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So it was conjectured that if all elliptic curves over $\mathbb{Q}$ are ordered by their heights, then the average rank is $\frac{1}{2}$.

Brummer initially showed assuming BSD and GRH that the average rank is bounded by 2.3. Since then many improvements have been made. In my search, I found the slides for a talk by Manjul Bhargava (linked here: http://www.dpmms.cam.ac.uk/research/BSD2011/bsd2011-Bhargava.pdf), where he talks about his result showing that the average rank is bounded by 1.5 unconditionally.

My question is has there been any improvement on his result since then? A reference to such a paper would be appreciated as well.

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    $\begingroup$ I think the general expectation is still (or again) in favor of the "minimalist"conjecture of average rank $1/2$, i.e. asymptotically 100% of curves have the smallest rank consistent with their parity, which is even or odd with equal probability. Meanwhile, last I heard Manjul Bhargava together with his student Arul Shankar have pushed the upper bound on the average rank down to $0.99$ . $\endgroup$
    – Noam D. Elkies
    Commented May 8, 2012 at 3:25
  • $\begingroup$ Noted. I've edited my question to reflect your comment. $\endgroup$
    – Eugene
    Commented May 8, 2012 at 3:26

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Color me surprised if people no longer believe that the distribution of elliptic curves is half rank zero, half rank one, and a density zero subset of higher rank curves. To my knowledge this is still a conjecture that people believe.

I believe the state of the art results are still due to Bhargava and Shankar, and are best summed up in the slides listed in your post. In particular:

  • The average size of a 3-Selmer group is 4
  • The average size of a 4-Selmer group is 7
  • The average size of a 5-Selmer group is 6
  • These all hold true up to a finite number of congruence conditions
  • A positive proportion of elliptic curves have rank zero
  • Assuming the finiteness of Sha, a positive proportion of elliptic curves have rank 1
  • Unconditionally, the average rank of an elliptic curve over $\mathbb{Q}$ is strictly less than one

There are lots of good expositions of this work (or at least the 2-Selmer result), for instance this Seminar Bourbaki article of Poonen ( http://www-math.mit.edu/~poonen/papers/Exp1049.pdf ) and this short note of Gross ( http://www.math.harvard.edu/~gross/preprints/manjul.pdf ) .

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  • $\begingroup$ I probably misunderstood what he wrote in the slides when he said, "Computations do not currently give much support to the conjecture either," to mean that it isn't widely believed anymore. I apologize for the error. $\endgroup$
    – Eugene
    Commented May 8, 2012 at 3:48
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Thank you everyone for the great references.

I also recently found this really good summary by Alice Silverberg about all things related to the rank of an elliptic curve if anyone is interested.

http://math.uci.edu/~asilverb/connectionstalk.pdf

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