Invitation to higher local fields, Part II, section 2: Adelic constructions for direct images of differentials and symbols

A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an algebraic proof of the 0-dimensional case of Grothendieck's anabelian conjecture (proven by Pop), which says that finitely generated infinite fields are determined up to purely inseparable extensions by their absolute Galois groups. As a secon...

This paper is the second part of arXiv:0707.1766. We develope harmonic analysis in some categories of filtered abelian groups and vector spaces over the fields R or C. These categories contain as objects local fields and adelic spaces arising from arithmetical surfaces. Some structure theorems are proven for quotients of the adelic groups of algebraic and arithmetical surfaces.

In this preprint our results obtained in arXiv.2304.09465v1 (Main Results) are extended to arbitrary semi-local regular base (with no restrictions on its residue fields).

This is a review of Parshin's higher local class field theory in characteristic p.

We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann-Roch formula as in Kapranov-Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P^1. Our set-up provides a...

The aim of this work is to provide a construction of generalized local symbols on algebraic curves as morphisms of group schemes. From a closed point of a complete, irreducible and non-singular curve $C$ over a perfect field $k$ as the only data, using theta groups over Picard schemes of curves, we offer a geometric construction that allows us to define generalizations of the tame symbol and the Hilbert norm residue symbol.

Notes for a course at the H.-C. R. I., Allahabad, 15 August 2008 -- 26 January 2009

A mixture of an historical article, and of a survey of recent developments, containing also a couple of new results.

This article is part introduction and part survey to the mathematical area centered around local cohomology.

This work introduces author's approach to harmonic analysis on algebraic groups over functional two-dimensional local fields. For a two-dimensional local field a Hecke algebra which is formed by operators which integrate pro-locally-constant complex functions over a non-compact domain is defined and its properties are described.