Erica Klarreich, Quanta, Mathematicians Shed Light on Minimalist Conjecture, here.
In 1922, Louis Mordell proved something remarkable: For any given elliptic curve, even one with infinitely many rational points, it’s possible to generate all the rational points by starting with just a finite handful of them and then connecting the dots again and again. When the number of rational points of an elliptic curve is infinite, the number of points in the smallest handful that can generate essentially all the rational points is called the curve’s rank. (When the number of rational points is finite, mathematicians say that the curve has rank 0.)
For decades, mathematicians have tossed around the so-called minimalist conjecture, a proposition about the rank of elliptic curves for which the evidence is decidedly mixed. The conjecture speculates that, statistically speaking, half of all elliptic curves have rank 0 (meaning that they have either finitely many rational points or none at all) and half have rank 1 (meaning that their infinite set of rational points can be generated essentially from just one point). According to the conjecture, all other possibilities are vanishingly rare. That doesn’t mean that exceptions never occur, or even that there are finitely many of them — just that as you look at bigger and bigger collections of elliptic curves, the ones that fall into other categories are a smaller and smaller percentage of the whole, approaching 0 percent.