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![]() It would be a particularly good book for a student to read before tackling the standard introductory texts in algebraic geometry (Hartshorne, for example). 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. The wealth of information and examples in this book give the reader a firm foundation and develop an intuition for the subject. This book is well-written and I greatly enjoyed reading it. Additionally, the current book has a more algebraic-geometric perspective, though the perspective is pre-Grothendieck (schemes do not appear) 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. The current book is filled with examples and explicit calculations that are not normally found in algebraic geometry textbooks.īoth books present the basic theory of algebraic curves and prove the Riemann hypothesis, but the second halves of the books differ. 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). In 1991, Moreno’s almost identically titled book Algebraic Curves Over Finite Fields appeared it is instructive to compare the two books. Part Three then surveys recent work in the field such as constructing curves with a maximal number of points or a given automorphism group Part Two proves the key theorems in the subject: the Stohr-Voloch and Hasse-Weil (“Riemann Hypothesis”) theorems and the Serre bound. Part One presents an introduction to the general theory of algebraic curves. The authors have done a remarkable job in presenting a concrete introduction to the subject. This book brings the reader to the forefront of current research in these areas 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. 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 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. 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.Īlgebraic curves were the first examples (and the simplest) studied classically in algebraic geometry. “Algebraic Curves over a Finite Field” is a rich, example-filled, comprehensive introduction to the subject.
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