On two approaches to classification of higher local fields

O. Ivanova ; S. Vostokov ; I. Zhukov

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 186-197

Résumé

This article links Kurihara's classification of complete discrete valuation fields and Epp's theory of elimination of wild ramification. For any complete discrete valuation field $K$ with arbitrary residue field of prime characteristic one can define a certain numerical invariant $\Gamma(K)$ which underlies Kurihara's classification of such fields into $2$ types: the field $K$ is of Type I if and only if $\Gamma(K)$ is positive. The value of this invariant indicates how distant is the given field from a standard one, i.e., from a field which is unramified over its constant subfield $k$ which is the maximal subfield with perfect residue field. (Standard $2$-dimensional local fields are exactly fields of the form $k\{\{t\}\}$.) We prove (under some mild restriction on $K$) that for a Type I mixed characteristic $2$-dimensional local field $K$ there exists an estimate from below for $[l:k]$ where $l/k$ is an extension such that $lK$ is a standard field (existing due to Epp's theory); the logarithm of this degree can be estimated linearly in terms of $\Gamma(K)$ with the coefficient depending only on $e_{K/k}$.

Détail

Keywords: higher local fields, wild ramification.