A Categorical Proof of the Parshin Reciprocity Laws on Algebraic Surfaces

The Two-dimensional Contou-carrère Symbol and Reciprocity Laws

We define a two-dimensional Contou-Carrère symbol, which is a deformation of the two-dimensional tame symbol and is a natural generalization of the (usual) one-dimensional Contou-Carrère symbol. We give several constructions of this symbol and investigate its properties. Using higher categorical methods, we prove reciprocity laws on algebraic surfaces for this symbol. We also relate the two-dim...

Logarithmic Functional and the Weil Reciprocity Law

This article is devoted to the (one-dimensional) logarithmic functional with argument being one-dimensional cycle in the group (C∗)2. This functional generalizes usual logarithm, that can be viewed as zero-dimensional logarithmic functional. Logarithmic functional inherits multiplicative property of the logarithm. It generalizes the functional introduced by Beilinson for topological proof of th...

Central Extensions and Reciprocity Laws on Algebraic Surfaces

Geometric Theory of Parshin Residues

This is a brief summary of results. More detailed papers are in preparation. Preliminary versions of the detailed papers are available on the arxiv.org ([MM1],[MM2]). We study the theory of Parshin residues from the geometric point of view. In particular, the residue is expressed in terms of an integral over a smooth cycle. Parshin-Lomadze Reciprocity Law for residues in complex case is proved ...

Harmonic analysis on local fields and adelic spaces I

We develop a harmonic analysis on objects of some category C 2 of infinite-dimensional filtered vector spaces over a finite field. It includes two-dimensional local fields and adelic spaces of algebraic surfaces defined over a finite field. The main result is the theory of the Fourier transform on these objects and two-dimensional Poisson formulas.