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\(L\)-functions of automorphic forms on simple classical groups. (English) Zbl 0566.10022
Modular forms, Symp. Durham/Engl. 1983, 251-261 (1984).
The authors announce the following result: Let \(G\) be a classical group over a global field \(k\) and \(\pi =\oplus_{\mathfrak p}\pi_{\mathfrak p}\) an automorphic cuspidal representation. Then the Langlands \(L\)-function \(L(\pi,r,s)\) (the function which corresponds to the natural representation \(r\) of the Langlands group \(L_ G)\) has a meromorphic continuation to \(\mathbb C\) with only a finite number of poles.
For these \(L\)-functions R. P. Langlands [Euler products. Yale Mathematical Monographs. 1. New Haven-London: Yale University Press (1971; Zbl 0231.20016)] proved the existence of a meromorphic continuation. In the case of holomorphic modular forms on the Siegel half-plane A. N. Andrianov [Usp. Mat. Nauk 34, No.1 (205), 67–135 (1979; Zbl 0418.10027)] proved the authors’ result with the precise location of poles. In their preprint [\(L\)-functions for the classical groups. 82 pp.] the authors proved their result for the symplectic and orthogonal groups. The authors’ method is based on a generalization of the Rankin-Selberg trick.
[For the entire collection see Zbl 0546.00010.]

MSC:
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms