Algebraic Curves over a Finite Field

✍ Scribed by J. W. P. Hirschfeld; G. Korchmáros; F. Torres

• Publisher: Princeton University Press
• Year: 2013
• Tongue: English
• Leaves: 716
• Series: Princeton Series in Applied Mathematics; 20
• Edition: Course Book
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents
Preface
PART 1. General theory of curves
Chapter One. Fundamental ideas
Chapter Two. Elimination theory
Chapter Three. Singular points and intersections
Chapter Four. Branches and parametrisation
Chapter Five. The function field of a curve
Chapter Six. Linear series and the Riemann–Roch Theorem
Chapter Seven. Algebraic curves in higher-dimensional spaces
PART 2. Curves over a finite field
Chapter Eight. Rational points and places over a finite field
Chapter Nine. Zeta functions and curves with many rational points
PART 3. Further developments
Chapter Ten. Maximal and optimal curves
Chapter Eleven. Automorphisms of an algebraic curve
Chapter Twelve. Some families of algebraic curves
Chapter Thirteen. Applications: codes and arcs
Appendix A. Background on field theory and group theory
Appendix B. Notation
Bibliography
Index