On the Ramanujan conjecture and finiteness of poles for certain \(L\)-functions. (English) Zbl 0654.10029

Shahidi, Freydoon

Ann. Math. (2) 127, No. 3, 547-584 (1988).

Let \(G\) be a quasi-split reductive algebraic group over a number field. Let \(P\) be a maximal parabolic subgroup of \(G\) and \(M\) a Levi component of \(P\). Given a cuspidal representation \(\pi\) of \(M\), one constructs an Eisenstein series on \(G\), which depends on one complex parameter s. There is a well-known expression for the constant term of this Eisenstein series, in which occur the (partial) \(L\)-functions attached to \(\pi\) and some finite-dimensional representations \(r_ i\) of the \(L\)-group of \(M\). The author gives a list of the highest weights of the representations \(r_ i\) extending Langland’s original list. Under the assumption that \(\pi\) is generic the author proves that the products defining these \(L\)-functions are absolutely convergent for \(\text{Re}(s)>2\). Under more restrictive assumptions local \(L\)-functions are defined for the ramified places and it is proved that the thus completed \(L\)-function extends to a meromorphic function on \({\mathbb C}\) with only finitely many poles and satisfies a functional equation.
There is an induction argument in the proofs. This induction is possible thanks to the important Lemma 4.2, whose proof involves case by case checking. The method of proof also gives an estimate for the eigenvalues of Hecke operators on generic cusp forms on certain quasi-split groups.

Reviewer: J.G.M.Mars

MSC:

11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

automorphic L-functions; Fourier coefficients of Maass wave forms; meromorphic continuation; quasi-split reductive algebraic group; cuspidal representation; Eisenstein series; constant term; finite-dimensional representations; highest weights; local L-functions; finitely many poles; functional equation; eigenvalues of Hecke operators; generic cusp forms; quasi-split groups