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3.8 out of 5 stars
3.8 out of 5 stars
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on 14 October 2013
I want to be frank: this book is a failure.

I do have a extensive and solid mathematics background far beyond high school mathematics, and yet there are many parts in the book that simply do not tell anything, and do not achieve anything for the reader, even one at my level. Some of the reasoning is so stripped down and filled with comments like "You now need to know this but -heck- we cannot explain it here because it is too advanced and would take up too much space if we did". I question there is value in books that bluntly admit defeat and rolls on regardless to the next incomprehensible bare-back statement of defeat.

The book is about the Birch-Dyer conjecture. It sketches the relevance and the content of this conjecture only in the final few pages, and gives a minute (or to be blunt, not even) application of it. The book should have been planned far more extensively and rewritten from scratch.

I would have started out with a few straightforward number theoretic problems, rational or integer arithmetic problems, that are recognisable to any determined reader with high-school mathematics, which is their professed target public. With that context, I would have motivated why high-school mathematics has difficultly handling such problems, either in the positive way (finding the solution) or in the negative way (proving there are no integer or rational solutions, only irrational and/or complex solutions). After creating that context, I would have introduced a plan or a strategy to find out more, and motivate why the strategy stands a chance of developping some meaningful, interesting mathematics. And then go into elliptic curves, group theory, function theory. Without such a framework or a plan, the book just babbles on but doesn't achieve anything, even for the determined reader of more mathematical development of their target public.

It should also have many more examples and many more graphical representations ... it hardly has any! Just google Birch-Dyer conjecture, and you will get a graph on the internet summarising the original work in the 50'ies and 60'ies. Look at WikiPedia and at some of the freely available pdfs, this will show you more compelling examples and more graphical insight and feel for the issues around elliptic curves, Birch-Dyer and other conjectures, related problems in Diophantine equations ...

From the comments on their previous book, "Fearless Symmetry", which I have not purchased, I can glance that that book suffers from the same ills. This book is a failure. In my opinion, authors Ash and Gross should replan and rewrite their popular mathematics books, or bin their ambition to produce any. Surpring that a reputed editor such as Princeton University Press puts this on the market.
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on 25 February 2014
An superbly accessible accouting of ellyptic curves and related concepts and theories. It sometimes seems that mathematicians hide their ideas behind mysterious words almost as Guild members would hide their skills. This approach of stripping the mathematics bare of the structure around which it is built (Bourbaki?) seems to me at the heart of the lack of interest in our education system and in our public at large. It seems to make mathematics special, mystical, and not for ordinary folk. These authors in their non-academic writings make a valiant attept to pull down the screens and show us what is behind - and they do it brilliantly. I lost count of the ideas this book clarified for me.
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on 1 October 2015
It promises an easy walk in the manner of the Canterbury tales. Actually I wode eqoth that Chaucer is a much easier read! The early part is easy going, but the terrain rapidly becomes waterlogged; and you have to pick up and carry bits of heavy apparatus on the way without a clear understanding of why. As you drive the pitons into the last few chapters the ascent is complete, and in a blue haze below is the stupendous, shimmering Birch Swinnerton Dyer conjecture, into which you must dive. As Butch Cassidy said, the fall alone will probably kill us.
To be less fatuous, I do now have some considerable inkling about elliptic curves, and why degree three is the interesting one. By the way, I have an MA in sums but I found the going fairly tough: one chapter a day was all I could take - but there are many actual example case studies to assist and they do explain everything clearly. It would help you to already know about groups, rings and fields.
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on 10 November 2016
This book is a very nice introduction to elliptic curves and counting problems for non-experts. I would say it provides material that help build some intuition behind the theory of elliptic curves and L-functions and I found it especially useful while reading it in parallel with Koblitz and some parts of Milne's book on elliptic curves. The book starts with very basic material from definitions of elliptic curves, to the definition of abelian groups etc nd goes all the way down to the point of exposing the BSD conjecture that relates the rank of the group E(Q) to the order of vanishing of the appropriate Hecke L-function. I would greatly recommend it to anyone who is interested in grasping these incredible ideas.
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on 2 December 2015
Very good. Otherwise, the Royal Mail is now very slow.
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on 15 March 2015
Lots of formulas, poorly explained concepts.
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on 8 September 2013
I recall being asked to find the turning points of an elliptic curve and sketch it in my first year as an undergraduate. Since then such curves have been deified by Andrew Wiles, the chappie who proved Fermat's Last Theorem, using them. They are now a hot topic in encryption; the graph is very seductive, a smooth curve with a little bubble that has broken away. It was the first time in my lfe that I got to use all that homogenious algebraic projective geometery, and at the age I am it was a great joy to find I could remember it all: if only I could work out what I did last Tuesday with the same facility!
Had to skip some passages - just too much!
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on 29 September 2015
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on 14 May 2013
The book is well structured, and is much easier to read than many on the subject. The main reason it did not get 5 stars is that it does not cover the extension of Mazur's theorem by fields of character 2 and 3.
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