Class field theory for curves over local fields
type: Item
instance of: scholarly article
title: Class field theory for curves over local fields
zbMATH Open document ID: 0599.14008
DOI: 10.1016/0022-314X(85)90011-3
author: Shuji Saito
published in: Journal of Number Theory
publication date: 1985-01-01T00:00:00Z
review text: Let \(X\) be a smooth projective geometrically irreducible curve over a local field \(k\). The purpose of this paper is to develop unramified class theory for \(X\), i.e. to try to approximate the abelian fundamental group \(\pi_ 1^{ab}(X)\) which controls the abelian étale covers of \(X\) by means of some group defined ``downstairs''. \textit{S. Bloch} [Ann. Math. (2) 114, 229--265 (1981; Zbl 0512.14009)] devised what this group should be, namely the group \(SK_ 1(X)\) which is the cokernel of the tame residue map going from \(K_ 2\) of the function field \(K=k(X)\) of \(X\) to the direct sum of the multiplicative groups \(k(x)^*\) of the residue fields \(k(x)\) at all closed points \(x\) of \(X\). There is a natural map from \(SK_ 1(X)\) to \(k^*\) induced by the norm maps \(k(x)^*\to k^*\). The kernel of this map is denoted \(V(X)\). The aim of the paper is to define and study a reciprocity map \(\sigma: SK_ 1(X)\to \pi_ 1^{ab}(X)\) and an induced map \(\tau: V(X)\to T_ G\) where \(T_ G\) is the module of coinvariants of the Tate module of the Jacobian of \(X\), which is known to control the abelian unramified extensions of \(X\) which do not come from \(k\) (\(T_ G\) is the kernel of the surjective map \(\pi_ 1^{ab}(X)\to \text{Gal}(k^{ab}/k))\). The map \(\sigma\) is defined in a rather elaborate manner via the class field theory of two-dimensional local fields [\textit{K. Kato}, J. Fac. Sci., Univ. Tokyo, Sect. IA 26, 303--376 (1979; Zbl 0428.12013); ibid 27, 603--683 (1980; Zbl 0463.12006)] and the reciprocity law for two-dimensional local rings, due to Kato and described in \S1 of the present paper. The map \(\sigma\) then induces the map \(\tau\), first introduced by Bloch for \(X(k)\neq \emptyset\). In the good reduction case, Bloch (loc. cit.) showed that \(\tau\) is a surjection onto a finite group. At least when \(\text{char}(k)=0\), the author shows: (i) the image of \(\tau\) is always finite; (ii) its kernel is the maximal divisible subgroup of \(V(X)…
Mathematics Subject Classification ID: 11G45
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 3966276
zbMATH Keywords: class field theory for curves over local fields
zbMATH Keywords: abelian fundamental group
zbMATH Keywords: class field theory of two-dimensional local fields
zbMATH Keywords: reciprocity law
reviewed by: Jean-Louis Colliot-Thélène
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3966276