Rubik’s cubes and Selmer groups
One of my biggest regrets when I left academic math and number theory behind in 2007 was that I never finished writing up and publishing some cool results I’d been working on with Manjul Bhargava about what we called “3x3x3 Rubik’s cubes”.
Just a teeny bit of background. Say you have a 3x3x3 matrix filled with numbers, including in the very center. So you have 27 numbers in a special 3-dimension configuration. Since there are three axis for such a cube, there are three ways of dividing such a cube into three 3×3 matrices and Once you do that you can get a cubic form by computing
which gives you a cubic equation in three variables, or in other words a genus one curve.
Actually you get three different genus one curves, since you do it along any axis. Turns out there are crazy interesting relationships between those curves, as well as in the space of all 3x3x3 cubes.
Just talking about that stuff gets me excited, because it’s first of all a really natural construction, second of all number theoretic, and third of all it actually makes me think of solving Rubik’s cubes, which I’ve always loved.
Anyhoo, I gave my notes to a grad student Wei Ho when I left math, and she and Manjul recently came out with this preprint entitled “Coregular Spaces and genus one curves”, which is posted on the mathematical arXiv.
First, what’s freaking cool about their paper, to me personally, is that my work with Manjul has been incorporated into the paper in the form of parts of sections 3.2 and 5.1.
But what’s even more incredibly cool, to the mathematical world, is that Wei and Manjul are going to use this paper as background to understand the average size of Selmer groups of elliptic curves, a really fantastic result. Here’s the full abstract of their paper:
A coregular space is a representation of an algebraic group for which the ring of polynomial invariants is free. In this paper, we show that the orbits of many coregular irreducible representations where the number of invariants is at least two, over a (not necessarily algebraically closed) field k, correspond to genus one curves over k together with line bundles, vector bundles, and/or points on their Jacobians. In forthcoming work, we use these orbit parametrizations to determine the average sizes of Selmer groups for various families of elliptic curves.
One last thing. I am lucky enough to be a neighbor of Wei right now, as she finishes up a post-doc at Columbia, and she’s agreed to explain this stuff to me in the coming weeks. Hopefully I will remember enough number theory to understand her!