Algebraic Curves Over A Finite Field

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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.

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Genre : Mathematics
Author : J. W. P. Hirschfeld
Publisher : Princeton University Press
Release : 2008-03-23
File : 716 Pages
ISBN-13 : 9780691096797


Algebraic Curves Over A Finite Field

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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.

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Genre : Mathematics
Author : J. W. P. Hirschfeld
Publisher : Princeton University Press
Release : 2013-03-25
File : 717 Pages
ISBN-13 : 9781400847419


Algebraic Curves Over Finite Fields

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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.

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Genre : Mathematics
Author : Carlos Moreno
Publisher : Cambridge University Press
Release : 1993-10-14
File : 264 Pages
ISBN-13 : 052145901X


Rational Points On Curves Over Finite Fields

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Discussion of theory and applications of algebraic curves over finite fields with many rational points.

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Genre : Computers
Author : Harald Niederreiter
Publisher : Cambridge University Press
Release : 2001-06-14
File : 260 Pages
ISBN-13 : 0521665434


Algebraic Curves And Finite Fields

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Algebra and number theory have always been counted among the most beautiful and fundamental mathematical areas with deep proofs and elegant results. However, for a long time they were not considered of any substantial importance for real-life applications. This has dramatically changed with the appearance of new topics such as modern cryptography, coding theory, and wireless communication. Nowadays we find applications of algebra and number theory frequently in our daily life. We mention security and error detection for internet banking, check digit systems and the bar code, GPS and radar systems, pricing options at a stock market, and noise suppression on mobile phones as most common examples. This book collects the results of the workshops "Applications of algebraic curves" and "Applications of finite fields" of the RICAM Special Semester 2013. These workshops brought together the most prominent researchers in the area of finite fields and their applications around the world. They address old and new problems on curves and other aspects of finite fields, with emphasis on their diverse applications to many areas of pure and applied mathematics.

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Genre : Mathematics
Author : Harald Niederreiter
Publisher : Walter de Gruyter GmbH & Co KG
Release : 2014-08-20
File : 254 Pages
ISBN-13 : 9783110317916


Applications Of Curves Over Finite Fields

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This volume presents the results of the AMS-IMS-SIAM Joint Summer Research Conference held at the University of Washington (Seattle). The talks were devoted to various aspects of the theory of algebraic curves over finite fields and its numerous applications. The three basic themes are the following: 1. Curves with many rational points. Several articles describe main approaches to the construction of such curves: the Drinfeld modules and fiber product methods, the moduli space approach, and the constructions using classical curves. 2. Monodromy groups of characteristic $p$ covers. A number of authors presented the results and conjectures related to the study of the monodromy groups of curves over finite fields. In particular, they study the monodromy groups from genus 0 covers, reductions of covers, and explicit computation of monodromy groups over finite fields. 3. Zeta functions and trace formulas. To a large extent, papers devoted to this topic reflect the contributions of Professor Bernard Dwork and his students. This conference was the last attended by Professor Dwork before his death, and several papers inspired by his presence include commentaries about the applications of trace formulas and L-function. The volume also contains a detailed introduction paper by Professor Michael Fried, which helps the reader to navigate the material presented in the book.

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Genre : Mathematics
Author : Michael D. Fried
Publisher : American Mathematical Soc.
Release : 1999
File : 254 Pages
ISBN-13 : 9780821809259


Codes On Algebraic Curves

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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.

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Genre : Mathematics
Author : Serguei A. Stepanov
Publisher : Springer Science & Business Media
Release : 2012-12-06
File : 352 Pages
ISBN-13 : 9781461547853


Algebraic Curves And Cryptography

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Focusing on the theme of point counting and explicit arithmetic on the Jacobians of curves over finite fields the topics covered in this volume include Schoof's $\ell$-adic point counting algorithm, the $p$-adic algorithms of Kedlaya and Denef-Vercauteren, explicit arithmetic on the Jacobians of $C_{ab}$ curves and zeta functions.

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Genre : Computers
Author : Vijaya Kumar Murty
Publisher : American Mathematical Soc.
Release : 2010
File : 142 Pages
ISBN-13 : 9780821843116


Finite Fields And Applications

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This volume represents the refereed proceedings of the Fifth International Conference on Finite Fields and Applications (F q5) held at the University of Augsburg (Germany) from August 2-6, 1999, and hosted by the Department of Mathematics. The conference continued a series of biennial international conferences on finite fields, following earlier conferences at the University of Nevada at Las Vegas (USA) in August 1991 and August 1993, the University ofGlasgow (Scotland) in July 1995, and the University ofWaterloo (Canada) in August 1997. The Organizing Committee of F q5 comprised Thomas Beth (

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Genre : Mathematics
Author : Dieter Jungnickel
Publisher : Springer Science & Business Media
Release : 2012-12-06
File : 500 Pages
ISBN-13 : 9783642567551


Algebraic Function Fields And Codes

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This book links two subjects: algebraic geometry and coding theory. It uses a novel approach based on the theory of algebraic function fields. Coverage includes the Riemann-Rock theorem, zeta functions and Hasse-Weil's theorem as well as Goppa' s algebraic-geometric codes and other traditional codes. It will be useful to researchers in algebraic geometry and coding theory and computer scientists and engineers in information transmission.

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Genre : Mathematics
Author : Henning Stichtenoth
Publisher : Springer Science & Business Media
Release : 2009-02-11
File : 360 Pages
ISBN-13 : 9783540768784