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Elliptic Tales: Curves, Counting, and Number Theory Hardcover – 12 Mar 2012

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Product details

  • Hardcover: 280 pages
  • Publisher: Princeton University Press (12 Mar 2012)
  • Language: English
  • ISBN-10: 9780691151199
  • ISBN-13: 978-0691151199
  • ASIN: 0691151199
  • Product Dimensions: 2.5 x 17.1 x 24.8 cm
  • Average Customer Review: 3.5 out of 5 stars  See all reviews (4 customer reviews)
  • Amazon Bestsellers Rank: 450,318 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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"The authors present their discussion in an informal, sometimes playful manner and with detail that will appeal to an audience with a basic understanding of calculus. This book will captivate math enthusiasts as well as readers curious about an intriguing and still unanswered question."--Margaret Dominy, Library Journal

"Minimal prerequisites and its clear writing make this book (which even has a few exercises) a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics."--Mathematics Magazine

"The authors of Elliptic Tales do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves."--Sungkon Chang, Times Higher Education

"One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles."--James Case, SIAM News

"Ash and Gross thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjection. . . . [A]sh and Gross deliver ample and current intellectual and technical substance."--Choice

"I would envision this book as an excellent text for an undergraduate 'capstone' course in mathematics; the book lends itself to independent reading, but topics may be explored in much greater depth and rigor in the classroom. Additionally, the book indeed brings together ideas from calculus, complex variables and algebra, showing how a single mathematical research question may require an integrated understanding of the various branches of mathematics. Thus, it encourages students to reinforce their understanding of these various fields, while simultaneously introducing them to an open question in mathematics and a vibrant field of study."--Lisa A. Berger, Mathematical Reviews Clippings

"The book is very pleasantly written, and in my opinion, the authors have done an admirable job in giving an idea to non-experts what the Birch-Swinnerton Dyer conjecture is about."--Jan-Hendrik Evertse, Zentralblatt MATH

"The book's most important contributions . . . are the sense of discovery, invention, and insight into the habits of mind used by mathematicians on this journey. I would recommend this book to anyone who wants to be challenged mathematically or who wants to experience mathematics as creative and exciting."--Jacqueline Coomes, Mathematics Teacher

"[T]his book is a wonderful introduction to what is arguably one of the most important mathematical problems of our time and for that reason alone it deserves to be widely read. Another reason to recommend this book is the opportunity to share in the readily apparent joy the authors have for their subject and the beauty they see in it, not least because . . . joy and beauty are the most important reasons for doing mathematics, irrespective of its dollar value."--Rob Ashmore, Mathematics Today

From the Inside Flap

"Assuming only what every mathematically inclined freshman should know, this book leads the reader to an understanding of one of the most important conjectures in current number theory--whose proof is one of the Clay Mathematics Institute's million-dollar prize problems. The book is carefully and clearly written, and can be recommended without hesitation."--Peter Swinnerton-Dyer, University of Cambridge

"The Birch and Swinnerton-Dyer Conjecture is one of the great insights in number theory from the twentieth century, and Ash and Gross write with care and a clear love of the subject. Elliptic Tales will have wide appeal."--Peter Sarnak, Princeton University

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Most Helpful Customer Reviews

1 of 1 people found the following review helpful By Albear on 25 Feb 2014
Format: Hardcover Verified Purchase
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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1 of 2 people found the following review helpful By charles.nightingale@btinternet.com on 8 Sep 2013
Format: Hardcover Verified Purchase
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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2 of 5 people found the following review helpful By E. Vynckier on 14 Oct 2013
Format: Hardcover Verified Purchase
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.
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1 of 4 people found the following review helpful By Simon Jackson on 14 May 2013
Format: Hardcover Verified Purchase
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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Most Helpful Customer Reviews on Amazon.com (beta)

Amazon.com: 17 reviews
71 of 77 people found the following review helpful
Explains the Birch and Swinnerton-Dyer Conjecture 7 Mar 2012
By Ed Pegg Jr - Published on Amazon.com
Format: Hardcover
I learned quite a bit from this book, and the authors use great effort to explain everything as coherently as possible. The Birch and Swinnerton-Dyer Conjecture (BSD) is difficult material, involving special functions, modular forms, group theory, and elliptic curves. I was able to follow the text. I'm also a mathematician, but that doesn't mean I understand everything (not hardly).

The main competition for this book is Andrew Wiles' (the guy that solved the Fermat Conjecture) write up of BSD for the Millennium Prize Problems. It's a free PDF. That paper was beyond my skills when I first looked at it. If you can understand Wiles, you don't need this book. If you can solve the problem, you get a million dollars. It won't be easy.

If you don't understand Wiles, but want to, then this is a great book for you.

I was annoyed in places. The authors also wrote Fearless Symmetry, as they constantly remind the readers. "This paragraph will only make sense to you if you read Fearless Symmetry or learned these concepts from somewhere else." Often, the authors refer to something short and elegant in either their own book or elsewhere. One rule of popular math -- If it's short and interesting, use it, don't tease it. Worse, the author frequently mention the wriggly graphs plotted by Birch and Swinnerton-Dyer. These graphs led to the original conjecture, but the authors here never show one of these graphs. That's another rule of popular math -- give the important pictures.

I was slightly annoyed that they didn't list some of the important elliptic curves, but some online research showed the small number in the Elkies curve is 20067762415575526585033208209338542750930230312178956502, many are worse, and all of them are easily found in long tables online. A nice short fun table is utterly impossible, so I can forgive them for not having it.

In short, it's a pretty good book, and recommendable. But the authors missed a chance on making it a great popular math book. If they make a revised version, it will have some of those pictures that are currently only described, and they will take out many areas where they are teasing the reader to go elsewhere.
47 of 54 people found the following review helpful
A warning to the readers 23 Jun 2012
By ScienceThinker - Published on Amazon.com
Format: Hardcover
Although this book is promoted as a trade (popular) book, it is not such a book. The claim is that a minimum background in calculus is necessary. However, a calculus course as taught in its simplest form (that is, a simple enumeration of basic facts without precise definitions and actual proofs of statements) is way far for what is required. A more precise statement is `A certain amount of mathematical sophistication is needed to read this book' which I took from the preface of the book. And the amount of sophistication is equal to that of a college math major. The book is packed with equations and material which are very deep and require equal deep concentration and training. Unfortunately, people who have no extensive math knowledge but rush to buy the book will end up with unpleasant feelings: frustration for the money they wasted and disappointment for their overestimated hope to understand current mathematical research.

Having being so critical with the misrepresenting-the-level promotion of the book, I would like to change mode and say that if you are a math major or a physics major hoping to learn some of the mathematics related to string theory or any other scientist whose background includes a good understanding of mathematics, you will find this book extremely valuable. Indeed you will learn a lot about some of the most important topics in current mathematical research. As the reviewer who wrote the review for the Mathematics Magazine observed it is 'a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics.' So use the book as a text for a seminar which you can run at your own leisure but, by no means, the experience will be similar to newspaper reading.

Depending who you are, you may or may not enjoy this book. Decide wisely before you buy it.
24 of 29 people found the following review helpful
An excellent overview for non-specialist readers 8 May 2012
By A Reader - Published on Amazon.com
Format: Hardcover
This book is a delight for the appropriate audience, but not appropriate for all readers.

The authors have organized the book with several reading strategies in mind. Their introduction makes it clear what can be profitably skipped by which readers, from most casual to most determined. This is reinforced by a "Roadmap" paragraph at the head of each chapter, reminding the reader how the current material builds on the preceding and supports the following chapters.

If you are a graduate-level mathematician, this is not the book for you, unless you want a very quick introduction to a previously unfamiliar area. Many important theorems are stated but not proven, and several are simply summarized rather than fully stated. Some references to other texts are given, but by no means does this contain a thorough survey of the literature.

The authors assume the reader has a more complete background in calculus than in algebra, spending time on introductory algebraic material, but in neither case is any deep knowledge required. However, the reader should have a healthy mathematical maturity, as considerable infrastructure is erected on the way to the main topic. Readers familiar with computer graphics or numerical analysis will nod their heads over the chapters discussing why elliptic curves are studied over the complex projective plane with counted root multiplicity. Some important algebraic simplifications through change of coordinates are slipped in along the way. Readers without an appreciation for this kind of technique for reducing problems to more manageable ones (by adding technicalities) may lose sight of the goal.

The authors are as concrete as it is reasonable to be when writing about rather abstract topics. There are diagrams when discussing geometry, and multiplication tables when discussing finite groups. Page count is spent on understanding concepts rather than deep or rigorous mathematical demonstrations.

It's a bit difficult to summarize the book's best audience. Someone interested in the interplay of geometry and algebra, or in the framework of mathematics generally. Someone more interested in the contents of American Scientist than Scientific American. Someone with mathematical maturity but not a professional mathematican. Someone who is content to read the map to the main results, without walking every step on the ground themselves (i.e., proving theorems) or enumerating every secondary result (i.e., cataloging lemmas).
15 of 18 people found the following review helpful
Excellent in every way 6 April 2012
By Dr. Lee D. Carlson - Published on Amazon.com
Format: Hardcover Verified Purchase
There is often a large gap between elementary texts or monographs and advanced treatises on a particular mathematical topic. This gap is due to the lack of motivation or "intuitive" guidance of the relevant concepts, and this is absolutely essential for real understanding of mathematics. Given their emphasis on rigor in presentation and argument, professional mathematicians are frequently antithetic to "handwaving" or pictures to convey mathematical ideas. But such methods enhance the understanding of mathematical concepts, and one might argue (with strong justification) would be impossible without it.

So even a statement to the effect that the Birch and Swinnerton-Dyer conjecture is an assertion that "the analytic and algebraic ranks of elliptic curves are equal" can be of enormous help to those who desire a solid in-depth understanding of this conjecture and elliptic curves in general. Early on in this book, the authors make such a statement, along with many others that enable readers to gain insight about elliptic curves that otherwise could only be obtained by painstaking research into some of the early papers on this subject (or be part of the oral tradition of the lineage of fine mathematicians doing research on elliptic curves).

Studying this book will therefore be invaluable to those who are just beginning to study elliptic curves, and have a real thirst for understanding their properties and applications. The applications of elliptic curves are immense, reaching into areas such as cryptography, dynamical systems, quantum field theory, and superstring theory for starters. But elliptic curves are fascinating in and of themselves, and their beauty alone justifies their study and their placement in pure mathematics.

Some of the mathematical problems that elliptic curves are used to resolve go way back, such as the congruent number problem, which has its origin in the mathematical musings of the ancient Greeks, and concerns the finding of right triangles with rational sides that have integral area. On the physics side, elliptic curves are used to solve the spinning top problem, the motion of the plane pendulum, and interestingly the dynamical behavior of the perihelion of Mercury. They are ubiquitous in exactly-solved models in statistical mechanics, and are used effectively in public key cryptography.

The authors don't mention all of these applications, but because of its emphasis on both the analytic and algebraic concepts behind elliptic curves, readers of this book will be amply prepared to understand them if they so desire. With more background in algebra and complex analysis (but not a whole lot really), they will also be able to move on to more advanced treatments of elliptic curves such as the parity and Shafarevich-Tate conjectures and Neron models. With considerably more preparation readers can tackle higher order generalizations of elliptic curves such as hyperelliptic curves and Abelian varieties, and eventually gain entrance into the somewhat mysterious subject of elliptic cohomology.
4 of 4 people found the following review helpful
A simplified look at current research in elliptic curves 4 Feb 2013
By bebopluvr - Published on Amazon.com
Format: Hardcover Verified Purchase
If anyone evers asks you (assuming you have a degree in math), "what do mathematicians actually do?" this is a good book to hand to them. It covers the basics of elliptic curves using as little machinery as possible. I agree with other reviewers that reading this book requires a certain level of mathematical maturity beyond what a typical student gets out of a basic calculus course, but anyone with enough gumption can work through the material.

The authors are careful to develop concepts as needed, but no more than that. For example, groups and finite fields get a nice elementary treatment. I found reading this book to be a real joy.

I also recommend it to anyone with a degree in math who is curious about why elliptic curves are so important in the field of algebraic geometry.

I deducted one star for overselling who can comfortably read this book, but I still highly recommend it for readers who want to know something about elliptic curves--they are, after all, how Fermat's Last Theorem fell! To be clear, you won't be anywhere near understading the proof Wiles developed for FLT, but the concept of rational points on elliptic curves is where it all begins.
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