Langlands, R. P. Representation theory and arithmetic. (English) Zbl 0655.10030 The mathematical heritage of Hermann Weyl, Proc. Symp., Durham/NC 1987, Proc. Symp. Pure Math. 48, 25-33 (1988). [For the entire collection see Zbl 0644.00001.] This paper is a report on recent progress made in the theory of Shimura varieties. The central problem is the expression of the Hasse-Weil zeta function of a Shimura variety in terms of automorphic L-functions. The author formulates this in terms of a trace formula. The crux of proving a formula of the expected form has recently been overcome by R. Kottwitz and the author is optimistic that positive, arithmetically useful results will soon be available. Reviewer: S.J.Patterson Cited in 1 Document MSC: 11F70 Representation-theoretic methods; automorphic representations over local and global fields 14K15 Arithmetic ground fields for abelian varieties 11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols Keywords:Hasse-Weil zeta function; Shimura variety; automorphic L-functions; trace formula Citations:Zbl 0644.00001 PDFBibTeX XML