Parshin, A. N. Representations of higher adelic groups and arithmetic. (English) Zbl 1266.11118 Bhatia, Rajendra (ed.) et al., Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. I: Plenary lectures and ceremonies. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency (ISBN 978-981-4324-30-4/set; 978-81-85931-08-3/hbk; 978-981-4324-31-1/hbk; 978-981-4324-35-9/ebook). 362-392 (2011). The paper under review discusses various problems and results concerning higher dimensional local fields and adelic groups. The main topics include: 1) measure theory and harmonic analysis on higher local fields and adelic groups, 2) extending of Tate-Iwasawa’s analytic method to higher dimensions, 3) behaviour of zeta- and \(L\)-functions,4) representations of discrete and adelic Heisenberg groups. The paper also includes a rich bibliography on these subjects.For the entire collection see [Zbl 1220.00031]. Reviewer: V. Popa (Chişinău) Cited in 2 ReviewsCited in 10 Documents MSC: 11R56 Adèle rings and groups 11S31 Class field theory; \(p\)-adic formal groups Keywords:n-dimensional local field; adelic group; harmonic analysis; discrete Heisenberg group PDFBibTeX XMLCite \textit{A. N. Parshin}, in: Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19--27, 2010. Vol. I: Plenary lectures and ceremonies. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency. 362--392 (2011; Zbl 1266.11118) Full Text: arXiv Link