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.



Average rank of elliptic curves over $\mathbb{Q}$
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> ) .


