Comparison of Classifications of Two-Dimensional Local Type II Fields

Abstract

The paper contributes to the theory of the elimination of wild ramification for two-dimensional fields and continues the research related to the classification of fields introduced in the work of Masato Kurihara. We consider two-dimensional mixed-characteristic local fields with the characteristic of the finite residue field not equal to 2. The structure of fields that are weakly unramified over their constant subfield, i.e., the so-called standard fields, is well known. It is also known that any field can be extended into the standard one by a finite extension of its constants subfield. In the general case, the question of the minimum degree of this extension remains open. In Kurihara’s paper, two-dimensional fields are subdivided into two types as follows. A linear relation between the differentials of local parameters is considered. If the valuation of the coefficient at the uniformizer is less than that before the second local parameter, the field belongs to type I; otherwise it belongs to type II. This paper is devoted to the fields of type II. For them, we consider an improved Kurihara invariant: for each field, we introduce a quantity Δ equal to the difference between the valuations of the coefficients in the relation for the differentials of the local parameters. The degree of the constant extension that eliminates the ramification is not less for any field than the ramification index over the constant subfield. However, not all the fields have an extension of this degree. It is proved that in order that the extension of the least possible degree may exist, it suffices for the absolute values of Δ to be sufficiently large. The corresponding estimate for Δ depends on the ramification index of the field over its constant subfield.