Talk:Birch and Swinnerton-Dyer conjecture
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Conjecture: If and only if L(C, 1) = 0 then C(Q) is infinite. 126.96.36.199 23:27, 31 May 2006 (UTC)
Does this conjecture have implication for elliptic curve cryptography? 188.8.131.52 12:46, 16 November 2006 (UTC)
- No. Why should it? Charles Matthews 18:47, 16 November 2006 (UTC)
Statement of conjecture
The article could be improved considerably by the addition of a statement of the conjecture. As it stands it is impossible to tell from the article what would qualify as a solution. The article talks informally about three conjectures without stating any one of them precisely and without saying which one has to be solved in order to win the prize. --Vaughan Pratt (talk) 22:56, 16 October 2008 (UTC)
- The statements in the History section are quite precise. However, for clarity, I have put a summary of the basic conjecture into the lead section. Gandalf61 (talk) 08:51, 17 October 2008 (UTC)
- Much appreciated. Unfortunately the article on L-functions that you linked to doesn't give an adequate definition of L(E,s), instead talking around the concept without nailing things down precisely. The problem seems to be that the editors of these two articles know the material so intimately that they seem unable to tell when they've left stuff out. But even if those lacunae are filled in, the article furthermore suffers from being pitched at an unnecessarily high level thereby greatly reducing its accessibility, as well as raising questions that aren't necessarily even relevant to the conjecture. I just sat through an hour lecture by Kenneth Ribet this afternoon about related stuff---the difference in clarity between what he presented and what's written here is night and day. (Incidentally I'd printed out the article just before the lecture, halfway through which I noticed your addition of the summary to the lead, thanks again.) For example nothing is lost by dropping the reference to Taylor expansions in the lead: it is just as precise to conjecture simply that L(C,s) is O((s-1)r), with the additional benefit of detaching the conjecture from the question of whether L(C,s) even has a meromorphic continuation. It would be great to have an elementary, precise, and accessible exposition of this conjecture. (If I knew what the conjecture was I'd write it myself.) --Vaughan Pratt (talk) 03:07, 18 October 2008 (UTC)
- Not sure that the conjecture is quite equivalent to "L(C,s) is O((s-1)r) as s approaches 1". This would be satisfied if the first non-zero term in the Taylor expansion were (s-1)r or a higher order term, whereas the conjecture is that the first non-zero term is precisely the (s-1)r term.
- However, as I am sure you know, Wikipedia is "the free encyclopedia that anyone can edit" - so if you (or indeed Ken Ribet) wish to improve this article or the L-function article, there is nothing standing in your way. Your knowledge in this area is clearly more than sufficient. Gandalf61 (talk) 10:02, 18 October 2008 (UTC)
I think that this article could be substantially improved in several directions. First of all and most importantly, throughout the article the words "elliptic curve" are used to mean "elliptic curves defined over the rational numbers". I think that the formulation should be made more precise. That brings me to the next point, namely that there is a totally analogous formulation of the conjecture for elliptic curves over other number fields than the rationals, which might be worth mentioning (and also over function fields but I don't insist that this belongs here). Thirdly, it might be worth updating the current state of knowledge. The conjecture on the order of vanishing (over the rationals) is now known modulo 2, i.e. the parity of the order of vanishing of the L-function is equal to the parity of the rank, provided one assumed the finiteness of Tate-Shafarevich groups . And lastly, since there is a section on background, it might be worth explaining why it is heuristically plausible that the value at s=1 is somehow related to the number of points on the elliptic curve, namely it is related to the number of points modulo p as p varies over primes. I will be happy to modify the article if the author wants me to, but I thought I would first share my thoughts here before changing anything. AlexBartel (talk) 22:53, 5 May 2009 (UTC)
The statement of the conjecture is by far the most important portion of the article. Currently, this is how the statement of the conjecture reads:
"The conjecture relates arithmetic data associated to an elliptic curve E over a number field K to the behaviour of the Hasse-Weil L-function L(E, s) of E at s = 1. More specifically, it is conjectured that the rank of the abelian group E(K) of points of E is the order of the zero of L(E, s) at s = 1, and the first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K."
This seems fine until "and the first non-zero coefficient . . .." But that last clause is totally vague and needs to be rewritten, replaced, or deleted. (What does "more refined arithmetic data" mean???)
Let C denote an elliptic curve and C(Q) its rational points. Let L(C,s) denote the incomplete L-series of C.
Then the Clay Math Institute's writeup of this problem, by Andrew Wiles, describes the conjecture itself in full as follows:
"Conjecture (Birch and Swinnerton-Dyer). The Taylor expansion of L(C, s) at s = 1 has the form
L(C, s) = c(s − 1)r + higher order terms
with c ≠ 0 and r = rank(C(Q)). In particular this conjecture asserts that L(C, 1) = 0 if and only if C(Q) is infinite."