Timeline for How many solutions are there to $\sum_{i=1}^3 x_i^2+x_iy_i+y_i^2=k$?
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jan 25, 2012 at 8:24 | comment | added | GH from MO | @Noam: You are absolutely right, I don't know how I missed that. I also posted a supplement to your answer. | |
| Jan 25, 2012 at 8:23 | answer | added | GH from MO | timeline score: 7 | |
| Jan 24, 2012 at 20:35 | vote | accept | emiliocba | ||
| Jan 24, 2012 at 16:16 | answer | added | Noam D. Elkies | timeline score: 16 | |
| Jan 24, 2012 at 15:51 | comment | added | Noam D. Elkies | @GH There are no cusp forms here. In any case I think if the form is unique in its genus the theta function must be in the Eisenstein subspace. | |
| Jan 24, 2012 at 5:02 | comment | added | GH from MO | @Noam: You have a point, but if we decompose spectrally (into Hecke eigenforms) we might get a contribution from cusp forms which is probably not so simple to describe. On the other hand the level is small here, so there might be no cusp forms at all. | |
| Jan 24, 2012 at 0:20 | answer | added | Greg Martin | timeline score: 4 | |
| Jan 23, 2012 at 22:49 | comment | added | Noam D. Elkies | @emiliocba: As usual I don't know a reference, and it would be easier to (re)construct the formula than to locate it in the literature. If you already know the formula it's just a matter of checking that it gives rise to a modular form in the appropriate space and that this space is small enough that there a unique candidate form. If not, itshould be possible to surmise the formula from the first few dozen coefficients of the theta function $\left(\sum_{x,y\in\bf Z} q^{x^2+xy+y^2}\right)^3$. | |
| Jan 23, 2012 at 18:40 | history | edited | emiliocba | CC BY-SA 3.0 |
added 133 characters in body
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| Jan 23, 2012 at 18:15 | comment | added | emiliocba | Thank Noam, I had not been able to explain it. Do you know some reference in where this formula "could be"? | |
| Jan 23, 2012 at 17:42 | comment | added | Noam D. Elkies | @GH: True, but here the number of variables is even, so the theta function is a modular form has integral weight and the formula for its coefficients should be reasonably simple. | |
| Jan 23, 2012 at 17:30 | comment | added | GH from MO | It is not clear what you mean by "formula". For example, the quadratic form $x^2+y^2+z^2$ is also alone in its genus, but there is no simple way to determine the number of representations $x^2+y^2+z^2=k$. There is a formula involving class numbers and the square part of $k$, but it only connects two subtle quantities, neither of which is simpler than the other. | |
| Jan 23, 2012 at 15:29 | history | asked | emiliocba | CC BY-SA 3.0 |