- Paperback: 260 pages
- Publisher: Cambridge University Press; New Ed edition (14 Oct 1993)
- Language English
- ISBN-10: 052145901X
- ISBN-13: 978-0521459013
- Product Dimensions: 2.3 x 1.5 x 0.2 cm
- Amazon Bestsellers Rank: 2,673,043 in Books (See Top 100 in Books)
- See Complete Table of Contents
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' … a careful and comprehensive guide to some of the most fascinating of plasma processes, a treatment that is both thorough and up-to-date.' The Observatory
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In this Tract Professor Moreno develops the theory of algebraic curves over finite fields, their zeta and L-functions, and, for the first time, the theory of algebraic geometric Goppa codes on algebraic curves. Amongst the applications considered are: the problem of counting the number of solutions of equations over finite fields; Bombieri's proof of the Reimann hypothesis for function fields, with consequences for the estimation of exponential sums in one variable; Goppa's theory of error-correcting codes constructed from linear systems on algebraic curves. There is also a new proof of the Tsfasman–Vladut–Zink theorem. The prerequisites needed to follow this book are few, and it can be used for graduate courses for mathematics students. Electrical engineers who need to understand the modern developments in the theory of error-correcting codes will also benefit from studying this work.
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First Sentence
In applications to arithmetical questions and coding theory, the basic field of constants will be the finite field k = F of characteristic p; in particular this will be apparent in the proof of the Riemann-Roch theorem as well as in the study of the zeta function of a curve. Read the first page
Front Cover | Copyright | Excerpt | Index | Back Cover
In applications to arithmetical questions and coding theory, the basic field of constants will be the finite field k = F of characteristic p; in particular this will be apparent in the proof of the Riemann-Roch theorem as well as in the study of the zeta function of a curve. Read the first page
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Front Cover | Copyright | Excerpt | Index | Back Cover
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2 reviews
6 of 7 people found the following review helpful
Informative, oddly organized
28 Oct 2001
Format:Paperback|Amazon Verified Purchase
This is really two booklets put together, one for mathematicians and one for computer scientists or electrical engineers, with the more accessible one put second. Throughout, the method is very algebraic, as opposed to geometrical. The author explains that he wants to require as little geometry as possible. And this book assumes you have your own motivation for the material. So it gives few examples or explanations until after the theorems are proved. Probably the best source for introductory examples and geometric motivation for Zeta functions on finite fields is chapters 7, 10, and 11 of Ireland and Rosen's CLASSICAL INTRODUCTION TO MODERN NUMBER THEORY.
Moreno's chapters 1-4 aim at math grad students. Chapters 1-3 prove the Riemann-Roch theorem for curves on finite fields, and then use it to give Bombieri's proof of the Riemann Hypothesis for those curves. The treatment is on the level of Hartshorne's ALGEBRAIC GEOMETRY and often refers to that book for an alternative account. But Moreno's book is just on dimension one, and only over finite fields. So he proves the Riemann Hypothesis for this case by page 69, while Hartshorne gives the proof as an exercise on page 368. Ireland and Rosen prove only some special cases, notably the case of elliptic curves on their page 302.
Chapter 4 continues the first three, with a very long and attractive discussion of L functions and exponential sums on these curves, with applications in number theory. I have not learned this material yet but I'll tell you this is the most encouraging treatment of it I have seen.
Chapter 5, which is over one third of the book, does not assume the earlier chapters. It aims at computer scientists interested in the theoretical limits on efficiency of Goppa codes and practical ways to approach those limits. Here are much more elementary explanations of the methods used in earlier chapters. It explains the "birational" viewpoint where a curve is studied by way of the field of rational functions on it, and in fact that field takes on a life of its own. The original curve is forgotten and "points" are defined to be "discrete valuation rings" of any function field. Moreno explains how, in clear cases, these correspond to the points of geometrical curves. But, as a key example, "points" at infinity are automatically included on this approach, even if you do not include them in the geometric curves. He explains what the Riemann-Roch theorem actually says about curves. This chapter does not give complete proofs, and indeed it cites other books for the proof of at least one theorem that was already proved in chapter 1.
You can learn a lot from this book but you'll have to dig for it.
2 of 3 people found the following review helpful
Very not self contained
6 Sep 2007
Format:Paperback
Hi,
I tried to read this book as I wanted to understand Bombieri's
proof of Weil's Theorem (Reimann hypothesis for function fields over
finite field)
For me this this book was very hard to read- I spent alot of time trying to
understand the first chapter-and did not get far.
The book is far from self contained and contains some errors
I think that this is far from inevitable- as there
are books who explain Bombieri's proof- the new vesion of
Schmidt's book equations over finite fields, and I think Stichentoff's
book algberaic function fields and codes- that are much more
accesible
Maybe this book has advantages that I currently cannot
appreciate- but If you're in a situation like mine-someone
that has undergraduate and partial graduate knowledge of
math , and you want to understand Bombieri's proof
and some concepts from function fields and algebraic geometry
- I recommend you look elsewhere
I tried to read this book as I wanted to understand Bombieri's
proof of Weil's Theorem (Reimann hypothesis for function fields over
finite field)
For me this this book was very hard to read- I spent alot of time trying to
understand the first chapter-and did not get far.
The book is far from self contained and contains some errors
I think that this is far from inevitable- as there
are books who explain Bombieri's proof- the new vesion of
Schmidt's book equations over finite fields, and I think Stichentoff's
book algberaic function fields and codes- that are much more
accesible
Maybe this book has advantages that I currently cannot
appreciate- but If you're in a situation like mine-someone
that has undergraduate and partial graduate knowledge of
math , and you want to understand Bombieri's proof
and some concepts from function fields and algebraic geometry
- I recommend you look elsewhere
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