In my second year of graduate school my advisor set to me to work on a project concerning the arithmetic of orders in quaternion algebras. At the time there were very few books that discussed quaternion algebras in any sort of depth, and far fewer that got into orders. The references I found most helpful were Marie-France Vignéras' *Arithmétique des algèbres de quaternions*, Colin Maclachlan and Alan Reid's book *The Arithmetic of Hyperbolic 3-Manifolds* and Irving Reiner's *Maximal orders*. These are wonderful books, but none give anything close to an encyclopedic account of quaternion algebras. Vignéras' book, for example, is barely 150 pages long, while Maclachlan and Reid only discuss the aspects of the theory that relate to the construction of hyperbolic manifolds. The focus of Reiner's book isn't quaternion algebras, but rather, central simple algebras. Quaternion algebras are examples of central simple algebras, but Reiner's more general focus means that many nuances particular to quaternion algebras are overlooked. Until very recently the literature on quaternion algebras hadn't changed all that much, and researchers interested in aspects of quaternion algebras not covered by a handful of narrowly focused texts remained in a difficult position. All that has changed however, with the publication of John Voight's *Quaternion Algebras*.

*Quaternion Algebras* is a massive tome that provides its readers with a comprehensive account of its subject. The book assumes surprisingly little by way of prerequisite knowledge (a first course in algebraic number theory, some linear algebra, and a bit of commutative algebra) and was written to introduce graduate students to quaternion algebras and some of the many areas of algebra, number theory and geometry in which they often arise. Although my bookshelf is buckling under the weight of *Quaternion Algebras* as I write this, the word that I would use to describe the book is not *encyclopedic*. It is *friendly*. In order to keep his book's prerequisites from escalating out of control, Voight gives many "mini-introductions" to objects that are relatively basic but of which readers with minimal prerequisites might be ignorant. For example, there are introductions to the \(p\)-adic numbers, to the adeles and ideles, to quadratic forms, to function fields, to classical modular forms, etc. This is not to say that *Quaternion Algebras* is an easy book, because it isn't. Voight clearly believes that the "best" perspective on quaternion algebras, the vantage point from which the landscape is clearest, is a fairly abstract one. (And he is likely correct.) Working in this generality is one of the reasons for the book's great length. (The other reason is that quaternion algebras are simply an exciting topic about which there is a great deal to say!) This level of abstraction, combined with the book's vast scope, will make it difficult reading for many newcomers. However, returning to the landscape metaphor, the journey may be difficult, but the views at the summit more than make up for it.

Many number theory texts are written in a very unbalanced way. They'll begin, for example, by writing as though the reader may be only vaguely aware of what a number field is and will give very careful proofs of even the most elementary results, but within 50 pages are throwing around terms from Galois cohomology as though the reader has been using them for years. *Quaternion Algebras*, by contrast, is written extremely evenly, with a level of sophistication assumed on the part of the reader that remains more of less constant throughout the book's 900 pages. Moreover, Voight has gone out of his way to make the book's structure more horizontal than vertical, with chapters that are as logically independent of each other as is possible. (Serge Lang's *Algebra* has a similar structure. Both books are written so that a reader with a basic knowledge of the subject can, in general, open the book to Chapter \(n\) and begin reading without being at an insurmountable disadvantage for not having previously read Chapter \(n-k\).)

*Quaternion Algebras* is organized into five parts, each centering on a different theme.

- Algebra
- Arithmetic
- Analysis
- Geometry and Topology
- Arithmetic Geometry

Part I: Algebra provides the reader with a overview of non-commutative algebra, with an emphasis on central simple algebras and quaternion algebras. This part contains a great deal of material that is often skipped by other texts. For example, there is an entire chapter devoted to quaternion algebras over fields of characteristic two.

Part II: Arithmetic is the heart of the book and concerns orders in quaternion algebras. This part contains excellent discussions of quaternion ideals, the relationship between quaternion orders and ternary quadratic forms, and a variety of important families of orders (hereditary orders, Bass orders, Gorenstein orders, etc), along with a nice proof of the classification of quaternion algebras over global fields. I am especially fond of Chapter 11: The Hurwitz order. This chapter, which appears directly after the chapter defining orders, gives the well-known proof of Lagrange's Four Squares theorem using the Hurwitz and Lipschitz orders of quaternions. When one encounters orders for the first time there is a danger of getting lost in a sea of abstractions and forgetting how concrete orders can be. Chapter 11 eliminates this danger by focusing on two very concrete orders and using them to prove a classical result from elementary number theory. This material is inherently interesting and allows the reader to focus on a concrete application before getting to the more abstract material contained in later chapters.

I'm not entirely convinced that all of the material in Part III: Analysis is technically analysis, but much of it is and this is in any event a minor quibble. Part III contains a lot of material on Eichler's mass formula and quaternionic zeta functions. Moreover, it is here that Voight introduces his readers to the quaternionic adeles and ideles. The adele ring of a quaternion algebra is useful for the same reason that the adele rings of number fields are useful: they allow one to consider all completions at once and are thus useful for transitioning between local and global settings. After defining the adeles Voight proves the Strong Approximation theorem and applies it to study the embedding theory of quaternion orders. (The general problem in this embedding theory is determining when a quadratic order embeds into a quaternion order.) These embedding results are proven in Chapters 30 and 31. Although this material gets quite technical, the reader is rewarded with a very interesting application: the construction of isospectral, nonisometric Riemannian manifolds.

In Part IV: Geometry and Topology Voight uses the unit groups of certain quaternion orders to construct discrete groups of isometries of hyperbolic spaces and related manifolds. These constructions are vast generalizations of the fact that the group \(\mathrm{SL}_2(\mathbf Z)\), which arises so naturally in algebra and number theory, is a discrete group of isometries of the hyperbolic plane and gives rise to the modular surface. The highlights of this part are Voight's thorough treatment of fundamental domains and his derivation of the formula for the covolume of principal arithmetic groups arising from quaternion algebras.

Part V: Arithmetic Geometry is the shortest of the book's five parts, but likely to be of intense interest. After giving a brief introduction to classical elliptic modular forms, Voight defines Brandt matrices in order to construct modular forms from definite quaternion algebras. This construction is at the center of Eichler's basis problem, and Voight has done a nice job of synthesizing results that have hitherto only appeared in the periodical literature. Voight rounds out his book with chapters on supersingular elliptic curves and abelian surfaces with QM (quaternionic multiplication).

John Voight's *Quaternion Algebras* is a wonderful book that will be treasured by generations of graduate students and researchers in non-commutative algebra and algebraic number theory. It introduces its readers to the structure of quaternion algebras and the arithmetic of their orders, to some of the many different areas of mathematics in which quaternion algebras arise, and exposits a number of important results that until now have never been treated in textbook form. I am certain that *Quaternion Algebras* will soon become the standard reference on its subject. I only wish that it was available when I was in graduate school.

Benjamin Linowitz ([email protected]) is an Assistant Professor of Mathematics at Oberlin College. His website can be found at http://www2.oberlin.edu/faculty/blinowit/.