Algebraic curves over a finite field / J. W. P. Hirschfeld, G. Korchmaros, F. Torres.

Format:

Book

Author/Creator:

Hirschfeld, J. W. P. (James William Peter), 1940- author.

Korchmáros, G., author.

Torres, F. (Fernando), author.

Series:

Princeton series in applied mathematics.

Princeton Series in Applied Mathematics

Language:

English

Subjects (All):

Curves, Algebraic.

Finite fields (Algebra).

Physical Description:

1 online resource (717 p.)

Edition:

Course Book

Place of Publication:

Princeton, New Jersey : Princeton University Press, 2008.

Language Note:

English

Summary:

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.

Contents:

Front matter

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

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781400847419

1400847419

OCLC:

884645547