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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

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/dp/0691151199 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-Undergraduate-Mathematics/dp/331918587X https://www.amazon.com/Rational-Points-Elliptic-Undergraduat... (Non-affiliate links, just so you know.)

mrkgnao 9y ago
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.
onuralp 9y ago
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 9y ago
  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 9y ago

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.

onuralp 9y ago

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 9y ago
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!