Elliptic Tales: Curves, Counting, and Number Theory

by Avner Ash, Robert Gross

"Elliptic Tales" describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. The Clay Mathematics Institute is offering a prize of $1 million to anyone who can discover a general solution to the problem. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.

The key to the conjecture lies in elliptic curves, which are cubic equations in two variables. These equations may appear simple, yet they arise from some very deep--and often very mystifying--mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and in the process venture to the very frontiers of modern mathematics. Along the way, they give an informative and entertaining introduction to some of the most profound discoveries of the last three centuries in algebraic geometry, abstract algebra, and number theory. They demonstrate how mathematics grows more abstract to tackle ever more challenging problems, and how each new generation of mathematicians builds on the accomplishments of those who preceded them. Ash and Gross fully explain how the Birch and Swinnerton-Dyer Conjecture sheds light on the number theory of elliptic curves, and how it provides a beautiful and startling connection between two very different objects arising from an elliptic curve, one based on calculus, the other on algebra.

Genres: Mathematics, Science, Nonfiction, Number Theory

280 pages, Hardcover

First published January 1, 2012

Reviews

[Darrenglass · Rating 4 out of 5]

Ash and Gross have taken on a task that I would have guessed was impossible -- a book for general audiences about the Birch and Swinnerton-Dyer Conjecture. That won't mean much to those of you who aren't mathematicians, but suffice it to say that the BSD Conjecture is one of the most famous open problems in mathematics (the Clay Math Institute has offered a million dollars for a solution) and it is extremely technical to state even to math grad students...making it all the more impressive that Ash and Gross have attempted a book-length description of it intended for people with little more than a high school math background. And they have largely succeeded, I think. Certainly any undergraduate math student could pick this book up and get all the big ideas -- and many of the details as well. I think they have done a wonderful job of knowing which details to drill down into and which to wave their hands at, and the exposition throughout the book is of a very high level.

That said, while this is a very well written exposition of mathematics, it IS a math book, filled with theorems and equations and pictures. My memory of their previous book (Fearless Symmetry) is that it weaved more of the history and cultural context into its mathematical explorations, and as such I was somewhat disappointed that this book didn't do more of that. But that probably stems from the fact that I knew all the math before I picked up Ash and Gross's book (I work in a closely related area) so I was hoping for more content that was new to me. But that is specific to my case and for anyone who is interested in learning some of the exciting mathematics at the forefront of current knowledge, I think Ash and Gross's book is a must read!

[Andrew Davis · Rating 3 out of 5]

Finished with mixed feelings. An interesting subject of elliptic curves and projective geometry. Struggled at times and had to refer to other online sources to grasp some of the concepts discussed, especially around the group law. Some of the other concepts, especially Riemann-Zeta functions need to be followed further in the other books, as not enough space allocated for this topic.

[Brandon · Rating 4 out of 5]

If you know what linear algebra does with Ax=c, then you can begin to imagine the implications of studying y^2=x^3+Ax+B. Enter elliptic curves over finite fields

[Eliot Long · Rating 5 out of 5]

As a college freshman who loves math and is always on the hunt for good resources, this book falls in the Goldilocks zone of teaching new concepts: it laid out enough that I could gain a comprehensive understanding of the topic while simultaneously including more advanced sections that took me a bit to wrap my head around. It's the perfect level of a challenge for one at my level, and the authors target the content towards one who has minimal prior knowledge of number theory, elliptic curves, etc., while at the same time going in depth. I don't have any complaints about their style; I love that they touch on so many different topics, but always make sure to connect it back to elliptic curves and the BSD conjecture.

The only area that could use improvement was the spell-checking, since there are an abnormal amount of typos — but it didn't affect my understanding, and after all, what do you expect from a couple of mathematicians? ; )

[Isaac van der Meulen · Rating 4 out of 5]

I liked the beginning portion of the book explaining how the complex projective plane can be used to ensure that certain curves always intersect at a certain number of points. The later part of the book reviewed some material I already knew (abelian groups, analytic continuation) and then contained some fairly advanced algebraic geometry that was quite difficult to understand. I guess that is to be expected when reading a book about an old unsolved math problem. I'm not sure how the book could have been better. It was well written, had plenty of examples (with solutions!), and was an appropriate length. It is just that the Birch--Swinnerton-Dyer conjecture is not something particularly easy to write a book about

[Colin · Rating 4 out of 5]

I made it all the way through this for the first time, having given myself permission to "just read" and not worry too much about grokking everything. I learnt a few things, which was the goal.

On the spectrum between pop maths and textbook, it's much more at the textbook end - but it's close to what I think a textbook ought to be like. The pace is not too fast, the writing is not overly formal, and it's not afraid to digress on interesting tangents. I would like to have seen a bit more by way of anecdote, but I enjoyed this.

[Bob Woodley · Rating 4 out of 5]

I wanted to learn about elliptic curves and complex projective geometry so I started this book. It did indeed provide visual intuitions in these areas. I was not interested in the Birch and Swinnerton-Dyer Conjecture at that time so I stopped half-way through, having extracted what I needed.

It is an easy an introduction are you're likely to get to these topics. Not for beginners, but not too advanced either, and minimal notation.

[Yi Li · Rating 3 out of 5]

A good popular math book, especially for the first two parts (the book has three parts). Despite my background in undergraduate and some postgraduate mathematics, I found it to be an enjoyable read. In particular, I like the presentation of the projective curves and projective planes, which is lucid and accessible. The exercises are well-crafted, which make a valuable addition to the book, aiding in comprehension. However, in the last part, particularly when delving into topics like modular forms and L-functions, I found the explanations to be a bit too terse for my taste. I felt that the inherent beauty of these theories wasn't fully conveyed.

[Emil Kerimov · Rating 5 out of 5]

Loved it.

[산 · Rating 4 out of 5]

I spent nearly a month with this so I don't care, I'm adding it to my goal.

[Rafal · Rating 4 out of 5]

Elliptic Curves are fascinating mathematical objects that appear in many areas of math. They are indispensable in securing our credit card numbers and other personal information over the internet traffic; they were key to solving perhaps the most famous problems in mathematics—Fermat's Last Theorem. They are also central to Birch and Swinnerton-Dyer's conjecture, which this book is about.

The book's target audience is people who have a somewhat reasonable understanding of mathematics, and that's probably not me. Thus, I felt pretty lost going over some of these topics. I found the book hard to follow, even for parts where my experience was better, like finite fields.

Fortunately, while reading this book, I caught up with some of the concepts online, which greatly helped.

[Achilleas · Rating 5 out of 5]

A challenging but rewarding book. Elliptic curves is a fascinating subject with surprisingly rich connections to many areas of maths. Highly recommended for those readers who would like to have an insight in this field.

[Chris · Rating 4 out of 5]

An accessible approach to one of the conjectures of number theory.

I found this book quite relaxing, even as it explains complex projective varieties over rings.

[Cold · Rating 3 out of 5]

As far as maths books go it's good. But maths really doesn't work in this format...