Category Archives: Elliptic curves

An April Fools’ Day to Remember

Today is the 10th anniversary of the death of Martin Gardner. His books on mathematics had a huge influence on me as a teenager, and I’m a fan of his writing on magic as well, but it was only last year that I branched out into reading some of his essays on philosophy, economics, religion, literature, etc. In this vein, I highly recommend “The Night Is Large”, a book of collected essays which showcases the astonishingly broad range of topics about which Martin had something interesting to say. It’s out of print, but it’s easy to find an inexpensive used copy if you search online.

Thinking back on my favorite Martin Gardner columns, my all-time favorite has to be the April 1975 issue of Scientific American. In that issue, Martin wrote an article about the six most sensational discoveries of 1974. The whole article was an April Fools’ Day prank: among the discoveries he reported were a counterexample to the four-color problem and an artificial-intelligence computer chess program that determined, with a high degree of probability, that P-KR4 is always a winning move for white. The article also contained the following:

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The BSD conjecture is true for most elliptic curves

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 \mathbf Q 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 \mathbf Q 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 →