Rational Points On Curves Over Finite Fields

Author by : Harald Niederreiter
Languange : en
Publisher by : Cambridge University Press
Format Available : PDF, ePub, Mobi
Total Read : 72
Total Download : 418
File Size : 45,9 Mb
GET BOOK

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


Lecture Notes Series

Author by :
Languange : en
Publisher by :
Format Available : PDF, ePub, Mobi
Total Read : 25
Total Download : 805
File Size : 42,9 Mb
GET BOOK

Description :


Algebraic Curves Over A Finite Field

Author by : J. W. P. Hirschfeld
Languange : en
Publisher by : Princeton University Press
Format Available : PDF, ePub, Mobi
Total Read : 50
Total Download : 509
File Size : 45,9 Mb
GET BOOK

Description : 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

Author by : Serguei A. Stepanov
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 42
Total Download : 365
File Size : 46,5 Mb
GET BOOK

Description : 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.


Applications Of Curves Over Finite Fields

Author by : Joint Summ Ams-Ims-Siam
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 85
Total Download : 986
File Size : 44,8 Mb
GET BOOK

Description : 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: 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; 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; and, 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 in the material presented in the book.


Frobenius Action On Jacobians Of Curves Over Finite Fields

Author by : Wanlin Li
Languange : en
Publisher by :
Format Available : PDF, ePub, Mobi
Total Read : 48
Total Download : 269
File Size : 50,5 Mb
GET BOOK

Description : This thesis focuses on studying the eigenvalues of the Frobenius action on the l-adic Tate modules of Jacobians of curves over finite fields. Some of the results have applications to answering questions in analytic number theory over function fields. The study of zeros of L-functions associated to Dirichlet characters has been a topic of interest in analytic number theory. Questions and conjectures arising there could also be studied in the function field setting. With the field of rational numbers replaced by the field of rational functions over a finite field, those questions are closely related to the study of the Frobenius action on the l-adic Tate modules of Jacobians of curves over finite fields. Chowla conjectured that the L-function of any quadratic Dirichlet character does not vanish at the central point s=1/2. Soundararajan showed that Chowla's conjecture holds for a positive proportion of quadratic characters ordered by conductor. Over the function field F_q(t), the analogous statement can be phrased but the situation can be very different. Quadratic characters correspond to hyperelliptic curves over F_q and their L-functions are closely related to the Hasse-Weil zeta functions of the curves. To construct quadratic characters whose L-functions vanish at the central point s=1/2 is equivalent to constructing hyperelliptic curves whose Jacobians admit sqrt(q) as an eigenvalue of the Frobenius action on its l-adic Tate module. Over any given finite field F_q, I use the Honda-Tate theory and other previous results to show the existence of such hyperelliptic curves which then give quadratic characters over the function field F_q(t) whose L-functions vanish at the central point s=1/2. This is in contrast with the situation over the rational numbers. Moreover, using a counting result of Poonen on the number of squarefree values of squarefree polynomials over the function field, I give a lower bound on the number of such characters which grows to infinity when the conductor is allowed to be arbitrarily large. Although the analogous statement of Chowla's conjecture does not hold over the function field, it is still believed that 100% of the quadratic characters satisfy the condition that their L-functions do not vanish at the central point s=1/2. So in order to approach this conjecture, joint with J. Ellenberg and M. Shusterman, we use the idea of reduction to give an upper bound on the number of quadratic characters whose L-functions vanish at a given point of the critical line. This upper bound gets better when the size of the constant field is large and the density of such characters goes to 0 when the size of the constant field grows to infinity. Geometrically, we realize the number of hyperelliptic curves whose Jacobians admit some fixed number as an eigenvalue of the Frobenius action on its l-torsion subgroup can be counted by the number of rational points of a twisted Hurwitz scheme over finite fields. Using an earlier result of Ellenberg--Venkatesh--Westerland on the homological stability for Hurwitz spaces, we give an upper bound on the number of rational points of the twisted Hurwitz scheme to get the result. The previous work are all related to studying Weil integers realized as Frobenius eigenvalues for curves over finite fields. From Honda-Tate theory, it is known that every Weil integer appears as a Frobenius eigenvalue for some abelian variety over finite fields. To show the same holds for Jacobian varieties, it suffices to show that every abelian variety over the finite field is covered by a Jacobian variety. This result can be deduced from Poonen's work on the Bertini theorem over finite fields. But there was not an effective bound on the dimension of the Jacobian variety with respect to the degree and dimension of the abelian variety and this is the topic of the last part of my thesis. Given an abelian variety in a projective space over a finite field, joint with J. Bruce, we show the existence of a smooth curve whose Jacobian admits a dominant map to the given abelian variety with an explicit upper bound on its genus. Applying this to simple abelian varieties combined with the theory of Honda-Tate, one can deduce the existence of smooth curves whose Jacobians admit some fixed Weil integer as an eigenvalue with an upper bound on its genus.


Algebraic Curves Over Finite Fields

Author by : Carlos Moreno
Languange : en
Publisher by : Cambridge University Press
Format Available : PDF, ePub, Mobi
Total Read : 73
Total Download : 445
File Size : 42,6 Mb
GET BOOK

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


Curves Over Finite Fields Of Characteristic Two With Many Rational Points

Author by : Gerard van der Geer
Languange : en
Publisher by :
Format Available : PDF, ePub, Mobi
Total Read : 21
Total Download : 835
File Size : 49,9 Mb
GET BOOK

Description : Abstract: "In this note we construct curves over finite fields in characteristic 2 with many rational points. The methods of construction are inspired by considerations from coding theory."


Algebraic Geometry In Coding Theory And Cryptography

Author by : Harald Niederreiter
Languange : en
Publisher by : Princeton University Press
Format Available : PDF, ePub, Mobi
Total Read : 46
Total Download : 319
File Size : 55,9 Mb
GET BOOK

Description : This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algebraic-geometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available. Introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory Provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields Includes applications to coding theory and cryptography Covers the latest advances in algebraic-geometry codes Features applications to cryptography not treated in other books


Rational Points On Curves Over Finite Fields

Author by : Jean-Pierre Serre
Languange : en
Publisher by :
Format Available : PDF, ePub, Mobi
Total Read : 94
Total Download : 737
File Size : 55,8 Mb
GET BOOK

Description : In 1985 Jean-Pierre Serre gave a series of lectures at Harvard University on the number of points of curves over finite fields. Based on notes taken at that time by F. Q. Gouvea, the present revised and completed documents provides an insightful introduction to this beautiful topic and to most of the ideas that have been developed in this area during the last 30 years.


Rational Points On Elliptic Curves

Author by : Joseph H. Silverman
Languange : en
Publisher by : Springer
Format Available : PDF, ePub, Mobi
Total Read : 16
Total Download : 938
File Size : 40,9 Mb
GET BOOK

Description : The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This volume stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make Rational Points on Elliptic Curves an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry. Most concretely, an elliptic curve is the set of zeroes of a cubic polynomial in two variables. If the polynomial has rational coefficients, then one can ask for a description of those zeroes whose coordinates are either integers or rational numbers. It is this number theoretic question that is the main subject of Rational Points on Elliptic Curves. Topics covered include the geometry and group structure of elliptic curves, the Nagell–Lutz theorem describing points of finite order, the Mordell–Weil theorem on the finite generation of the group of rational points, the Thue–Siegel theorem on the finiteness of the set of integer points, theorems on counting points with coordinates in finite fields, Lenstra's elliptic curve factorization algorithm, and a discussion of complex multiplication and the Galois representations associated to torsion points. Additional topics new to the second edition include an introduction to elliptic curve cryptography and a brief discussion of the stunning proof of Fermat's Last Theorem by Wiles et al. via the use of elliptic curves.


The Arithmetic Of Elliptic Curves

Author by : Joseph H. Silverman
Languange : en
Publisher by : Springer
Format Available : PDF, ePub, Mobi
Total Read : 99
Total Download : 232
File Size : 40,5 Mb
GET BOOK

Description : The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.


Higher Dimensional Geometry Over Finite Fields

Author by : D. Kaledin
Languange : en
Publisher by : IOS Press
Format Available : PDF, ePub, Mobi
Total Read : 36
Total Download : 576
File Size : 52,8 Mb
GET BOOK

Description : Number systems based on a finite collection of symbols, such as the 0s and 1s of computer circuitry, are ubiquitous in the modern age. Finite fields are the most important such number systems, playing a vital role in military and civilian communications through coding theory and cryptography. These disciplines have evolved over recent decades, and where once the focus was on algebraic curves over finite fields, recent developments have revealed the increasing importance of higher-dimensional algebraic varieties over finite fields. The papers included in this publication introduce the reader to recent developments in algebraic geometry over finite fields with particular attention to applications of geometric techniques to the study of rational points on varieties over finite fields of dimension of at least 2.


The Hasse Weil Theorem Elliptic Curves And Cryptography

Author by : Carollan B. Ehn
Languange : en
Publisher by :
Format Available : PDF, ePub, Mobi
Total Read : 99
Total Download : 619
File Size : 49,5 Mb
GET BOOK

Description : In this thesis, we examine the theory of algebraic function fields related to the Hasse-Weil Theorem, following primarily the book of Stichtenoth [11]. We also study elliptic curves, their relationship with function fields, and how they are used in cryptography. We study the Riemann-Roch Theorem, which states that, for D a divisor and W a canonical divisor of F/K, dim D = deg D + 1 - g +dim(W - D). This result is fundamental to the proof of the Hasse-Weil Theorem. It can also be used as a tool to determine the genus of a function field. For example, we can use it to show that an elliptic function field has genus one. We examine the fact that if y2 = f(x) defines an elliptic curve over a field K (Where char K [is not equal to] 2 and f(x) is a separable cubic over K), then K (x,y) / K is an elliptic function field. Rational points on the curve correspond to places of K (y,x) / K of degree one. Let F be an algebraic function field of one variable over a finite constant field [double-struck capital F] [subscript q], where q is a power of a prime. We define the Zeta function [zeta]F over F, an analogue of the classical Riemann [zeta]-function. The Hasse-Weil theorem states that the roots t [element of] [double-struck capital C] of [zeta] [subscript capital F] (t) = 1/2. This theorem is often referred to as the "Riemann Hypothesis" for algebraic function fields over finite constant fields for this reason. The Hasse-Weil bound is an important consequence of the Hasse-Weil Theorem with applications to elliptic curves. let N represent the number of rational points on an elliptic curve over a finite field [double-struck capital F] [subscript q]. Then [vertical line] N - (q + 1) [vergical line] [less than or equal to] 2[subscript q] [superscript 1/2]. We discuss the discrete logarithm problem based on the addition of rational points on an elliptic curve. Given rational points P and Q on an elliptic curve, the problem is to find an integer k such the kQ = P. It is computationally difficult to solve the discrete logarithm problem defined over an elliptic curve if the number of rational points on the curve has a "large" prime factor. Relatively recently, mathematicians have developed elliptic curve cryptosystems whose security depends on the difficulty of solving the discrete logarithm problem.


Rational Points And Arithmetic Of Fundamental Groups

Author by : Jakob Stix
Languange : en
Publisher by : Springer
Format Available : PDF, ePub, Mobi
Total Read : 66
Total Download : 743
File Size : 46,5 Mb
GET BOOK

Description : The section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birational analogue in lieu of the fundamental group extension.