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.

中文翻译：

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二维局部II型场的分类比较

摘要

本文为消除二维字段的野生分枝理论做出了贡献，并继续进行了与Masato Kurihara的工作中引入的字段分类有关的研究。我们考虑具有有限残差场不等于2的特征的二维混合特征局部场。众所周知，在其恒定子场上微弱地无分支化的场的结构，即所谓的标准场。还已知任何字段都可以通过其常数子字段的有限扩展而扩展为标准字段。在一般情况下，此扩展的最小程度的问题仍然存在。在Kurihara的论文中，二维字段分为以下两种类型。考虑局部参数的微分之间的线性关系。如果在均化器上的系数估值小于第二个局部参数之前的估值，则该字段属于类型I；否则，它属于II型。本文致力于II型领域。对于它们，我们考虑改进的Kurihara不变性：对于每个字段，我们引入的数量Δ等于局部参数微分关系中系数的估值之间的差。对于任何场，消除分支的恒定扩展程度不小于在恒定子场上的分支索引。但是，并非所有字段都具有此程度的扩展。已经证明，为了存在最小可能度的扩展，Δ的绝对值足够大是足够的。