On the Ramanujan Conjecture and Finiteness of Poles for Certain L- Functions

Abstract

In this paper we prove a number of general results about automorphic L-functions which appear in the constant terms of Eisenstein series for an arbitrary quasi-split connected reductive algebraic group over a number field. These L-functions were first introduced by Langlands for Chevalley groups [20] and through them he was led to make some of his deep conjectures [21]. One significance of these L-functions is that all the automorphic L-functions studied so far are among them. There are three general results. First, we obtain a uniform line of absolute convergence for all of these L-functions (Theorem 5.1). A uniform estimate for the Hecke eigenvalues of generic cusp forms on many absolutely simple quasi-split connected reductive algebraic groups over number fields (Corollary 5.4) and an improvement on the best available estimate for the Fourier coefficients of Maass wave forms (Corollary 5.5) also follow. Next, we establish a meromorphic continuation and functional equation for each of these L-functions (Theorem 6.1). Finally, in Theorem 6.2, we prove finiteness of poles on the whole complex plane for an important class of these L-functions (Corollaries 6.6 through 6.10). More precisely, let G be a quasi-split connected reductive algebraic group over a number field F. Set G = G(A F)' where A F is the ring of adeles of F. Fix a Borel subgroup B of G over F and let U be its unipotent radical. Let P be a maximal F-parabolic subgroup of G with P D B. In the context of the problems studied here, nothing new will be obtained if one drops the maximality condition on P. Write P = MN, a Levi decomposition, N C U. and let B, U, P, M, and N be the corresponding groups of adelic points. For every place v of F. let GV = G(FJ). Similarly we have Bv, Us, Pv, Me, and Nv. Let X = ?~ Xv be a generic character of U(F) U (cf. Section 3). Then each Xv is generic. Let r = Ov 7Tv be a cusp form on M. We shall say 7T is