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Covering Data and Higher Dimensional Global Class Field Theory. arXive: math/0804.3419
"... Abstract: For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism ρX: CX → πab 1 (X), which is surjective and whose kernel is the connected component of the identity. The (topological) group CX is explicitly given and built solely out of data a ..."
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Abstract: For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism ρX: CX → πab 1 (X), which is surjective and whose kernel is the connected component of the identity. The (topological) group CX is explicitly given and built solely out of data attached to points and curves on X. A similar but weaker statement holds for smooth varieties over finite fields. Our results are based on earlier work of G. Wiesend. To the memory of Götz Wiesend 1 The aim of global class field theory is the description of abelian extensions of arithmetic schemes (i.e. regular schemes X of finite type over Spec(Z)) in terms of arithmetic invariants attached to X. The solution of this problem in the case dim X = 1 was one of the major achievements of number theory in the first part of the previous century. In the 1980s, mainly due to K. Kato and S. Saito [8], a generalization to higher dimensional schemes has been found. The description of
Higher class field theory and the connected component
, 2008
"... Abstract. In this note we present a new selfcontained approach to the class field theory of arithmetic schemes in the sense of Wiesend. Along the way we prove new results on space filling curves on arithmetic schemes and on the class field theory of local rings. We show how one can deduce the more ..."
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Abstract. In this note we present a new selfcontained approach to the class field theory of arithmetic schemes in the sense of Wiesend. Along the way we prove new results on space filling curves on arithmetic schemes and on the class field theory of local rings. We show how one can deduce the more classical version of higher global class field theory due to Kato and Saito from Wiesend’s version. One of our new result says that the connected component of the identity element in Wiesend’s class group is divisible if some obstruction is absent.
Regular singular stratified bundles and tame ramification, arXiv:1210.5077
, 2012
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Tame Class Field Theory for Singular Varieties over Finite Fields
, 2014
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Communicated by Takeshi Saito
, 2013
"... Abstract. Let X be a smooth, connected, projectivevariety overan ..."
Independence of `adic representations of geometric Galois groups
, 2013
"... Let k be an algebraically closed field of arbitrary characteristic, let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime `, the absolute Galois group of K acts on the `adic etale cohomology modules of X. We prove that this family of r ..."
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Let k be an algebraically closed field of arbitrary characteristic, let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime `, the absolute Galois group of K acts on the `adic etale cohomology modules of X. We prove that this family of representations varying over ` is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ` become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations. 1