Relative K-theory and class field theory for arithmetic surfaces

Details Relative K-theory and class field theory for arithmetic surfaces Relative K-theory and class field theory for arithmetic surfaces

2002-04-29

arxiv.org/abs/math/0204330v1

Mathematics

Number Theory

32 pages

Scientific paper

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 \pi_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 \pi_1^t(X,Y)^{ab}. Both groups are finite.

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