# zbMATH — the first resource for mathematics

$$L$$-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