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Title: Algebraic Curves over Finite Fields : Error-Correcting Codes and Exponential Sums by Carlos Moreno ISBN: 0-521-45901-X Publisher: Cambridge University Press Pub. Date: 14 October, 1993 Format: Paperback Volumes: 1 List Price(USD): $32.00

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Average Customer Rating: 4 (1 review)

Rating: 4
Summary: Informative, oddly organized
Comment: 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.

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