Higher Dimensional Class Field Theory Research Articles

Bloch’s formula for 0-cycles with modulus and higher-dimensional class field theory

We prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse Q ¯ ℓ \overline {\mathbb {Q}}_{\ell } -sheaves. This was originally solved by Kerz and Saito in characteristic ≠ 2 \neq 2 .

Iterated Laurent series over rings and the Contou-Carrère symbol

This article contains a survey of a new algebro-geometric approach for working with iterated algebraic loop groups associated with iterated Laurent series over arbitrary commutative rings and its applications to the study of the higher-dimensional Contou-Carrère symbol. In addition to the survey, the article also contains new results related to this symbol. The higher-dimensional Contou-Carrère symbol arises naturally when one considers deformation of a flag of algebraic subvarieties of an algebraic variety. The non-triviality of the problem is due to the fact that, in the case , for the group of invertible elements of the algebra of -iterated Laurent series over a ring, no representation is known in the form of an ind-flat scheme over this ring. Therefore, essentially new algebro-geometric constructions, notions, and methods are required. As an application of the new methods used, a description of continuous homomorphisms between algebras of iterated Laurent series over a ring is given, and an invertibility criterion for such endomorphisms is found. It is shown that the higher- dimensional Contou-Carrère symbol, restricted to algebras over the field of rational numbers, is given by a natural explicit formula, and this symbol extends uniquely to all rings. An explicit formula is also given for the higher-dimensional Contou-Carrère symbol in the case of all rings. The connection with higher-dimensional class field theory is described. As a new result, it is shown that the higher-dimensional Contou-Carrère symbol has a universal property. Namely, if one fixes a torsion-free ring and considers a flat group scheme over this ring such that any two points of the scheme are contained in an affine open subset, then after restricting to algebras over the fixed ring, all morphisms from the -iterated algebraic loop group of the Milnor -group of degree to the above group scheme factor through the higher-dimensional Contou-Carrère symbol. Bibliography: 67 titles.

REFINED SWAN CONDUCTORS OF ONE-DIMENSIONAL GALOIS REPRESENTATIONS

For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.

HIGHER IDELES AND CLASS FIELD THEORY

We use higher ideles and duality theorems to develop a universal approach to higher dimensional class field theory.

Chow group of 0-cycles with modulus and higher-dimensional class field theory

One of the main results of this paper is a proof of the rank one case of an existence conjecture on lisse l-adic sheaves on a smooth variety over a finite field due to Deligne and Drinfeld. The problem is translated into the language of higher dimensional class field theory over finite fields, which describes the abelian fundamental group by Chow groups of zero cycles with moduli. A key ingredient is the construction of a cycle theoretic avatar of refined Artin conductor in ramification theory originally studied by Kazuya Kato.

The status quo of higher dimensional class field theory

The status quo of higher dimensional class field theory

Some consequences of Wiesend’s higher dimensional class field theory

We use a result of G. Wiesend to establish the relation between the integral singular homology in degree zero and the abelianized tame fundamental group of a regular, connected scheme of finite type over Spec\((\mathbb{Z})\) .

Characteristic classes and multidimensional reciprocity laws

In this note, we formulate and prove a non-commutative generalization of the reciprocity laws of Parshin and Kato in higher dimensional class field theory [Kat][P1][P4]. This can be regarded as a geometric realization of the reciprocity laws implicit in the Gersten-Quillen complex in algebraic K-theory [Q]. We expect this work to be related to the “higher dimensional Langlands program”, which has yet to be formulated precisely (although see [Kap]). Let X be an k-dimensional complex analytic space (possibly singular). Let F := (C0 ⊂ C1 ⊂ · · · ⊂ Ck = X) be a flag of irreducible subspaces in X, with dim Ci = i (we often omit Ck from the notation). For any analytic open set V ⊂ X, we constructed in [Br-M1] a homology class κF,V ∈ Hk(V ;Z) and showed that it satisfies the homological reciprocity law described in Theorem 2.1. Let ĉp denote the characteristic class of Deligne-Beilinson [Be] which refines the usual pth Chern class cp. In section 3, we define the non-commutative symbol F,V at the “place” F , by pulling back ĉk+1 via the evaluation map BGL(n,O(V )) × V → BGL(n,C) and taking the “slant-product” of the resulting class with κF,V . Here O(V ) denotes the algebra of holomorphic functions on V and we are viewing GL(n,O(V )) as a discrete group. It is important to note that the non-commutative symbol is not just a number, rather it is the cohomology class in Hk+1(BGL(n,O(V )),C∗) of a degree k + 1 group cocycle on GL(n,O(V )). Since the homology class κF,V satisfies a reciprocity law, we obtain