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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 answer to the question in the title would seem to be "no". And to the question in the body of the post "If you know more than we do, claim your $1m". As it stands neither is really a proper question, and this ought to be closed - except that the comments are interesting.2011-09-26

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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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    @Alex: I'll have a look at that, but of what is available online: in the 75th birthday volume for Swinnerton-Dyer, Birch writes of the positive rank leading coefficient conjecture that the 1965 doctoral thesis of his student Nelson Stephens "is where it is first precisely stated". Stephens' 1966 paper "Conjectures concerning Elliptic Curves", taken from his thesis, formulates the precise conjecture (for twists of the Fermat cubic) in terms of heights and says "this is in essence a formula suggested by Tate and others, by analogy with the work of Ono [on Tamagawa numbers]."2011-09-27