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
If you want