When I was a student, my own introduction and initiation into the theory of modular forms came in four stages: I had a mathematical life-changing experience in a reading course on part of Weil’s gorgeous book, Elliptic Functions According to Eisenstein and Kronecker, with V. S. Varadarajan. Then I attended a seminar on modular forms brilliantly run by Basil Gordon. My PhD advisor, Audrey Terras, had me learn Hecke theory from his famous 1938 IAS lecture notes. Finally, I wrote my thesis with Terras on modular forms, Dirichlet series, and representation theory. The upshot of this personal history is that I have a great love for these miraculous things to this day, even as I have not had occasion to work with them lately, at least not directly.
Still, modular forms are everywhere, even if they manage to disguise themselves a bit. Perhaps the most telling example of this surreptitiousness is the fact that they manage to sneak into representation theory in subtle and varied ways, and then proceed to make their presence felt. Here there are three things that we need to mention right off, namely, the prelude to everything presented in Tate’s 1950 thesis written with Emil Artin (included as the last chapter of Cassels-Fröhlich, Algebraic Number Theory), the nigh-on indispensable discussion of this business by Gel’fand, Graev, and Piatetskii-Shapiro in Representation Theory and Automorphic Forms, and then the Langlands Program. This covers a huge amount of very serious and deep modern number theory — and then some: the old borders are getting harder and harder to find.
The book under review is an excellent introduction to this part of number theory, geared to graduate students or accelerated and enthusiastic advanced undergraduates: Deitmar stresses the need for “some knowledge of algebra and complex analysis … [acquaintance] with group actions and the basic theory of rings … [the ability to] apply the residue theorem … [and (recommended)] knowledge of measure and integration theory.” He employs the usual device of “collect[ing] these facts [about measure and integration] in an appendix. So the stage is set, and the audience has been defined.
What does the reader get? Well, Deitmar is explicit about the fact that he proposes to focus on “the interrelation between automorphic functions and L-functions” and the book indeed moves in this direction very effectively. In three initial chapters, Deitmar starts with a compact and complete discussion of the basic themes of doubly periodic functions, modular forms for SL2(Z), and representations of SL2(R), and in the process manages to do justice to, e.g., Eisenstein series, Hecke operators, congruence subgroups, and Maass wave forms. The third chapter on SL2(R) is in itself a particularly welcome discussion: it is very proper to have this material included early on, woven into the fabric of the presentation. It obviates the reader having to go to, e.g., Lang’s book (titled SL2(R), in fact), where the coverage is obviously far more expansive. Additionally, Deitmar is keen to put in a proper dose of Lie theory, which is of course the right move.
Right after all this, Deitmar hits locally compact groups, i.e. p-adic analysis. After discussing the adèles and idèles and Fourier analysis, he goes on to Tate’s famous thesis: L-functions ascendant. The book’s final two chapters throw out all the stops: automorphic repesentations of GL2(A), the 2-dimensional general linear group over the adèles, and, lastly, automorphic L-functions. What a line-up!
It is worth mentioning that one major reason for the burgeoning (and undiminished) popularity of modular forms (and, more generally, automorphic representations), in addition to their role in the Langlands Program, is the mid-1990s proof of Fermat’s Last Theorem by Andrew Wiles, or, more precisely, his (and Taylor’s) settling of the Shimura-Taniyama-Weil Conjecture to the effect that all rational elliptic curves are modular. This opened up the flood-gates, of course, and the ensuing cataract of mathematical activity is still going strong. Although Deitmar’s book does not get into Wiles’ work, and only mentions the Langlands Program en passant (p. 77), these deep parts of modern mathematics are unquestionably knocking on the door.
The book comes equipped with nice exercise sets and a collection of strategically placed “Remarks” which will guide the reader to more advanced sources and provide him a broader and more organic perspective on the field. Accordingly, a novice who works his way through Automorphic Forms will find himself more than prepared for the next phase of work in this irresistibly beautiful part of mathematics.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.