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Modern Analysis of Automorphic Forms by Example, Two Volumes

Paul Garrett
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Studies in Advanced Mathematics 173
[Reviewed by
Michael Berg
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Some time ago (last academic year, in fact), I had occasion to teach a senior seminar in my department, and I chose modular forms as my subject; more broadly, automorphic forms and functions were also on the agenda.  The course was a qualified success, or not, depending on your point of view. A few students, including a particularly strong one (now in graduate school in the UK), both enjoyed the course, and did well; however, at the other end of the bell curve a few students floundered rather dramatically, in part because they disinvested early on and then failed to regroup.  My collusion in all this had to do with what, in retrospect, were rather poor text choices: I used an old book by the late Marvin Knopp, and, as a supplement, part of Serre’s unsurpassed but austere, Course of Arithmetic.  It was my old undergraduate professor, the late Basil Gordon, who once characterized Serre’s book as austere (his words), and so it is, despite its great elegance. And then Knopp’s book, while a classic, and presenting the subject from the workbench of a master craftsman, so to speak, proved altogether inappropriate for all but two of my seniors: they simply weren’t up to the intensive labor called, for, specifically the blood, sweat, and tears, of developing the subject in Knopp’s old-fashioned primarily complex analytic way.  In choosing the book I had made the mistake on focusing on the structure and landscape of the subject, and quickly ran into time trouble in my lectures. So, yes, my collusion with my students, as far as the difficulties they had, must be counted as non-trivial. Mea culpa.

But hindsight is 20-20, as the saying goes, and so I can say that if I had it to do over again, I would have used another approach altogether.  And this is where the books under review enter the discussion. Paul Garrett is a serious modular former, whose work I have been aware of for quite a while; to wit, he wrote an excellent book on Hilbert modular forms back in the early 1990s (here’s a BAMS review: ), covering some very useful and important more general stuff, too.  So it’s exciting to have, now, the two books, Modern Analysis of Automorphic Forms by Example I,II, in the game, by the same author.  And, yes, I think that the best way to present this material, central to modern number theory, is by means of well-chosen examples, amenable to proper generalization.  After all, even the Riemann zeta function, clearly the biggest silver-back gorilla in the zoo, is an example of, e.g. and L-function.  But after that, the generalizations are indispensable, and the ultimate goal must be to build a coherent theory, in all its glory.  Well, Garrett does all that.

 Specifically, Part I starts off with “Four Small Examples” that carry the germ of some magnificent and very deep more general stuff: of course they’re all about sundry special linear groups (and a symplectic one), and Garrett wastes no time in using this raw material to hit, e.g., Cartan and Iwasawa decompositions, invariant Laplacians, Eisenstein series, and even some spectral theory.  He continues with this philosophy in his second chapter where, at least to my rather older eyes, he gets pretty modern, pretty fast: ideles, adeles, and the according general linear groups (always of dimension 2) appear, and Eisenstein series stay on the stage, now joined by such things as Maass-Selberg relations, and, yes, more spectral theory. And on and on it goes.

 It is marvelous to see how Garrett goes about presenting such deep and broad material in what is certainly a superbly holistic manner.  It’s really a wonderful example of what I think is the right pedagogy for this part of number theory. The examples he uses are lynchpins for an increasingly elaborate development of the subject, and the reader has a number of accessible places to hang his hat as the story unfolds.  In order to get a more specific idea of what Garrett is up to, what his more explicit goals are, as well as his a pretty detailed preview of the ensuing so-many pages, consult his “Introduction and Historical Notes,” available in both parts of the set.

It is worth stressing that Garrett indeed covers a vast landscape, worthy of a number of courses in sequence.  The diligent student comes out at the other end as a serious and able modular or automorphic former, schooled in the thick of what this business is about --- in the modern sense.  Thus, to return briefly to my earlier comment about my recent senior seminar, I guess that in that setting I’d have to restrict myself to Chapter 1 of Part I. But what beautiful material Garrett already covers in these fifty-some pages.  After this, the call has to made whether the kids are ready for non-archimedean valuations, ideles, adeles, &c. I guess a very strong honors class might be the right place. It’s really on to graduate school at this stage.

But, to be fair, what Garrett offers in these two well-written and very scholarly works (it’s really one big whole, cut in two, of course) is far, far more than a textbook, even for graduate students.  In point of fact, these fabulous books qualify as a very good source for all the indicated material for anyone about to do --- or already doing --- research in this field. When I young, taking my first seminar in this subject, my professor, V. S. Varadarajan, used the now-classic book, Introduction to the Arithmetic Theory of Automorphic Functions, by Goro Shimura.  I think that Garrett’s books need to take their place on the shelf right next to this book, and all three of these books should be rife with dog-eared pages --- as also other books on the same shelf: Hecke, Serre, Gelfand-Graev-Piatestkii-Shapiro, and so on.

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

1. Four small examples
2. The quotient Z+GL2(k)/GL2(A)
3. SL3(Z), SL5(Z)
4. Invariant differential operators
5. Integration on quotients
6. Action of G on function spaces on G
7. Discrete decomposition of cuspforms
8. Moderate growth functions, theory of the constant term
9. Unbounded operators on Hilbert spaces
10. Discrete decomposition of pseudo-cuspforms
11. Meromorphic continuation of Eisenstein series
12. Global automorphic Sobolev spaces, Green's functions
13. Examples – topologies on natural function spaces
14. Vector-valued integrals
15. Differentiable vector-valued functions
16. Asymptotic expansions.