...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?
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).
I think understanding that sentence reduces to parsing jargon. I'll provide an explanation I hope isn't in any way condescending.
The main activity of research mathematicians is theorem proving. That is, turning conjectures into theorems. In the jargon, proven theorems (and equations, bounds, counterexamples etc.) are often called 'results'. Essentially, the author is saying that with very few constructs and tools, number theorists have been able to pose a conjecture that is comparatively simple to state and understand, but is frustratingly as yet unproven, in spite of decades of effort by many, very good mathematicians. So far, many special cases ('partial results') have been proven, but the highly desired goal that is the proof of the BSD conjecture, in full generality, has not yet been achieved. The consensus is that mathematicians are quite far away from resolving the BSD conjecture, but very strongly believe it is true.
>> ... 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?
It's not clear what you mean by "this". You might mean this small section that you quoted, or you might mean the entire article to which this is an opening preamble.
If it's the first of these, then here's a free translation:
A talk like this can convince an audience that the
conjecture is just a wild, crazy thing, produced by
wild crazy people, and it comes from nowhere, without
motivation, without provenance. If so, you'd have to
believe that number theorists are wasting their lives
chasing bizarre and pointless things, and really, not
many of them either.
I hope that this article *fails* to make you think
that. I hope this article shows you that the BSD
conjecture is a part of a larger, meaningful context.
On the other hand, if your "this" refers to the article itself, then it's a rather larger task to help access it. I attended a talk a while ago that gave an excellent, high-level overview of this conjecture and the work being done on it, and I can see the structure of the arguments here.
But it's not intended to be read like a novel by people other than experts in the area. If you're not an expert - and very few people are - it's an article you need to wrestle with, work on, and dive into.
It's math, it's not a graphic novel, and math is not a spectator sport. It's requires participation and effort.
So, is it the article itself for which you need an ELI5? If so, seriously, good luck. This is material at the very limits of modern math.
Thank your for the comments. Yes, it's the article which I would need help understanding. I don't plan to put in enough work to understand it well; the explanation from jordigh and others here was good enough. I do like the self-deprecating nature of the "dry, ridiculous" statement from the authors, and the valiant efforts of others to explain it!
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".
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.
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?