Average rank of elliptic curves over $\mathbb{Q}$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T12:05:28Z http://mathoverflow.net/feeds/question/96285 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96285/average-rank-of-elliptic-curves-over-mathbbq Average rank of elliptic curves over $\mathbb{Q}$ Eugene 2012-05-08T03:08:36Z 2012-05-09T18:36:21Z <p>So it was conjectured that if all elliptic curves over $\mathbb{Q}$ are ordered by their heights, then the average rank is $\frac{1}{2}$. </p> <p>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: <a href="http://www.dpmms.cam.ac.uk/research/BSD2011/bsd2011-Bhargava.pdf" rel="nofollow">http://www.dpmms.cam.ac.uk/research/BSD2011/bsd2011-Bhargava.pdf</a>), where he talks about his result showing that the average rank is bounded by 1.5 unconditionally. </p> <p>My question is has there been any improvement on his result since then? A reference to such a paper would be appreciated as well.</p> http://mathoverflow.net/questions/96285/average-rank-of-elliptic-curves-over-mathbbq/96287#96287 Answer by stankewicz for Average rank of elliptic curves over $\mathbb{Q}$ stankewicz 2012-05-08T03:40:08Z 2012-05-08T03:40:08Z <p>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.</p> <p>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:</p> <ul> <li>The average size of a 3-Selmer group is 4</li> <li>The average size of a 4-Selmer group is 7</li> <li>The average size of a 5-Selmer group is 6</li> <li>These all hold true up to a finite number of congruence conditions</li> <li>A positive proportion of elliptic curves have rank zero</li> <li>Assuming the finiteness of Sha, a positive proportion of elliptic curves have rank 1</li> <li>Unconditionally, the average rank of an elliptic curve over $\mathbb{Q}$ is strictly less than one</li> </ul> <p>There are lots of good expositions of this work (or at least the 2-Selmer result), for instance this Seminar Bourbaki article of Poonen ( <a href="http://www-math.mit.edu/~poonen/papers/Exp1049.pdf" rel="nofollow">http://www-math.mit.edu/~poonen/papers/Exp1049.pdf</a> ) and this short note of Gross ( <a href="http://www.math.harvard.edu/~gross/preprints/manjul.pdf" rel="nofollow">http://www.math.harvard.edu/~gross/preprints/manjul.pdf</a> ) .</p> http://mathoverflow.net/questions/96285/average-rank-of-elliptic-curves-over-mathbbq/96491#96491 Answer by Eugene for Average rank of elliptic curves over $\mathbb{Q}$ Eugene 2012-05-09T18:36:21Z 2012-05-09T18:36:21Z <p>Thank you everyone for the great references. </p> <p>I also recently found this really good summary by Alice Silverberg about all things related to the rank of an elliptic curve if anyone is interested.</p> <p><a href="http://math.uci.edu/~asilverb/connectionstalk.pdf" rel="nofollow">http://math.uci.edu/~asilverb/connectionstalk.pdf</a></p>