Schmid’s Formula for Higher Local Fields

Schmidt, Matthew

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.

Classification : 11R37, 11S15
Mots clés : Artin–Schreier–Witt, Schmid–Witt, higher local field, ramification groups

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