In this paper we investigate the Kummer pairing associated to an arbitrary (one-dimensional) formal group. In particular, we obtain formulae describing the values of the pairing in terms of multidimensional p-adic differentiation, the logarithm of the formal group, the generalized trace and the norm on Milnor K-groups. The results are a generalization to higher-dimensional local fields of Kolyvagin's explicit reciprocity laws. In particular, they constitute a generalization of the reciprocity laws of Artin-Hasse, Iwasawa and Wiles.

中文翻译：

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较高局部场的范数残差符号

在本文中，我们研究了与任意（一维）形式群相关的 Kummer 配对。特别是，我们获得了用多维p进微分、形式群的对数、广义迹线和 Milnor K 群的范数来描述配对值的公式。结果是对 Kolyvagin 显式互易律的高维局部域的推广。特别是，它们构成了对 Artin-Hasse、Iwasawa 和 Wiles 互惠定律的概括。