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A. N. Parshin

Publications and source records attributed to A. N. Parshin.

15 recordsLinked to original sources
arXiv2020

Harmonic analysis on the rank-$2$ value group of a two-dimensional local field

In this work we construct harmonic analysis on free Abelian groups of rank $2$, namely: we construct and investigate spaces of functions and distributions, Fourier transforms, actions of discrete and extended discrete Heisenberg groups. In case of the rank-$2$ value group of a two-dimensional local field with finite last residue field we connect this harmonic analysis with harmonic analysis on the two-dimensional local field, where the latter harmonic analysis was constructed in earlier works by the authors.

math.NT↗
arXiv2013

Questions and Remarks to the Langlands Program

A brief survey is given of the classical Langlands correspondence between n-dimensional representations of Galois groups of local and global fields of dimension 1 and irreducible representations of the groups GL(n). A generalization of the Langlands program to fields of dimension 2 is considered and the corresponding version for 1-dimensional representations is described. We formulate a conjecture on a direct image (=automorphic induction) of automorphic forms which links the Langlands correspondences in dimension 2 and 1. The direct image conjecture implies the classical Hasse-Weil conjecture on the analytical behaviour of the L-functions of curves defined over global fields of dimension 1.

math.NT↗
arXiv2011

Harmonic analysis and the Riemann-Roch theorem

This paper is a continuation of papers: arXiv:0707.1766 [math.AG] and arXiv:0912.1577 [math.AG]. Using the two-dimensional Poisson formulas from these papers and two-dimensional adelic theory we obtain the Riemann-Roch formula on a projective smooth algebraic surface over a finite field.

math.AG↗
arXiv2011

Representations of Higher Adelic Groups and Arithmetic

We discuss the following topics: n-dimensional local fields and adelic groups; harmonic analysis on local fields and adelic groups for two-dimensional schemes (function spaces, Fourier transform, Poisson formula); representations of discrete Heisenberg groups; adelic Heisenberg groups and their representations arising from two-dimensional schemes.

math.NT↗
arXiv2010

Harmonic analysis on local fields and adelic spaces II

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.

math.AG↗
arXiv2010

Notes on the Poisson formula

These notes are a part of my lectures on representations of adelic groups attached to two-dimensional schemes. They contain a study of the one-dimensional case as a preliminary step to the case of dimension two. We consider the following issues: the Tate--Iwasawa method for algebraic curves; a discrete version and holomorphic duality; the Poisson formula and residues; explicit formulas; relation with the Artin representation; analogues for the number fields. With appendix on the Dedekind zeta-functions by Irina Rezvjakova.

math.NT↗
arXiv2009

Numbers as functions: the development of an idea in the Moscow school of algebraic geometry

This is expanded text of a lecture delivered by the author at the conference "Matériaux pour l'Histoire des Mathématiques au XXème siècle", which took place in Nice in January 1996. The task was to describe one area in the development of arithmetical algebraic geometry in Moscow during the 1950s and 1960s. We shall begin by explaining the meaning of the analogy between numbers and functions, starting with the simplest concepts. In the second part we study a nontrivial example: the explicit formula for the law of reciprocity. In the third part we shall become acquainted with certain aspects of the "social" life of the Moscow school, in particular, with certain seminars, lectures, and books. In the final part we shall examine another example of this analogy: arithmetical surfaces and Arakelov theory.

math.AG↗
arXiv2009

Integrable systems and local fields

In 70's there was discovered a construction how to attach to some algebraic-geometric data an infinite-dimensional subspace in the space k((z)) of the Laurent power series. The construction is known as the Krichever correspondence. It was applied in the theory of integrable systems, particularly, for the KP and KdV equations. We show that the Krichever construction can be generalized to the case of dimension 2. We also include a known description of connection between the KP hierarchy in the Lax form and the vector fields on infinite Grassmanian manifolds and a construction of the semi-infinite monomes for the field k((z)) which is an important part of the theory of Sato Grassmanian. The text was published in Communications in Algebra, 29(2001), No.9, 4157-4181. This version includes a corrected proof of the proposition 2. Also, we include some additional remarks on the deduction of concrete equations from the Lax hierarchy and appendix 2.

math.AG↗
arXiv2008

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.

math.AG↗
arXiv2000

Invitation to higher local fields, Part II, section 1: Higher dimensional local fields and L-functions

This work describes several first steps in extending Tate-Iwasawa's analytic method to define an L-function in higher dimensions. For generalizing this method the author advocates the usefulness of the classical Riemann-Hecke approach, his adelic complexes together with his generalization of Krichever's correspondence. He analyzes dimension 1 types of functions and discusses properties of the lattice of commensurable classes of subspaces in the adelic space associated to a divisor on an algebraic surface.

math.NT↗
arXiv1999

Krichever Correspondence for Algebraic Surfaces

In 70's there was discovered a construction how to attach to some algebraic-geometric data an infinite-dimensional subspace in the space k((z)) of the Laurent power series. The construction was successfully used in the theory of integrable systems, particularly, for the KP and KdV equations. There were also found some applications to the moduli of algebraic curves. But there remained a hard restriction by the case of curves, so by dimension 1. Recently, it was pointed out by the author that there are some connections between the theory of the KP-equations and the theory of n-dimensional local fields. From this point of view it becomes clear that the Krichever construction should have a generalization to the case of higher dimensions. This generalization is suggested in the paper for the case of algebraic surfaces.

math.AG↗
arXiv1999

On a Ring of Formal Pseudo-differential Operators

We study the notion of non-commumative higher dimensional local fields. A simplest example is the ring P of formal pseudo- differential operators. As an application we extend the KP hierarchy to the space $P^n$.

math.AG↗
arXiv1996

Vector Bundles and Arithmetical Groups I. The higher Bruhat-Tits tree

We define and study a simplicial complex which is a homogeneous space for the group $PGL(2, K)$ over a two-dimensional local field $K$. The complex is a generalization of the tree studied by F. Bruhat, J. Tits, J.-P. Serre and P. Cartier in the 60's and early 70's. Such complex can be canonically attached to the triples $x \in C \subset X$ where $X$ is an algebraic surface, $C$ is an irreducible curve and $x$ is a smooth point on $C$ and $X$. This construction can be used for a description of the isomorphism set of vector bundles on $X$.

alg-geom↗