. Math. 176 (1937), 161-167 · Zbl 0016.05205
[13] J.-P. Serre, Corps Locaux. Hermann, Paris (1962). [English translation : Local Fields. Graduate Texts in Math. 67, Springer, New York (1979)]. · Zbl 0137.02601
[14] S.S. Shatz, Profinite groups, arithmetic and geometry. Princeton university press and university of Tokyo press (1972). · Zbl 0236.12002
[15] O. Teichmüller, Zerfallende zyklische \(p\)-Algebren. J. Reine Angew. Math., 174 (1936), 157-160. · Zbl 0016.05201
[16] L. Thomas, Arithmétique des extensions d’Artin-Schreier-Witt, Ph.D. (2005). Université Toulouse II Le Mirail.
[17] E. Witt, Zyklische Körper und Algebren der Charakteristik vom Grad \(p^n\). J. Reine Angew. Math., 174 (1936), 126-140. · Zbl 0016.05101
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