Non-abelian class field theory for arithmetic surfaces
Let \(K\) be a number field with ring of integers \({\mathcal O}\), let \(X\) be a smooth projective curve with function field \(F\) and field of constants \(K\) and let \(\pi:{\mathcal X}\to \text{Spec}({\mathcal O})\) be an integral, normal proper and flat model; the scheme \({\mathcal X}\) is called an arithmetic surface. Let \(U\subset X\) be an open subscheme which is assumed to be regular. The main result of this paper gives a description of the étale fundamental group \(\pi_1({\mathcal U})\) of \({\mathcal U}\), and the projective limit of \(\pi_1({\mathcal U})\) over all such \({\mathcal U}\) is used to give a description of the absolute Galois group of \(F\). These results are applied to develop abelian class field theory for arithmetic surfaces using \(K_0\)- and \(K_1\)- groups of local and global fields. In the last section the authors give a comparison of their approach with the class field theory of \textit{K. Kato} and \textit{S. Saito} [in: Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 103--152 (1983; Zbl 0544.12011)].
scientific article; zbMATH DE number 2165577
scientific article