Author: Avner Ash and Robert Gross Publisher: Princeton University Press Pages: 280 ISBN: 9780691151199 Aimed at: Adventurous mathematicians Rating: 4.5 Pros: Fascinating excursion into advanced topics Cons: Too fast and too tough in places Reviewed by: Mike James
This is a real math book  can you cope?
I'm reviewing this book as a nonmath specialist and the review is aimed at the same sort of readership. You need to be happy with math and know something about groups fields and so on to get anything much from this book.
The subtitle is: Curves, Counting, and Number Theory and it is an introduction to the theory of Elliptic curves taking you from an introduction up to the statement of the Birch and SwinnertonDyer (BSD) Conjecture. One reason for interest in the BSD conjecture is that the Clay Mathematics Institute is offering a prize of $1 million to anyone who can discover a general solution to the problem, and this book at least helps you to understand the problem!
If you are a nonmathematician, then elliptic curves are interesting and there are practical applications in cryptography, but to be honest this is a book you read to find out about the math, so be warned.
Most math books that attempt to introduce a difficult and advanced topic usually start off slow enough for you to follow, but there is nearly always a point where the average reader gets lost. This is usually where the author has grown tired of the effort and moved on a little too fast. Alternatively, it is the point where authors miss that they are using some idea which is obvious to them after so many years working on the subject but completely nonobvious to the innocent reader. This is how it is with this book, but with two authors the energy tends not to flag early on.
The first part of the book is about the problem of defining the degree of a curve. The topics introduced include the algebraic closure of a field and the projective plane. I had encountered the projective plane a number of times before, but never really understood what the point (pun intended) of it all was. So you add some additional points to the plane an make infinity manageable. I found the explanations in this book went a long way to making me understand the value of the projective plane formalism. However, this might not work for every reader unless they have some background in projective geometry taught from another perspective (pun intended again).
Although this wasn't the specific objective of the book, there were lots of places where it managed almost by accident to illuminate some peripheral area of math.
Part II is where the book lost me. After going over some group theory, which should be no problem to most readers, the pace changes and the transition to the standard form of the elliptic curve and its group was too fast. Despite reading it through several times, there are various points at which the description seems to be discontinuous. This is the most difficult part of the book.
If you do find Part II difficult, keep going to Part III where we have a minor restart. The first few chapters make a very interesting course on analytic functions, including the zeta function. This leads up to the final chapter of the book which is the assault on the BSD Conjecture. Again the material speeds up and you stand a reasonable chance of losing the thread between Chapter 13 and the end of the book Chapter 15. It is tough going.
Overall the book has a lot to give even if you aren't a novice mathematician. If you have a grasp of math at the engineering, physics or computer science level, then this book gives you a view of what pure math is all about. There is a lot to think about as a result  for example I had never considered the idea of a rational parametrization which is introduced on page 10. That is, an equation for a curve that provides all of the rational points on that curve. For a circle for example:
(x,y)=( 1m^{2} 2m ) (  ,  ) ( 1+m 1+m^{2} )
gives all of the rational points for m = 1,2,3,...
What curves have rational parametrization? Does this have anything to do with Kolmogorov complexity? What does it have to do with computability etc...
You can tell that my thoughts haven't much to do with the BSD conjecture, but they are generated by reading the book.
Sadly, I can't give this book an excellent rating because it, like most math books, doesn't quite manage to sustain its clear and introductory exposition from start to finish. Some parts of it are the clearest, most straightforward writing on advanced math that I have read but at several points it just goes too fast and loses the nonexpert reader.
I also now see the connection between elliptic curves and number theory which before was a great mystery to me. I didn't learn much that helps me with the practical problems of elliptic curve cryptography, but what I already knew makes more sense.
Even so, I recommend it if you have an interest in modern math and know algebra, calculus and have bumped into groups, fields and so on. It is a very good book and it only misses being an excellent book by a tiny margin.
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