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Author: Harald Niederreiter
Publisher: Cambridge University Press
Published: 2001-06-14
Total Pages: 260
ISBN-13: 9780521665438
DOWNLOAD EBOOK →Discussion of theory and applications of algebraic curves over finite fields with many rational points.
Author: Joseph H. Silverman
Publisher: Springer Science & Business Media
Published: 2013-04-17
Total Pages: 292
ISBN-13: 1475742525
DOWNLOAD EBOOK →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.
Author: J. W. P. Hirschfeld
Publisher: Princeton University Press
Published: 2013-03-25
Total Pages: 717
ISBN-13: 1400847419
DOWNLOAD EBOOK →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.
Author: Jean-Pierre Serre
Publisher:
Published: 2020
Total Pages: 187
ISBN-13: 9782856299234
DOWNLOAD EBOOK →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.
Author: Carlos Moreno
Publisher: Cambridge University Press
Published: 1993-10-14
Total Pages: 264
ISBN-13: 9780521459013
DOWNLOAD EBOOK →Develops the theory of algebraic curves over finite fields, their zeta and L-functions and the theory of algebraic geometric Goppa codes.
Author: Henri Darmon
Publisher: American Mathematical Soc.
Published: 2004
Total Pages: 146
ISBN-13: 0821828681
DOWNLOAD EBOOK →The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 7-9, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and Swinnerton-Dyer conjecture.
