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Elliptic Tales: Curves, Counting, and Number Theory 1st Edition

4.2 out of 5 stars 19 customer reviews
ISBN-13: 978-0691151199
ISBN-10: 0691151199
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Editorial Reviews


"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

"This book has many nice aspects. Ash and Gross give a truly stimulating introduction to elliptic curves and the BSD conjecture for undergraduate students. The main achievement is to make a relative easy exposition of these so technical topics."--Jonathan Sanchez-Hernandez, Mathematical Society

From the Back Cover

"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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Product Details

  • Hardcover: 280 pages
  • Publisher: Princeton University Press; 1st edition (March 12, 2012)
  • Language: English
  • ISBN-10: 0691151199
  • ISBN-13: 978-0691151199
  • Product Dimensions: 1 x 6.8 x 9.8 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (19 customer reviews)
  • Amazon Best Sellers Rank: #513,629 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

By Ed Pegg Jr VINE VOICE on March 7, 2012
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.
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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.
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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.
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