This past weekend I had the privilege to speak at the Southern California Number Theory Day along with Manjul Bhargava, Elena Fuchs, and Chris Skinner. Manjul and Chris spoke about a series of remarkable results which, when combined, prove that at least 66.48% of elliptic curves over satisfy the (rank part of the) Birch and Swinnerton-Dyer (BSD) Conjecture (and have finite Shafarevich-Tate group). Bhargava’s work with Arul Shankar also proves that at least 20.6% of elliptic curves over have rank 0, at least 83.75% have rank at most 1, and the average rank is at most 0.885. Conjecturally, 50% of elliptic curves have rank 0, 50% have rank 1, and 0% have rank bigger than 1, and thus the average rank should be 0.5. (And conjecturally, 100% of elliptic curves satisfy the BSD conjecture. :)) Before the work of Bhargava-Shankar and Bhargava-Skinner (which makes use of recent results of Skinner-Urban. Wei Zhang, and the Dokchitser brothers among others), the best known unconditional results in this direction were that at least 0% of elliptic curves have rank 0, at least 0% have rank 1, the average rank is at most infinity, and at least 0% of curves satisfy the BSD conjecture.
I will attempt to briefly summarize some of the main ideas from their talks; see these papers by Bhargava-Skinner and Bhargava-Shankar for more details and references. (The paper of Bhargava, Skinner, and Wei Zhang showing 66.48% is forthcoming. [Note added 7/8/14: that paper has now appeared at http://arxiv.org/abs/1407.1826.]) Continue reading