Kevin Hartnett in Quanta:
Elliptic curves seem to admit infinite variety, but they really only come in two flavors. That is the upshot of a new proof from a graduate student at Harvard University.
Elliptic curves may sound exotic, but they’re unspectacular geometric objects, as ordinary as lines, parabolas or ellipses. In a paper first posted online last year, Alexander Smith proved a four-decade-old conjecture that concerns a fundamental trait of elliptic curves called “rank.” Smith proved that, within a specific family of curves, and with one qualification, half of all curves have rank 0 and half have rank 1.
The result establishes baseline characteristics of objects that have intrigued mathematicians for centuries, and that have increased in importance in recent decades.
“We’ve been thinking about this for over 1,000 years, and now we have some probabilistic sense about [elliptic curves]. That’s super important,” said Shou-Wu Zhang, a mathematician at Princeton University who advised Smith at the outset of the work, when Smith was an undergraduate at Princeton.