The average rank of elliptic curves over Q ordered by height is at most 1.5

You need to know: For integers a,b, notation $a|b$ if b is divisible by a, and $a\nmid b$ if not, notation ${\mathbb Q}$ for the set of rational numbers, elliptic curve over ${\mathbb Q}$ , rank $\text{rank}(E)$ of elliptic curve E, limit superior $\limsup$ , notation $|S|$ for the number of elements in any finite set S.

Background: Any elliptic curve E over ${\mathbb Q}$ can be written in the form $y^2 = x^3+ax+b$ , where $a,b$ are integers such that if $p^4|a$ for some prime p, then $p^6\nmid b$ . The height of E is $\max\{4|a^3|, 27b^2\}$ . For any h>0, let $S(h)$ be the set of elliptic curves of height at most h.

The Theorem: On 4th June 2010 Manjul Bhargava and Arul Shankar submitted to arxiv a paper in which they proved that $\limsup\limits_{h\to\infty}\left(\frac{1}{|S(h)|}\sum\limits_{E\in S(h)}\text{rank}(E)\right) \leq 1.5$ .

Short context: Quantity $\frac{1}{|S(h)|}\sum\limits_{E\in S(h)}\text{rank}(E)$ is the average rank of elliptic curves of height at most h, and the Theorem states that average rank of all elliptic curves over Q ordered by height is at most 1.5. It is an old conjecture that 50% of all elliptic curves over Q have rank $0$ and 50% have rank 1, which would imply that the average rank is 0.5. However, before 2010, no-one could prove that the average rank is bounded from above by any finite constant whatsoever.