HN Books @HNBooksMonth

The best books of Hacker News.

Hacker News Comments on
Elliptic Tales: Curves, Counting, and Number Theory

Avner Ash, Robert Gross · 2 HN comments
HN Books has aggregated all Hacker News stories and comments that mention "Elliptic Tales: Curves, Counting, and Number Theory" by Avner Ash, Robert Gross.
View on Amazon [↗]
HN Books may receive an affiliate commission when you make purchases on sites after clicking through links on this page.
Amazon Summary
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. 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 may appear simple, but 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.
HN Books Rankings

Hacker News Stories and Comments

All the comments and stories posted to Hacker News that reference this book.
I can recommend https://www.amazon.com/Elliptic-Tales-Curves-Counting-Number.... It goes in-depth into the motivating use cases for elliptic curves (as well as their group law) and explains some very elegant theorems (Bézout's theorem) as well as open problems in Maths (the Birch-Swinnerton-Dyer conjecture).
For anyone looking to learn more about elliptic curves, which are astoundingly "well-connected" as math topics go, here's a good book that should be accessible to anyone with some calculus under their belt:

https://www.amazon.com/Elliptic-Tales-Curves-Counting-Number...

It's not a textbook, which is both good and bad. In my case, it did a good job whetting my appetite for more!

A really well-written and not-extremely-difficult undergrad textbook on elliptic curves:

https://www.amazon.com/Rational-Points-Elliptic-Undergraduat...

(Non-affiliate links, just so you know.)

onuralp
These look fascinating, thanks.

I tried to crack this problem, and ended up (very naively) resorting to substituting division with modulo operation that is, a%(b+c) + b%(a+c) + c%(a+b) = 4 of which the min solution is a=1, b=2, and c=4. I am glad that the true solution involved EC which, if my understanding is correct, is basically modular operations in high-dimensional space.

mrkgnao
Elliptic curves are "generalized modular arithmetic" only in the sense that they involve a different kind of addition law (the set of points on a nice EC is a group).

Also, you can look at the points on an elliptic curve where you allow the coordinates to be real, complex, rational, or even the integers mod p (any field will do), so the last choice gives you a closer link with modular arithmetic. It's best to treat ECs as their own weird, wonderful beasts!

mrkgnao
So what do I mean by "well-connected"? Elliptic curves are associated with

a) number theory - many questions about number theory boil down to finding points on elliptic curves with rational coordinates

b) algebraic geometry - elliptic curves are very "nice" from this point of view, and they are complicated enough that you can say interesting things, but not so complicated that you can't say anything

c) complex analysis - over the complex plane, an elliptic curve is a torus. You might have seen how a torus can be formed by identifying opposite edges of a square, and an elliptic curve is hence "a quotient of C by a square lattice". Modular forms, which are creatures of complex analysis with deep applications to number theory, are naturally defined in this framework.

and many more things I don't know about. They're in a sort of "sweet spot" and act as a bridge between multiple parts of mathematics.

HN Books is an independent project and is not operated by Y Combinator or Amazon.com.
~ yaj@
;laksdfhjdhksalkfj more things
yahnd.com ~ Privacy Policy ~
Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.