An elliptic curve of rank ≥ 30(elliptic-rank.icarm.cloud) |
An elliptic curve of rank ≥ 30(elliptic-rank.icarm.cloud) |
This morning, a mysterious user named "ranksunbounded" submitted the linked curve, which has rank at least thirty. This breaks the previous record of 29 found by Elkies and Klagsbrun in 2024.
Nobody knows whether it is possible to achieve arbitrarily high ranks.
Within academia, it is to your definite advantage to have your name associated with breakthroughs, for obvious reasons. Outside academia I don't have firsthand experience, but I'd have to imagine the same holds.
> it was Claude, with Levent Alpöge and Ava Howell!
Alpöge is the same guy who prompted Claude for a proof of the Jacobian conjecture.
I know your reaction is the normal one these days, but I’d like to normalize the ones that don’t care about society petting them on the head and saying atta boy. They just deliver the work and retreat into the darkness. I find it highly inspiring. The alternative.. not so much.
https://en.wikipedia.org/wiki/Superpermutation#Lower_bounds,...
BSD = https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture
GRH = https://en.wikipedia.org/wiki/Generalized_Riemann_hypothesisIt sounds like the big deal is you have at least 30 points in a graph that aren’t related and can be expressed as rational numbers (not necessarily integers so you get giant fractions). Being very humble here and invite people more advanced in mathematics to chime in where I’m wrong.
So like this is one of the x,y coordinate points that is two rational numbers.
x=30786757706172245427369935940751/4
y=58841476683002984849182029306774218124047405249/8
So there’s at least 30 points on the graph like this, which is a big deal because before there were only 29 independent points verified in any elliptic curve equation.
“Independent” seems to mean something precise here too but I’ll let someone else explain that because I’m not quite sure, somehow the points have to be unrelated to one another.
Bumping the rank from 28->29 previously took over ten years, so it’s a big deal in mathematics apparently.
[1] There are some details: https://en.wikipedia.org/wiki/Elliptic_curve#Group_law
There are different types of addition, though. A rank two addition would mean it looks like (x, y) + (x', y') = (x + x', y + y'). A rank three addition would mean it looks like (x, y, z) + (x', y', z') = (x + x', y + y', z + z'). Here they found the first example of an elliptic curve where the rank is 30.
To really quantify how exotic, it's conjectured that curves with rank 2 or greater have an asymptotic density of zero. That doesn't mean they don't or can't exist, but it does mean they become vanishingly rare, so finding even individual examples of high-rank curves has been absurdly hard.
The record before this morning was 29 and people suspected that was as high as it got. When we learn how this curve was obtained that might change.
Exactly this. A fundamental question in the subject is, whether elliptic curve ranks are bounded. Contrast with e.g. prime numbers, of which are known to be infinitely many. If you set a new record for the largest known prime, then that's cool but everyone knew there were plenty out there to discover.
This paper, by leading experts,
https://arxiv.org/abs/1602.01431
made a significant impact in the field, coming up with a heuristic argument for why ranks of elliptic curves should be bounded. The same heuristic suggests, albeit more loosely, that we should perhaps be a little bit surprised to see a curve with rank at least 30. So it's mild evidence that the heuristic itself could be mistaken.
https://www.quantamagazine.org/without-a-proof-mathematician...