Parshin, A. N. Higher dimensional local fields and \(L\)-functions. (English) Zbl 1008.11060 Fesenko, Ivan (ed.) et al., Invitation to higher local fields. Extended version of talks given at the conference on higher local fields, Münster, Germany, August 29-September 5, 1999. Coventry: Geometry and Topology Publications, Geom. Topol. Monogr. 3, 199-213 (2000). Summary: 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.For the entire collection see [Zbl 0954.00026]. Cited in 1 ReviewCited in 8 Documents MSC: 11S40 Zeta functions and \(L\)-functions 11R56 Adèle rings and groups 11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) Keywords:Tate-Iwasawa’s analytic method; \(L\)-function in higher dimensions PDFBibTeX XMLCite \textit{A. N. Parshin}, Geom. Topol. Monogr. 3, 199--213 (2000; Zbl 1008.11060) Full Text: EMIS