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Relative K-theory and class field theory for arithmetic surfaces

Alexander Schmidt
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Abstract

In this paper we extend the unramified class field theory for arithmetic surfaces of K. Kato and S. Saito to the relative case. Let X be a regular proper arithmetic surface and let Y be the support of divisor on X. Let CH_0(X,Y) denote the relative Chow group of zero cycles and let \tilde π_1^t(X,Y)^ {ab} denote the abelianized modified tame fundamental group of (X,Y) (which classifies finite etale abelian covings of X-Y which are tamely ramified along Y and in which every real point splits completely). THEOREM: There exists a natural reciprocity isomorphism rec: CH_0(X,Y) --> \tilde π_1^t(X,Y)^{ab}. Both groups are finite.

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Alexander Schmidt. 2002-04-29. Relative K-theory and class field theory for arithmetic surfaces. https://arxiv.org/abs/math/0204330

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