Class field theory for curves over local fields
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 \(V(X)\); (iii) the cokernel \(\tau\) is isomorphic to \(\widehat{\mathbb Z}^ r\) (\(\widehat{\mathbb Z}\) is the profinite completion of \(\mathbb Z\)), where \(r\) is some integer which may be computed from the structure of the special fibre of a regular proper model of \(X\) over the ring of integers of \(k\), is at most equal to the genus of \(X\) and is zero if \(X\) has good reduction. Similar results hold for \(SK_ 1(X)\). In the good reduction case over a \(p\)-adic field \(k\), part (ii) was also obtained by \textit{K. R. Coombes} [``Local class field theory for curves'', in Contemp. Math. 55, part I, 117--134 (1986; Zbl 0601.14025)], and it is shown by the reviewer and \textit{W. Raskind} [Math. Ann. 270, 165--199 (1985; Zbl 0536.14004)] that the kernel of \(\tau\) is uniquely divisible prime-to-\(p\). In view of Bloch's remark that the map \(\tau\) may be viewed as some analogue of the norm residue symbol \(K_ 2k\to \mu (k)\), one may wonder whether in the general case more can be said about torsion of \(\text{ker}\,\tau\). That the cokernel of \(\tau\) might be non-trivial reflects a quite interesting geometric fact which makes a big difference with classical local class field theory: there may exist étale covers \(Y/X\) which are completely split, i.e. the fibre of \(Y/X\) over any closed point \(x\) of \(X\) consists of copies of \(\text{Spec}\,k(x)\). Such covers are not controlled by the \(K\)-theory of \(X\). Among the ingredients used for the proof are the higher local class field theory of Kato, results of Y. Ihara and of \textit{H. Miki} [J. Fac. Sci., Univ. Tokyo, Sect. I A 21, 377--393 (1974; Zbl 0301.12003)], and the Merkur'ev-Suslin theorem [\textit{A. S. Merkur'ev} and \textit{A. A. Suslin}, Math. USSR, Izv. 21, 307--340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011--1046 (1982; Zbl 0525.18008)]. Some results of this paper are used, and sometimes given a slightly different approach by \textit{K. Kato} and the author in Ann. Math. (2) 118, 241--275 (1983; Zbl 0562.14011) and Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 103--152 (1983; Zbl 0544.12011) -- this last paper also considers ramified abelian coverings of \(X\). Extensions to higher dimensional varieties over a local field k are considered by the author in Ann. Math. (2) 121, 251--281 (1985; Zbl 0593.14001).
class field theory for curves over local fields
class field theory of two-dimensional local fields
scientific article; zbMATH DE number 3966276
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