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Eisenstein Series and Automorphic L-Functions

Freydoon Shahidi
American Mathematical Society
Publication Date: 
Number of Pages: 
Colloquium Publications 58
[Reviewed by
Michael Berg
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Written by one of the main players in the game, Eisenstein Series and Automorphic L-functions is nigh on indispensable for anyone wishing to get into this beautiful and exciting part of number theory.

A glance at the table of contents of Freydoon Shahidi’s book shows that it is something of a pedagogical dream: it seems to cover just about everything one might want to learn about the interface between automorphic representations and L-functions — a theme of huge importance in number theory in the last 50 years. Of course, there is more to the subject than can be covered in the book’s 200 pages, but Shahidi touches on a great number of critically important topics, arranges them extremely efficiently, and then expounds them brilliantly. Additionally, as the book’s back-cover explains, the Langlands-Shahidi method, addressed in the text under review,

when combined with the converse theorems of Cogdell and Piatetski-Shapiro, has been quite successful in establishing a number of new cases of Langlands functoriality… These results have led to far-reaching new estimates for Hecke eigenvalues of Mass forms, as well as definitive solutions to certain problems in analytic and algebraic number theory.

This is accordingly a very significant contribution to the literature, particularly as far as properly educating fledgling automorphic formers is concerned.

Ostensibly addressed to rookies, i.e. graduate students, and “researchers who are interested in the Langlands program … and its connections with number theory,” Eisenstein Series and Automorphic L-functions starts off with a discussion of reductive groups, as of course is proper. By way of an example of his attention to detail it is worth noting that Shahidi makes very sure that adèlization is properly taken care of as early as p. 11. This is a welcome feature, seeing that this technique often appears as something of a curiosity in other writings only to appear later as being of critical importance. Shahidi discusses the matter at a pretty austere level, to be sure, referring to the indicated books by Tonny Springer, Jacques Tits, and André Weil for further elucidation, but the careful reader can get it all from the text under review.

It is not the case, however, that the proper audience for this book should include raw recruits: the reader had better be on the ball as far as his background work for number theory. Regarding the aforementioned adèles, for example, Shahidi races through material on local fields, parabolic subgroups, Iwasawa decomposition, and so on, in the first chapter. It is strongly indicated that the reader should have read, e.g., Serre’s Corps Locaux and Borel’s Linear Algebraic Groups. (By the way, given the book’s appendix on Dynkin diagrams, Lie groups would also be highly desirable as part of the reader’s tool-kit.)

So, the graduate student coming to this book should beware of what he is in for. Modulo proper preparation on the reader’s part, however, or his willingness to go slowly and do some supplementary reading, Eisenstein Series and Automorphic L-functions can serve as a springboard to working on some very evocative avant garde stuff.

Eisenstein Series and Automorphic L-functions starts off with a discussion of one of the mainstays of the Langlands program, the use of Eisenstein series vis à vis the determination of the continuous spectrum of the underlying group “as well as all the non-cuspidal representations appearing discretely in the spectrum as residues of these series, which are meromorphic functions of several variables.” Shahidi goes on to say that this subject “has its roots in the classical Eisenstein series on the [complex] upper half plane... [and] covers both the holomorphic and real analytic cases ... and mainly builds modular forms on the corresponding Riemannian symmetric spaces which account for the continuous part of their spectrum.” This lays out in a very compact way much of what the theory is all about and sets the stage for what follows.

Shahidi subsequently appends a very nice sketch of what happens in the case of the all-important group SL(2, Z), the paradigm par excellence, finishing with the observation that with L(s) being the completed L-function for Riemann’s zeta function,

the non-constant Fourier coefficients of [the attendant Eisenstein series] are basically ... the inverse ... of the L-function L(2s) which appears and controls the constant term via the quotient L(2s-1)/L(2s). We note that the poles of [said Eisenstein series] as a function of s are basically controlled by those of L(2s-1) and are precisely those of the constant term.

And then:

This phenomenon is quite general and in fact constant terms of general Eisenstein series are ratios of L-functions, while their non-constant Whittaker Fourier coefficients are inverses of these L-functions and here lies the point of inception of this book.

And not too long after that it’s off to the races. Satake in Chapter 2, Whittaker in Chapter 3, then material on intertwining operators and on local coefficients, Eisenstein in Chapter 6, Fourier coefficients in Chapter 7, and the, in the last three chapters, functional equations, more on L-functions, and finally applications to functoriality. Under the last heading we encounter Rankin-Selberg L-functions, a converse theorem, and Ramanujan-Selberg estimates for Maas forms. What a wealth of deep mathematics!

Finally, Shahidi provides the reader with a wonderful bibliography. I cannot imagine a better trajectory toward learning this material than to go through the pages of Eisenstein Series and Automorphic L-functions with a lot of note paper handy and a comfortable writing pen, to follow up on Shahidi’s numerous cross-references when so indicated (and to the right degree, so to speak), and of course to fill out the sketches and do the exercises that the author presents. This very beautiful mathematics is well worth the effort.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.