This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves.
The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.
"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."--Thomas Hagedorn, MAA Reviews
"Very useful both for research and in the classroom. The main reason to use this book in a classroom is to prepare students for new research in the fields of finite geometries, curves in positive characteristic in a projective space, and curves over a finite field and their applications to coding theory. I think researchers will quote it for a long time."--Edoardo Ballico, University of Trento
"This book is a self-contained guide to the theory of algebraic curves over a finite field, one that leads readers to various recent results in this and related areas. Personally I was attracted by the rich examples explained in this book."--Masaaki Homma, Kanagawa University
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