“Algebraic Curves over a Finite Field” is a rich, example-filled, comprehensive introduction to the subject. As an easy-to-read introductory book that presents the general theory of algebraic curves over finite fields, it fills a large gap in the literature.

Algebraic curves were the first examples (and the simplest) studied classically in algebraic geometry. Over a finite field, the theory is particularly rich, and the area has been a relatively accessible place where more general conjectures in algebraic geometry and number theory can be checked. For example, the analogue of the classical Riemann conjecture, the Riemann hypothesis for algebraic varieties over finite fields, was established for algebraic curves by Hasse and Weil in 1940, three decades before the general case was established by Dwork and Deligne.

Over the past decade, the field of algebraic curves over a finite field has been very active as the curves have found applications in number theory (construction of curves with a large number of points), coding theory (the construction of error-correcting codes), and finite geometry. This book brings the reader to the forefront of current research in these areas

The authors have done a remarkable job in presenting a concrete introduction to the subject. The book is divided into three parts. Part One presents an introduction to the general theory of algebraic curves. Part Two proves the key theorems in the subject: the Stohr-Voloch and Hasse-Weil (“Riemann Hypothesis”) theorems and the Serre bound. Part Three then surveys recent work in the field such as constructing curves with a maximal number of points or a given automorphism group

In 1991, Moreno’s almost identically titled book *Algebraic Curves Over Finite Fields* appeared; it is instructive to compare the two books. Both are self-contained and discuss a similar list of topics, but the current book has a more relaxed style (and so is longer: 696 pages versus 246 pages). The current book is filled with examples and explicit calculations that are not normally found in algebraic geometry textbooks.

Both books present the basic theory of algebraic curves and prove the Riemann hypothesis, but the second halves of the books differ. Moreno includes L-functions, estimates for exponential sums, and a chapter on Goppa curves and modular curves, whereas the emphasis of the second half of the current book is on finding optimal curves with a maximal number of points or a large automorphism group, with a short section on error-correcting codes and finite geometries concluding the book. Additionally, the current book has a more algebraic-geometric perspective, though the perspective is pre-Grothendieck (schemes do not appear)

This book is well-written and I greatly enjoyed reading it. The wealth of information and examples in this book give the reader a firm foundation and develop an intuition for the subject. The authors have used it as a textbook for a two-year course, and it would be a fine introduction to any advanced undergraduate or graduate student wanting to learn this subject. It would be a particularly good book for a student to read before tackling the standard introductory texts in algebraic geometry (Hartshorne, for example). Or, conversely, a student in an abstract algebraic geometry course could use this book to make the general theory more concrete.

Researchers in the subject will also find much useful information. This reviewer enjoyed the authors’ emphasis on non-classical curves that have properties that do not appear when working with real or complex curves. Also, introductions to topics such as dual curves are not easy to find in the literature. Their inclusion here is greatly welcomed. The bibliography and mathematical notes are also quite extensive and are useful pointers for someone looking to find research problems

Thomas Hagedorn is an Associate Professor of Mathematics and Statistics at The College of New Jersey. He can be reached at hagedorn at tcnj.edu.