Algebraic Curves Over a Finite Field

Algebraic Curves Over a Finite Field
Title Algebraic Curves Over a Finite Field PDF eBook
Author J. W. P. Hirschfeld
Publisher Princeton University Press
Pages 716
Release 2008-03-23
Genre Mathematics
ISBN 0691096791

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This title provides a self-contained introduction to the theory of algebraic curves over a finite field, whose origins can be traced back to the works of Gauss and Galois on algebraic equations in two variables with coefficients modulo a prime number.

Algebraic Curves Over Finite Fields

Algebraic Curves Over Finite Fields
Title Algebraic Curves Over Finite Fields PDF eBook
Author Carlos Moreno
Publisher Cambridge University Press
Pages 264
Release 1993-10-14
Genre Mathematics
ISBN 9780521459013

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Develops the theory of algebraic curves over finite fields, their zeta and L-functions and the theory of algebraic geometric Goppa codes.

Algebraic Curves over a Finite Field

Algebraic Curves over a Finite Field
Title Algebraic Curves over a Finite Field PDF eBook
Author J. W. P. Hirschfeld
Publisher Princeton University Press
Pages 717
Release 2013-03-25
Genre Mathematics
ISBN 1400847419

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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.

Codes on Algebraic Curves

Codes on Algebraic Curves
Title Codes on Algebraic Curves PDF eBook
Author Serguei A. Stepanov
Publisher Springer Science & Business Media
Pages 372
Release 1999-07-31
Genre Computers
ISBN 9780306461446

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This is a self-contained introduction to algebraic curves over finite fields and geometric Goppa codes. There are four main divisions in the book. The first is a brief exposition of basic concepts and facts of the theory of error-correcting codes (Part I). The second is a complete presentation of the theory of algebraic curves, especially the curves defined over finite fields (Part II). The third is a detailed description of the theory of classical modular curves and their reduction modulo a prime number (Part III). The fourth (and basic) is the construction of geometric Goppa codes and the production of asymptotically good linear codes coming from algebraic curves over finite fields (Part IV). The theory of geometric Goppa codes is a fascinating topic where two extremes meet: the highly abstract and deep theory of algebraic (specifically modular) curves over finite fields and the very concrete problems in the engineering of information transmission. At the present time there are two essentially different ways to produce asymptotically good codes coming from algebraic curves over a finite field with an extremely large number of rational points. The first way, developed by M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is rather difficult and assumes a serious acquaintance with the theory of modular curves and their reduction modulo a prime number. The second way, proposed recently by A.

Rational Points on Curves Over Finite Fields

Rational Points on Curves Over Finite Fields
Title Rational Points on Curves Over Finite Fields PDF eBook
Author Harald Niederreiter
Publisher Cambridge University Press
Pages 260
Release 2001-06-14
Genre Computers
ISBN 9780521665438

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Ever since the seminal work of Goppa on algebraic-geometry codes, rational points on algebraic curves over finite fields have been an important research topic for algebraic geometers and coding theorists. The focus in this application of algebraic geometry to coding theory is on algebraic curves over finite fields with many rational points (relative to the genus). Recently, the authors discovered another important application of such curves, namely to the construction of low-discrepancy sequences. These sequences are needed for numerical methods in areas as diverse as computational physics and mathematical finance. This has given additional impetus to the theory of, and the search for, algebraic curves over finite fields with many rational points. This book aims to sum up the theoretical work on algebraic curves over finite fields with many rational points and to discuss the applications of such curves to algebraic coding theory and the construction of low-discrepancy sequences.

Algebraic Functions and Projective Curves

Algebraic Functions and Projective Curves
Title Algebraic Functions and Projective Curves PDF eBook
Author David Goldschmidt
Publisher Springer Science & Business Media
Pages 195
Release 2006-04-06
Genre Mathematics
ISBN 0387224459

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This book gives an introduction to algebraic functions and projective curves. It covers a wide range of material by dispensing with the machinery of algebraic geometry and proceeding directly via valuation theory to the main results on function fields. It also develops the theory of singular curves by studying maps to projective space, including topics such as Weierstrass points in characteristic p, and the Gorenstein relations for singularities of plane curves.

Rational Points on Elliptic Curves

Rational Points on Elliptic Curves
Title Rational Points on Elliptic Curves PDF eBook
Author Joseph H. Silverman
Publisher Springer Science & Business Media
Pages 292
Release 2013-04-17
Genre Mathematics
ISBN 1475742525

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The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.