
What is the Birch and Swinnerton-Dyer conjecture? - subnaught
http://www.math.harvard.edu/~chaoli/doc/BSD.html
======
chrispeel
_...such a talk can easily succeed in convincing the audience that there is a
dry, ridiculous, but famous conjecture coming out of nothing; crazy number
theorists have wasted their life to contribute to the list of partial results,
which cannot even be claimed to be a long list.

Hopefully I have failed in that way._

Ummm, I guess that even among techies on HN, there are only a small fraction
who could understand this. So the author failed with me. Any one want to do an
ELI5?

~~~
jordigh
There are certain objects called elliptic curves. In a form called "affine",
they look like y^2 = (some degree three polynomial in x). When you let x and y
take values in several different fields (a field is an algebraic object in
which you can add, subtract, divide, and multiply), we call solutions (x,y) of
the elliptic curve _points on the curve_. After some work, you can determine
that points on the curve form an _abelian group_ , that is, an algebraic
structure with a single invertible operation that looks like addition (i.e.
a+b=b+a for points a and b on the curve). After more work, we have concluded
that the structure of this abelian group looks like some finite copies of the
integers plus some finite copies of the integers modulo m. The copies of the
integers modulo m is called the _torsion_ subgroup and the number of copies of
the integers is called the _rank_ of the elliptic curve.

From a very different direction, to each elliptic curve we can associate
something called an L-function, which is analogous to the Riemann zeta
function. Through a bunch of other work, we have determined that the L
function of an elliptic curve looks like a certain infinite product that in
fact converges, and this L-function has an analytic continuation. If we expand
this L-function as an infinite polynomial around 1 (a Taylor expansion you may
rememember from calculus), then it looks like c(s-1)^r + (higher order terms).
This "r" in the expansion is called the _analytic rank_ of the elliptic curve.

The BSD conjecture is that the analytic rank (the exponent in this L series
expansion) equals the rank (the number of copies of the integers in the
elliptic curve's abelian group).

~~~
subnaught
Thank you for taking the time to do that, that was great.

------
mjcohen
This is quite advanced. At least graduate math level.

This is supposed to be a "elementary introduction to the Birch and Swinnerton-
Dyer conjecture". Ha.

~~~
cperciva
_This is quite advanced. At least graduate math level._

Yes, I learned most of this when I took graduate mathematics courses. Soon
thereafter I decided that I'd be better off studying computer science...

 _This is supposed to be a "elementary introduction to the Birch and
Swinnerton-Dyer conjecture". Ha._

I don't think the word "elementary" means what you think it means. In number
theory, "elementary" means "not involving complex analysis"; it does not mean
"straightforward".

~~~
jordigh
In this context, they are not shying away from using complex analysis. They
really do mean "elementary" as in "not for number theory experts", but for a
general mathematical audience.

~~~
cperciva
They mention complex analysis in passing, but I wouldn't say that they _use_
it here.

