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Analytic properties of automorphic L-functions. (English) Zbl 0654.10028

Perspectives in Mathematics, 6. Boston, MA etc.: Academic Press, Inc. vii, 131 p. $ 18.75 (1988).
This report is concerned with the meromorphic continuation and the functional equation of automorphic L-functions. It gives a short and clear survey of the methods used by several mathematicians to study the problem.
The authors start with an exposition of Hecke theory for GL(2). For reductive groups G we have Langland’s method which consists in evaluating the constant term of an Eisenstein series on a reductive group H which contains G as Levi component of a maximal parabolic subgroup. This method has been taken up by Shahidi. He computes Fourier coefficients of Eisenstein series on H. Here Whittaker models are used (i.e. one considers generic automorphic representations). Shahidi even proved that, in the ccases considered, the L-function has only finitely many poles in the plane [see F. Shahidi, Ann. Math., II. Ser. 127, No.3, 547-584 (1988; see the following review Zbl 0654.10029)].
A different approach is the construction of L-functions by means of zeta- integrals (“Tate theory”). The method was developed for GL(n) by Godement and Jacquet and later generalized by Piatetski-Shapiro and Rallis. As this method does not make use of Whittaker models, it can be applied more generally than the other main method.
Several applications and variants of these methods are discussed: integrals of Rankin-Selberg type, the work of Shimura, Hecke theory for GL(n) (in connection with converse theorems), L-functions for GL(n)\(\times GL(m)\) (and \(G\times GL(n))\). Also, concrete examples are treated. The topics are treated in chronological order.
There is a historical comment to be made on the appendix to section (1.2) in chapter II. There Jacquet’s theory for GL(2)\(\times GL(2)\) is discussed. The Eisenstein series used by Jacquet was studied long before by R. Godement [Sémin. Bourbaki 17 (1964/65), No.278, 26 p. (1966; Zbl 0226.20047)].
Reviewer: J.G.M.Mars

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields