Schmid’s Formula for Higher Local Fields
Matthew Schmidt1
1 Department of Mathematics State University of New York Buffalo, NY 14260, USA
Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 355-371.

En théorie du corps de classes local, le symbole de Schmid–Witt encode des données intéressantes sur la ramification des p-extensions de K et peut, par exemple, être utilisé pour calculer les groupes de ramification supérieurs de telles extensions. En 1936, Schmid a découvert une formule explicite pour le symbole de Schmid–Witt pour les extensions d’Artin–Schreier des corps locaux. Sa formule a été ensuite généralisée au cas des extensions d’Artin–Schreier–Witt, toujours pour les corps locaux. Dans cet article, nous généralisons la formule de Schmid pour calculer le symbole d’Artin–Schreier–Witt–Parshin pour les extensions d’Artin–Schreier–Witt des corps locaux de dimension 2 de caractéristique positive.

In local class field theory, the Schmid–Witt symbol encodes interesting data about the ramification theory of p-extensi-ons of K and can, for example, be used to compute the higher ramification groups of such extensions. In 1936, Schmid discovered an explicit formula for the Schmid–Witt symbol of Artin–Schreier extensions of local fields. Later, his formula was generalized to Artin–Schreier–Witt extensions, but still over a local field. In this paper we generalize Schmid’s formula to compute the Artin–Schreier–Witt–Parshin symbol for Artin–Schreier–Witt extensions of two-dimensional local fields of positive characteristic.

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DOI : 10.5802/jtnb.1125
Classification : 11R37, 11S15
Mots clés : Artin–Schreier–Witt, Schmid–Witt, higher local field, ramification groups
Matthew Schmidt 1

1 Department of Mathematics State University of New York Buffalo, NY 14260, USA
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Matthew Schmidt. Schmid’s Formula for Higher Local Fields. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 355-371. doi : 10.5802/jtnb.1125. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1125/

[1] Ivan Fesenko Abelian local p-class field theory, Math. Ann., Volume 301 (1995) no. 3, pp. 561-586 | DOI | MR | Zbl

[2] Ivan Fesenko Sequential topologies and quotients of Milnor K-groups of higher local fields, St. Petersbg. Math. J., Volume 13 (2002) no. 3, pp. 485-501 | MR | Zbl

[3] Osamu Hyodo Wild ramification in the imperfect residue field case, Galois representations and arithmetic algebraic geometry (Advanced Studies in Pure Mathematics), Volume 12, North-Holland, 1987, pp. 287-314 | DOI | MR | Zbl

[4] Kazuya Kato A generalization of local class field theory by Using K-groups I, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 26 (1979), pp. 303-376 | MR | Zbl

[5] Kazuya Kato A generalization of local class field theory by Using K-groups II, J. Fac. Sci., Univ. Tokyo, Sect. I A (1980), pp. 603-683 | Zbl

[6] Kazuya Kato A generalization of local class field theory by Using K-groups III, J. Fac. Sci., Univ. Tokyo, Sect. I A (1982), pp. 31-43 | Zbl

[7] Nicholas M. Katz Witt Vectors and a Question of Keating and Rudnick, Int. Math. Res. Not., Volume 2013 (2014) no. 16, pp. 3613-3638 | DOI | Zbl

[8] Yukiyosi Kawada; Ichiro Satake Class formations II, J. Fac. Sci. Univ. Tokyo, Sect. I, Volume 7 (1956), pp. 353-389 | MR | Zbl

[9] Michiel Kosters; Daqing Wan On the Arithmetic of ℤ p -Extensions (2016) (https://arxiv.org/abs/1607.00523)

[10] Michiel Kosters; Daqing Wan Genus growth in ℤ p towers of function fields, Proc. Am. Math. Soc., Volume 146 (2018) no. 4, pp. 1481-1494 | DOI | MR | Zbl

[11] V. G. Lomadze On the Ramification Theory of Two-dimensional Local Fields, Math. USSR, Sb., Volume 37 (1980) no. 3, pp. 349-365 | DOI | MR | Zbl

[12] Matthew Morrow An introduction to higher dimensional local fields and adeles (2012) (https://arxiv.org/abs/1204.0586)

[13] Alexei N. Parshin Local Class Field Theory, Tr. Mat. Inst. Steklova, Volume 165 (1985), pp. 157-185 | MR | Zbl

[14] Alexei N. Parshin Galois Cohomology and the Brauer Group of Local Fields, Tr. Mat. Inst. Steklova, Volume 183 (1991) no. 4, pp. 191-201 | MR | Zbl

[15] Hermann L. Schmid Über das Reziprozitätsgesetz in relativ-zyklischen algebraischen Funktionenkörpern mit endlichem Konstantenkörper, Math. Z., Volume 40 (1936) no. 1, pp. 94-109 | DOI | MR | Zbl

[16] Jean-Pierre Serre Local Fields, Graduate Texts in Mathematics, 67, Springer, 1979 | Zbl

[17] Lara Thomas Arithmétique des extensions d’Artin-Schreier-Witt, Ph. D. Thesis, Université de Toulouse II le Mirail (France) (2005)

[18] Lara Thomas Ramification groups in Artin-Schreier-Witt extensions, J. Théor. Nombres Bordeaux, Volume 17 (2005) no. 2, pp. 689-720 | DOI | Numdam | MR | Zbl

[19] Ernst Witt Zyklische Körper und Algebren der Charakteristik p vom Grad p n , J. Reine Angew. Math., Volume 176 (1937), pp. 126-140 | Zbl

[20] Igor Zhukov Higher dimensional local fields, Invitation to higher local fields (Geometry and Topology Monographs), Volume 3, Geometry & Topology Publications, 2000 | MR | Zbl

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