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Russian Mathematical Surveys, 2012, Volume 67, Issue 3, Pages 509–539
DOI: https://doi.org/10.1070/RM2012v067n03ABEH004795
(Mi rm9479)
 

This article is cited in 4 scientific papers (total in 5 papers)

Questions and remarks to the Langlands programme

A. N. Parshin

Steklov Mathematical Institute, Russian Academy of Sciences
References:
Abstract: A brief survey is given of the classical Langlands programme to construct a correspondence between $n$-dimensional representations of Galois groups of local and global fields of dimension 1 and irreducible representations of the groups $\operatorname{GL}(n)$ connected with these fields and their adelic rings. A generalization of the Langlands programme to fields of dimension 2 is considered and the corresponding version for 1-dimensional representations is described. A conjecture on the direct image of automorphic forms is stated which links the Langlands correspondences in dimensions 2 and 1. In the geometric case of surfaces over a finite field the conjecture is shown to follow from Lafforgue's theorem on the existence of a global Langlands correspondence for curves. The direct image conjecture also implies the classical Hasse–Weil conjecture on the analytic behaviour of the zeta- and $L$-functions of curves defined over global fields of dimension 1.
Bibliography: 57 titles.
Keywords: Langlands correspondence, automorphic forms, $L$-functions, two-dimensional local fields, adeles, $K$-groups, class field theory, direct images.
Funding agency Grant number
Russian Foundation for Basic Research 11-01-00145-а
11-01-12098-офи-м
Ministry of Education and Science of the Russian Federation НШ-5139.2012.1
Received: 30.12.2011
Bibliographic databases:
Document Type: Article
UDC: 511.68+512.626
MSC: Primary 11F70, 11R39, 11S37; Secondary 22E50
Language: English
Original paper language: Russian
Citation: A. N. Parshin, “Questions and remarks to the Langlands programme”, Russian Math. Surveys, 67:3 (2012), 509–539
Citation in format AMSBIB
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\by A.~N.~Parshin
\paper Questions and remarks to the Langlands programme
\jour Russian Math. Surveys
\yr 2012
\vol 67
\issue 3
\pages 509--539
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  • https://doi.org/10.1070/RM2012v067n03ABEH004795
  • https://www.mathnet.ru/eng/rm/v67/i3/p115
  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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