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I read today that in 2010 Manjul Bhargava with Arul Shankar proved the conjecture basing upon the work of Kolyvagin. Is it right? Does it satisfy for all elliptic curves, or is it limited to some part? Can anyone tell the status of the conjecture?

The reference where I found this information is this.

Thanks a lot.

(I think intervention of some experts in Iwasawa theory is needed as it is about Iwasawa theory, I think that it will be nice if it is posted at MO.)

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The article you link doesn't say anyone's claimed a proof of Birch and Swinnerton-Dyer. It says there's a claimed proof that "a positive proportion of elliptic curves over Q" satisfy the conjecture. –  Chris Eagle Sep 26 '11 at 16:23
Even the Wikipedia link does not assert they proved the entire conjecture; rather, that they only established that a "positive proportion of elliptic curves over $\mathbb{Q}$[...] satisfy the Birch and Swinnerton-Dyer conjecture". That means that it holds for a nontrivial "amount" of elliptic curves, but not necessarily for all of them (e.g., a positive proportion of all rational primes are congruent to $1$ modulo $4$; that doesn't mean all rational primes are congruent to $1$ modulo $4$, nor that there are only finitely many exceptions). –  Arturo Magidin Sep 26 '11 at 16:23
I heard Bhargava repeat a joke auidience-question: Since he proved BSD holds for 10% of elliptic curves, shouldn't Clay offer him $100K? :-) –  Joseph O'Rourke Sep 26 '11 at 16:29
Iyengar: Please type more carefully. It is really not much to ask that you make an effort to have what you write minimally resemble the rest of the site. I did not downvote but given the complete absence of care with respect to spelling, capitalization and punctuation which you insistently display, I can imagine a few reasons why someone would think your post to be of bad quality. –  Mariano Suárez-Alvarez Sep 26 '11 at 16:54
I have given you a perfectly reasonable reason for your question to be downvoted. Please do not use these comments to discuss this, as it is completely off-topic: if you feel you must, start a meta thread. Here this discussion only makes more noise. –  Mariano Suárez-Alvarez Sep 26 '11 at 17:00

2 Answers 2

up vote 18 down vote accepted

Bhargava and Shankar have proved results about average $3$-Selmer ranks of elliptic curves. (See this arxiv preprint, which is the same paper cited by the Wikipedia article linked in the OP.) Their argument is via geometry of numbers (so to speak).

In fact, they are able to construct families such that exactly half of them have positive sign in their functional equation, and with average $3$-Selmer rank bounded by $7/6$. (This can be achieved by imposing appropriate conditions on the coefficients of the elliptic curve, and computing the root number as a product of local roots numbers.) Now work of the Dokchitser brothers on the parity conjecture implies that for elliptic curves with sign $+1$, the rank of the $3$-Selmer group is even. When combined with the bound of $7/6$, they deduce that the $3$-Selmer groups of the curves with sign $+1$ that lie in their family must be trivial.

Now (under some additional assumptions about the $3$-torsion, and some other technical assumptions, which they are able to impose on their family) by applying the results of Skinner and Urban on the Main Conjecture (which lets one pass from triviality of a Selmer group to non-vanishing of the $L$-function) they deduce that the curves in their family having sign $+1$ also have non-vanishing $L$-value at $s = 1$.

Now a positive proportion of elliptic curves overall lie in their family, and so putting all this together, one finds that a positive proportion of elliptic curves have both $3$-Selmer rank zero (and in particular, Mordell--Weil rank zero) and also analytic rank zero. Thus, a positive proportion of elliptic curves satisfy (the rank part of) BSD.

From my brief reading of the paper, the work of Kolyvagin and Gross--Zagier is not actually used (at least explicitly); Wikipedia seems to be in error on this point.

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"Birch and Swinnerton Dyer conjecture" usually refers to an amazing formula that predicts exactly the leading term of the L-function at $s=1$ (a real number $c$ and an integer $k$ such that the leading term is $c(s-1)^k$). The prediction of $k$ only, the conjecture that it is an "analytic rank" equal to the rank of the group of rational points on the curve, is the BSD rank conjecture. The million dollar prize is for the full conjecture with both $c$ and $k$, but the rank conjecture is a celebrated problem in its own right.

The Bhargava-Shankar paper proves, among other things, that a positive fraction of elliptic curves satisfy the rank prediction of BSD. In fact their ArXiv paper uses "BSD" to refer to the rank conjecture only. This is a loose but not uncommon use of the term.

I don't think the full BSD conjecture with the exact prediction of the leading coefficient of the L-function, is known for more than a finite number of curves.

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Dear zyx, Work of William Stein, Amod Agashe, and others, is pretty close to establishing, in the case when $L(E,1) \neq 0$, that the precise conjectured formula of BSD holds. So I don't think one is too far from improving Bhargava and Shankar's result to getting the full BSD conjecture for the rank $0$ members of their family. But you may well be right that at the moment one doesn't know full BSD for more than finitely many curves. Regards, –  Matt E Sep 27 '11 at 0:56
As far as I know, the Million Dollar prize is for the rank part of the conjecture (not that it matters). Moreover, it is historically fairly accurate to refer to the rank part as "BSD". The precise conjecture on the leading coefficient was formulated by Tate. Birch and Swinnerton-Dyer had given a formulation for rank 0 curves (in which case we can talk about the value instead of the leading coefficient), and had also indicated that in the case of positive rank, heights of the generators of the Mordell-Weil group seemed to play a role, but they never mentioned the regulator explicitly. –  Alex B. Sep 27 '11 at 0:58
@Alex B, thanks. About the leading coefficient, Tate's paper on B-SD calls the refined conjecture, for positive rank and using the height pairing, "the second conjecture [i.e., from the second paper] of Birch and Swinnerton". The same story appears in Milne's notes on elliptic curves. I thought Tate's contributions had to do with isogeny invariance, a formulation over function fields, and the generalization to Abelian varieties, but I don't know the history well at all. Is there a good reference? –  zyx Sep 27 '11 at 3:52
@Matt E, thanks for the information about the rank 0 case. –  zyx Sep 27 '11 at 4:11
@xyz Yes, there is: the original papers of Birch and Swinnerton-Dyer. It is very easy to convince oneself that they didn't have a precise expression for the leading coefficient for positive rank when they formulated the conjecture. In "Conjectures concerning elliptic curves", Proc. Symp. Pure Math. Vol. VIII, Birch explicitly credits Tate with the formulation we know today (penultimate paragraph). –  Alex B. Sep 27 '11 at 5:25

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