Goro Shimura is one of the world’s premier arithmeticians, with his name attached to a number of marvelous things. Just consider the seminal Shimura-Taniyama-Weil conjecture (now a theorem, courtesy of Wiles, Wiles-Taylor, and, Breuil-Conrad-Diamond-Taylor) [and which implies Fermat’s Last Theorem] and the Shimura correspondence in the theory of modular forms; and then there are Shimura varieties, of course. All very important and beautiful stuff.

I had the good fortune and great pleasure of being pointed in the direction of Shimura’s masterful monograph, *Introduction to the Arithmetic Theory of Automorphic Functions*, by my professor, V. S. Varadarajan, in a number theory seminar in the late 1970s. With my developing some mathematical maturity I only came to appreciate the contents of this wonderful book more and more, and concomitantly came to appreciate Shimura’s expository gifts. His writings are characterized by not only by depth, precision, and completeness, they ultimately evince considerable accessibility, too (modulo commitment and staying power on the reader’s part) — no mean feat.

The book under review is another case in point: it presents a lot more than its title suggests and does so magnificently. Divided into two parts, the first “is preliminary and consists of algebraic number theory and the theory of semisimple algebras.” The remainder of the book is subsequently devoted to the title’s promise, the arithmetic of quadratic forms. But there is a lot more to say: there’s a lot more to the promise…

The best thing to do is just to quote Shimura:

To classify a quadratic form φ over an algebraic number field F, almost all previous authors followed the methods of Helmut Hasse. Namely, one first takes φ in the diagonal form and associates an invariant to it at each prime spot of F, using the diagonal entries. A superior method was introduced by Martin Eichler in 1952, but strangely is was almost completely ignored, until I resurrected it in one of my recent papers. We associate an invariant to φ at each prime spot, which is the same as Eichler’s, but we define it in a different and more direct way, using Clifford algebras. In Sections 27 and 28 [of the book under review] we give an exposition of this theory. At some point we need the Hasse norm theorem for a quadratic extension of a number field, which is included in class field theory. We prove it when the base field is the rational number field to make the book self-contained in that case.

So we see right off (the above is the second paragraph of the book’s preface) what Shimura is up to, put in crystal clear terms, with no wasted language. This is descriptive of the whole book, in fact, and, indeed, the presentation Shimura crafted of algebraic number theory is a master’s compact development of the subject aiming all the time at his greater goal. The opening chapters take the reader from quadratic reciprocity to the arithmetic of number fields (featuring p-adic analysis and valuations, of course) and the attendant ideal theory; and then, quickly, Shimura gets to adèles, idèles, and cyclotomy and Galois extensions. The fourth chapter is devoted to algebras over a field, with the all-important Clifford algebras appearing five pages into the next chapter, the preceding subsection being titled, “Quadratic forms over a field.” We’re home and the party is about to begin.

*Caveat lector*, however: as I hinted earlier, commitment is certainly a prerequisite for any one attempting to learn what Shimura is teaching. The present fifth chapter, containing the aforementioned section 27, “The Hasse Principle” (Yes — nothing less than that!), is to be dissected carefully and slowly. Shimura’s proofs are beautiful, but they are often very spare: George Shearing not Oscar Peterson, if I may use the metaphor of jazz piano in the present context.

And, to be sure, Shimura’s timing and sense of balance is as exquisite as that of a great jazz pianist. Here is the last paragraph of Chapter V:

We mention here a result due to the author … with the hope that future researchers will find an elementary proof. The original proof requires a rather involved analysis.

Theorem 27.16. Every positive integer is the sum of three integers each of which is of the form (3m^{2}–m)/2 with m ∈ **Z**.

Traditionally a number of the form (3m^{2}–m)/2 with 0 < m ∈ **Z **is called a pentagonal number, but the case m ≤ 0 is included in the above theorem.

Of course this result follows on the heels of Shimura’s discussion and proof of one of the most famous classical theorems in all of arithmetic, one that allegedly caused Gauss, who first proved it, to opt for a life as a mathematician instead of a life as a philologist (as per E. T. Bell, of course), namely the beautiful fact that every positive integer is the sum of three triangular numbers.

And then it’s on to Chapter VI and the vaunted section 28. Well, the chapter is titled “Deeper Arithmetic of Quadratic Forms,” and § 28 is nothing less than the “Classification of quadratic spaces over local and global spaces.” This is of course very serious and austere mathematics and, while irresistible to even relative newcomers to the higher arithmetic, requires a lot of, well, yes, commitment. It is here that we encounter, for instance, Shimura’s discussion of strong approximation (for algebraic groups).

Finally, Chapter VII, “Quadratic Diophantine Equations,” includes a discussion of the classification of binary forms, and also new mass formulas for algebraic groups recalling the famous work of Eisenstein, Minkowski, and C. L. Siegel, “but here we employ the formulation introduced [by Shimura in papers dating to 1997 and 1999].”

Shimura’s *Arithmetic of Quadratic Forms* is another very important monograph by this fine scholar and, to use a hackneyed but apt phrase, will richly repay the reader who invests his time in a careful study of its pages.

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