## 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!

Cathy,

I hope you don’t regard your decision to “leave math behind” as a permanent state. The good news is that wonderful adventures are there any time you want to go exploring. I came back after some 20+ years hiatus, and the difference now is that I don’t get paid for doing math. (I do a small amount of teaching, but not for the money.) This is freeing.

Fred Dashiell

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Yeah, but if leaving math behind means engaging the “real world,” and leaving the Ivory Tower behind, then this can be viewed as a good thing. Years ago, we never even heard of Selmer Groups in grad math.

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Very cool to see ideas come to fruition when you thought they would wither and perish. On a tangential note, does anyone know the origin of the term “variety”? To my ear it is a completely random pairing of English noun and mathematical notion. These things bother me.

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I’m curious to know what a mathematician thinks about a (trivial) grammatical question posed in the 2nd sentence of the abstract. A picky editor might correct “correspond” to “corresponds” to agree with the subject “number.” However, “two or more invariants” is equivalent to “the number of invariants is at least two” and would agree with “correspond.” Is “correspond” then preferable? Awesome abstract!

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Orbits.

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? (I’m a cube.)

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Totally off-topic:”Economics students aim to tear up free-market syllabus” http://www.theguardian.com/business/2013/oct/24/students-post-crash-economics (a few days old, but I don’t know if you read The Guardian)

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I remember having a really intense conversation with you about this while waiting in a long line at the Chinese consulate in New York! Must have been 2005. Manjul was giving lectures about this stuff in Princeton. I remember I thought the right way to think about it was in terms of understanding what kind of integral models corresponded to the different orbits, but based on the talks I’ve seen Wei give about this, that turned out not to be the way they went about things at all. I think I had no conception at the time that you could sieve down to those orbits that corresponded to Selmer cubics (made-up notation, I mean those cubics with rational points everywhere locally.)

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That was one nasty consulate.

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I didn’t even notice because we were having so much fun talking number theory!

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