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Uspekhi Mat. Nauk, 2012, Volume 67, Issue 3(405), Pages 115–146 (Mi umn9479)  

This article is cited in 3 scientific papers (total in 4 papers)

Questions and remarks to the Langlands programme

A. N. Parshin

Steklov Mathematical Institute, Russian Academy of Sciences

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


DOI: https://doi.org/10.4213/rm9479

Full text: PDF file (804 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2012, 67:3, 509–539

Bibliographic databases:

Document Type: Article
UDC: 511.68+512.626
MSC: Primary 11F70, 11R39, 11S37; Secondary 22E50
Received: 30.12.2011

Citation: A. N. Parshin, “Questions and remarks to the Langlands programme”, Uspekhi Mat. Nauk, 67:3(405) (2012), 115–146; Russian Math. Surveys, 67:3 (2012), 509–539

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    This publication is cited in the following articles:
    1. S. V. Vostokov, S. O. Gorchinskiy, A. B. Zheglov, Yu. G. Zarkhin, Yu. V. Nesterenko, D. O. Orlov, D. V. Osipov, V. L. Popov, A. G. Sergeev, I. R. Shafarevich, “Aleksei Nikolaevich Parshin (on his 70th birthday)”, Russian Math. Surveys, 68:1 (2013), 189–197  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. D. V. Osipov, “The unramified two-dimensional Langlands correspondence”, Izv. Math., 77:4 (2013), 714–741  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. A. N. Parshin, “On the direct image conjecture in the relative Langlands programme”, Russian Math. Surveys, 70:5 (2015), 961–963  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. D. V. Osipov, “Adelic quotient group for algebraic surfaces”, St. Petersburg Math. J., 30 (2019), 111–122  mathnet  crossref  isi  elib
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