Parshin, A. N. Questions and remarks to the Langlands programme. (English. Russian original) Zbl 1318.22010 Russ. Math. Surv. 67, No. 3, 509-539 (2012); translation from Usp. Mat. Nauk 67, No. 3, 115-146 (2012). The author explores the possible existence of Langlands correspondence over 2-dimensional fields and the functoriality phenomena implied by the existence of such a correspondence. After a brief review on the classical Langlands correspondence, the author gives a rough form of such a correspondence over 2-dimensional fields, using the language of higher adeles rings and higher representations. He then goes on to discuss the functoriality given by push-forward of 1-dimensional local systems from schemes of dimension 2 to schemes of dimension 1, which he calls the “direct image conjecture”. He also outlines a proof of this conjecture when both schemes are defined over finite fields, using the Langlands correspondence for function fields of one variable established by Laurent Lafforgue. Finally, for the special case of the structure map of an arithmetic surface, the author explains the relation between the direct image conjecture and the Hasse-Weil conjecture. Reviewer: Zongbin Chen (Lausanne) Cited in 1 ReviewCited in 3 Documents MSC: 22E57 Geometric Langlands program: representation-theoretic aspects 11R39 Langlands-Weil conjectures, nonabelian class field theory 11S37 Langlands-Weil conjectures, nonabelian class field theory 11R70 \(K\)-theory of global fields 14D24 Geometric Langlands program (algebro-geometric aspects) Keywords:Langlands correspondence; 2-dimensional class field theory; functoriality; higher adeles ring; higher \(K\)-group PDFBibTeX XMLCite \textit{A. N. Parshin}, Russ. Math. Surv. 67, No. 3, 509--539 (2012; Zbl 1318.22010); translation from Usp. Mat. Nauk 67, No. 3, 115--146 (2012) Full Text: DOI arXiv