# Robert F. Coleman 1954-2014

I am very sad to report that my Ph.D. advisor, Robert Coleman, died last night in his sleep at the age of 59.  His loving wife Tessa called me this afternoon with the heartbreaking news.  Robert was a startlingly original and creative mathematician who has had a profound influence on modern number theory and arithmetic geometry.  He was an inspiration to me and many others and will be dearly missed.

Robert and Bishop in Paris

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.