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Rational Points on Curves over Finite Fields: Theory and Applications (London Mathematical Society Lecture Note Series), by Harald Niederr
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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.
- Sales Rank: #2773110 in eBooks
- Published on: 2001-06-14
- Released on: 2001-06-14
- Format: Kindle eBook
Review
'... the book under review develops many techniques that are not covered in the existing texts. I highly recommend it.' Steven D. Galbraith, Royal Holloway, University of London
'Because of the carefully selected contents and lucid style, the book can be warmly recommended to mathematicians interested in the above-mentioned topics or in algebraic curves over finite fields with many rational points.' EMS
'The book is very clearly written. It is warmly recommended to anyone who is interested in nice mathematical theories and/or in the recent applications.' Acta. Sci. Math.
About the Author
fm.author_biographical_note2
Chaoping Xing is Full Professor in the Department of Physical and Mathematical Sciences at Nanyang Technological University, Singapore.
Biography of J.W.S. Cassels
J. W. S. Cassels (known to his friends by the Gaelic form "Ian" of his first name) was born of mixed English-Scottish parentage on 11 July 1922 in the picturesque cathedral city of Durham. With a first degree from Edinburgh, he commenced research in Cambridge in 1946 under L. J. Mordell, who had just succeeded G. H. Hardy in the Sadleirian Chair of Pure Mathematics. He obtained his doctorate and was elected a Fellow of Trinity College in 1949. After a year in Manchester, he returned to Cambridge and in 1967 became Sadleirian Professor. He was Head of the Department of Pure Mathematics and Mathematical Statistics from 1969 until he retired in 1984.
Cassels has contributed to several areas of number theory and written a number of other expository books:
- An introduction to diophantine approximations
- Rational quadratic forms
- Economics for mathematicians
- Local fields
- Lectures on elliptic curves
- Prolegomena to a middlebrow arithmetic of curves of genus 2 (with E. V. Flynn).
Hitchin is with Oxford University.
Chaoping Xing is Full Professor in the Department of Physical and Mathematical Sciences at Nanyang Technological University, Singapore.
Most helpful customer reviews
2 of 2 people found the following review helpful.
An advanced informative text
By Rafael M. Rubio
Rational Points On Points Over Finite Fields is an advanced text for a readers with an extensive mathematical background. You should already have experience with algebraic function fields, the Riemann-Roch theorem, class field theory, ramification theory of Galois extensions, Goppa algebraic-geometric codes, and cryptography. The book recommends Stichtenoth's Algebraic Function Fields and Codes as ideal preparation for this text.
The first 2 chapters quickly review the relevant background material. The next 3 chapters dive into important function fields, useful techniques for getting general results, and current research (at the time) on global asymptotic behavior.
The final 3 chapters are remarkably accessible to a reader with only basic knowledge of number theory, abstract algebra, and cryptography. It covers different type of codes, perfect hash families, and surprisingly, applications to quasi-Monte Carlo methods.
I originally found Rational Points On Points Over Finite Fields when doing self study on permutation polynomials and I came across the name Harald Niederreiter. Although this book does not cover that subject, it is an interesting text in the realm of modern mathematics applied to coding theory and cryptography. I will admit that I probably wasn't the intended audience for this book as I am not an expert in the field already. For me, it provides valuable insights into what to study next if I wish to continue understanding modern developments in the field.
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