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MAA review
Automorphic Forms
Springer New York 2012
Deitmar, Anton
252 p.
Presents an elementary introduction that requires only few pre-requisites
Introduces a host of different techniques such as representation theory, adeles and ideles, and the methods of Tate's thesis
Combines the classical and analytical viewpoint and the modern representation-theoretic approach and reveals their interplay
Automorphic forms are an important complex analytic tool in number theory and modern arithmetic geometry. They played for example a vital role in Andrew Wiles's proof of Fermat's Last Theorem.
This text provides a concise introduction to the world of automorphic forms using two approaches: the classic elementary theory and the modern point of view of adeles and representation theory. The reader will learn the important aims and results of the theory by focussing on its essential aspects and restricting it to the 'base field' of rational numbers.
From the MAA review:
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.
FROM Philosophy, Religion and Science Book Reviews:
This book is an elementary introduction to the realm of Automorphic forms. There are many books on Automorphic forms but this is the first book that requires only few pre-requisites and definitely is suitable for advanced undergraduate classes and for graduate students. I really recommend this book if you are interested in to start learn this amazing subject.
The first chapter of the book introduces periodic functions and Doubly Periodic Functions and give the main properties. Also this chapter presents the P-Function of Weierstrass, Eisenstein Series and special values of the Zeta Function. This chapeter is very introductory but it is very well written and in the end of chapter you can find good problems to work with the theory learned in the chapter.
The second chapter starts the theory of Modular forms for SL(2,Z) this is also an introductory chapter and here we can see the ability of the author to introduce the main topics in a very elegant way. This chapter brings the modular group, modular forms, Fourier coefficients, L-functions, Hecke Operators… . Indeed this chapter is a mini-course in Modular forms and it is presented in a lovely way.
The chapter 3 lead us to a different topic, is about representations for SL(2,R) and introduces a little bit the representation theory and the exponential map. It is an interesting chapter mainly because of the discussion of the Modular Forms and representation vectors. You really can learn some representation theory from this 20 pages.
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Chapter 6 is one of the most important chapters in the book and is about Tate’s thesis. The idea is proof functional equation and analytic continuation for zeta functions in the adelic setting. Is a really beautiful theory and I found amazing to have this in this introductory book. This chapter is well-written but is short, definitely I recommend the reader to search for other sources when reading this chapter, but definitely you will have fun reading about Galois representations and L-functions.
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The last chapter is about Automorphic L-functions and is in my opinion the best chapter in the book it presents automorphic L-functions, Local factors and make a detailed discussion about Global L-functions. And one thing that is really nice about this chapter is that the author presents examples of classical cusp forms. The book finishes in a marvelous way.
Overall I would recommend this book if you are interested in starting learn Automorphic forms or you will give a first course on this subject. The book is very recent, very well-written and presents very nice exercises and remarks. Have fun!