Elliptic curves over Q, and other news

[Posted on february 10, 2016]

Last week, a paper on the arxiv by Jennifer Park, Bjorn Poonen, John Voight and Melanie Matchett Wood presented heuristics which “suggests that there are only finitely many elliptic curves of rank greater than 21“, something which divides experts as I once read on MO. And just a few days ago, a new type of database of elliptic curves over \mathbb{Q} was announced by Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan, Simon Spicer, William Stein and James Weigandt, which seems to go in the same direction as the aforementioned heuristic.

One thing leading to another, I wondered what related topic could be made into a good polymath project (i.e. with a full proof not too distant a prospect). One candidate may be the characterization of non-torsion rational points on elliptic curves over \mathbb{Q} of rank \geq 2 (the torsion part is obviously well understood, I see there is an undergraduate book by Joseph H. Silverman and John Tate which should be a nice read, and lots of stuff on the web exists, e.g. see this short undergraduate presentation).

It seems to have all the ingredients required (computational aspect, wide-audience topic, …). Five years ago Harald Helfgott asked questions on that topic on MO, and while it initially attracted interest, there has been no news since, unfortunately. That area of math is extremely active obviously, and has been for a long time, e.g. there’s a conference for database pioneer John Cremona’s 60th birthday that will take place next april in Warwick. So there would surely be enough people interested, while lots of software exists obviously which was used to compile the recent database, and also here’s a standalone plotter by Stefan Kebekus under the GPL just to mention one more item.

Update (february 15): well it wouldn’t have been wise to start a polymath on that topic just now, as today appeared a preprint by Sara Checcoli, Francesco Veneziano and Evelinda Viada which appears to solve this (if I understand correctly…), and much more! This paper looks highly important indeed.

In other news :

  • a day of festivities around the issue of a stamp celebrating Sophie Germain will be take place at IHP on march 18
  • the next day a Séminaire Bourbaki is scheduled, including a talk by Geordie Williamson. It is noticeable that the list of authors following the customary “[d’après…]” tend to be longer and longer these days
  • mathblogging.org had a change of URL
  • Cédric Villani has found yet another idea to make mathematians look like they’re not necessarily elitist ivory tower types, reviewing a concert of a symphonic metal band

Libros Libres, by Alan Levine on flickr

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