Timeline for How to distinguish division algebras from matrix algebras?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 11, 2011 at 8:31 | vote | accept | Tim Dokchitser | ||
| Mar 10, 2011 at 12:42 | answer | added | Henri Johnston | timeline score: 8 | |
| Mar 10, 2011 at 11:21 | comment | added | Tim Dokchitser | Thanks, Martin! I asked Tom, and his approach is to compute the maximal order (there is an implementation on Magma by Michael Stoll). If the maximal order has discriminant 1, then $D$ is a matrix algebra. They even go further, and use LLL on the maximal order to find an explicit isomorphism with a matrix algebra (the shortest vector is a zero divisor). As Claus says in his edit, this avoids Galois cohomology and, especially, $S$-unit calculations. Unfortunately, at the moment it is only implemented when the centre of $D$ is ${\mathbb Q}$. | |
| Mar 8, 2011 at 1:03 | answer | added | Claus Fieker | timeline score: 5 | |
| Mar 7, 2011 at 14:26 | comment | added | Martin Bright | Have you tried using the Galois cohomology routines in Magma? Assuming you can get a 2-cocycle representing your algebra, have a look at SUnitCohomologyProcess and IsGloballySplit, which will test an explicit 2-cocycle to see whether it's a coboundary. I must admit I've only tried it for quaternion algebras. There is a paper by Claus Fieker describing the algorithm (essentially an S-unit calculation, but being cunning to reduce S as much as possible). Alternatively there might be something in Cremona, Fisher, O'Neil, Simon, Stoll's papers on explicit descent. (You could ask Tom.) | |
| Mar 7, 2011 at 13:25 | history | edited | Tim Dokchitser | CC BY-SA 2.5 |
Expanded last paragraph on possible approaches
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| Mar 6, 2011 at 8:48 | comment | added | Tim Dokchitser | @Dror: Thank you! It looks like this is way too many variables and equations for the Groebner basis machinery to handle, but I'll think a bit more about this... | |
| Mar 5, 2011 at 23:32 | comment | added | Dror Speiser | Not efficient, but an idea: assume it is a matrix algebra, i.e. there are matrices $A$ and $B$, satisfying cubics which you know, and a product relation. Let the coefficients of $A$ and $B$ be 9 variables each. From the relations you get $9+9+9$ equations, that give an ideal in $\mathbb{Q}[x_1,...,x_{18}]$. Now you can try to use Groebner bases to find a solution, or prove that one doesn't exist. | |
| Mar 5, 2011 at 23:16 | answer | added | Mikhail Bondarko | timeline score: 3 | |
| Mar 5, 2011 at 22:24 | comment | added | Tim Dokchitser | Rachel Newton is doing explicit computations with cup products in class field theory, and one of the things she wants to test boils down to just this. So these are really just central simple algebras, honestly given by generators and relations. | |
| Mar 5, 2011 at 21:28 | comment | added | William Stein | What is your application? E.g., are you interested in whether or not an abelian variety is absolutely simple? I've thought about this problem in the context of endomorphism rings of modular abelian varieties; there one can tell whether or not the algebra is a matrix algebra using results of Ribet and Lario (?), but finding an explicit representation as a matrix algebra is harder. | |
| Mar 5, 2011 at 16:43 | history | asked | Tim Dokchitser | CC BY-SA 2.5 |