Relative K-theory and class field theory for arithmetic surfaces

2002/04/29 by Alexander Schmidt, Schmidt, Alexander
Mathematics · #11R37 #19F05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.AG #math.NT #msc:11R37 #msc:19F05

paper · pdf · doi:10.48550/arxiv.math/0204330

32 pages

arxiv created 2002/04/29 · openalex publication_date 2002/04/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 CH₀(X,Y) denote the relative Chow group of zero cycles and let π₁^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: CH₀(X,Y) --> π₁^t(X,Y)^ab. Both groups are finite.

Related

• Geometric class field theory with bounded ramification
• Tame class field theory over local fields
• Class field theory for open curves over local fields
• A cycle class map from Chow groups with modulus to relative K-theory
• Zero-cycles with modulus and relative K-theory