Langlands, R. P. Problems in the theory of automorphic forms. (English) Zbl 0225.14022 Lectures in modern analysis and applications. III, Lect. Notes Math. 170, 18-61 (1970). Let \(G\) be a connected reductive algebraic group defined over a global field \(F\). Let \(\mathbb A(F)\) be the adele ring of \(F\). \(G_{\mathbb A(F)}\) is a locally compact topological group with \(G_F\) as a discrete subgroup. The group \(G_{\mathbb A(F)}\) acts on \(L^2(G_F\backslash G_{\mathbb A(F)})\). Let \(\pi\) be an irreducible representation of \(G_{\mathbb A(F)}\) which occurs in \(L^2(G_F\backslash G_{\mathbb A(F)})\). To a given \(G\) the author introduces a complex analytic group \(\hat G_F\) and to each complex analytic representation \(\sigma\) of \(\hat G_F\) and each \(\pi\) he attaches an \(L\)-function \(L(s,\sigma,\pi)\) defined by an Euler product of the local \(L\)-functions at “unramified” primes of \(F\). Under some natural assumptions the author proves that the Euler product converges in a half-plane.The author’s problems are mainly concerned with some fundamental properties of the \(L\)-functions:– Are the \(L\)-functions meromorphic in the entire complex plane with only a finite number of poles and do they satisfy the functional equation of the usual form?– Are there relations between the \(L\)-functions of different \(G\)?– Is there a relation of the \(L\)-functions to the \(L\)-functions associated to non-singular algebraic varieties (especially for \(G= \mathrm{GL}(2)\) and elliptic curves)?The problems are posed in some reasonable precise manner. Some remarks are made about the cases where some of these problems are proved or may be proved \((G= \mathrm{GL}(1). \mathrm{GL}(2))\).For the entire collection see [Zbl 0213.00101]. Reviewer: A. N. Andrianov Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 10 ReviewsCited in 88 Documents MSC: 11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations 11F12 Automorphic forms, one variable 11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Citations:Zbl 0225.14023; Zbl 0213.00101 PDFBibTeX XML Full Text: DOI