There's been a breakthrough in elliptic curves - a subject important in number theory and cryptography! An

**elliptic curve** is defined by an equation like

y² = x³ + ax + b

for some choice of a and b. If we graph the

*real numbers* x and y that solve this equation, we get pictures like those here. But the true beauty of an elliptic curve is revealed when we draw the

*complex* solutions: we get a shape like the surface of a doughnut with one point removed, and people usually add in that extra point to make it nicer.

If a and b are

*rational* numbers it's very interesting to look for solutions where both x and y are

*rational:* these are called

**rational points** of the elliptic curve. They're hard to find, but if you find two there's a cool way to get more of them. Sometimes there are infinitely many, sometimes just finitely many.

Now Manjul Bhargava (one of the youngest people to become a full professor at Princeton University) and Arul Shankar (his grad student) have shown that if you randomly choose rational numbers a and b, an elliptic curve has just finitely many rational points at least 12% of the time!

This is a step toward proving the

**minimalist conjecture**, which says - among other things - that in fact this happens exactly 50% of the time.

I'm leaving out a lot of fun and interesting details, just trying to give you a precise statement of what was discovered, with a minimum of fancy jargon. If you want more, this popularization is great:

https://www.simonsfoundation.org/features/science-news/mathematicians-shed-light-on-minimalist-conjecture/Thanks to

+David Roberts for pointing this out!

If you want even more, try this expository account:

• Bjorn Poonen, Average rank of elliptic curves,

http://arxiv.org/abs/1203.0809.

#spnetwork #recommend #numberTheory #ellipticCurves #expository arXiv:1203.0809

If you want