Timeline for Lorentzian characterization of genus
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 20, 2011 at 17:42 | vote | accept | Will Jagy | ||
| Jul 19, 2011 at 17:55 | comment | added | Will Jagy | Yes, I'm the one who is out of step. Kap always wrote quadratic forms as homogeneous polynomials. Indeed, in JKS (1997), we called forms odd or even, precisely backwards of what Conway and most of the world would. Then, if you talk to Wai Kiu Chan or anyone in the J. S. Hsia circle, fields vary, everything is scaled as needed anyway, none of this terminology matters. | |
| Jul 19, 2011 at 15:58 | answer | added | Abhinav Kumar | timeline score: 7 | |
| Jul 19, 2011 at 15:18 | comment | added | S. Carnahan♦ | I see. Somehow, I got used to the convention that one starts with quadratic forms as fundamental objects, and derives even bilinear forms by the rule $B(v,w) = Q(v+w)-Q(v)-Q(w)$. This would suggest that the hyperbolic plane is written as $xy$, your ternary quadratic form produces an even lattice, and the odd bilinear form giving the sum of $k$ squares is derived from $\frac12 \sum_{i=1}^k x_i^2$. It looks like you (and perhaps many others) are viewing integral bilinear forms as fundamental. At any rate, doubling doesn't lose any information with respect to integral equivalence or genera. | |
| Jul 19, 2011 at 4:31 | history | edited | Will Jagy | CC BY-SA 3.0 |
added 139 characters in body
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| Jul 19, 2011 at 3:51 | comment | added | Will Jagy | Oh, most of the root lattices are written as even. The most important single item in the present project is $E_8,$ let me find a link, math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E8.html if you look at the Gram matrix it is integral with all even numbers on the diagonal. So it is "even," and half of it is still an "integer-valued" quadratic form. | |
| Jul 19, 2011 at 3:46 | comment | added | Will Jagy | In short, I do not know why Conway mixes terminology. If I am to consider the genus of my ternary above, by adding in variables $u,v,$ is he talking about $f(x,y,z) + 2 u v$ or $f(x,y,z) + u v?$ I do not know. | |
| Jul 19, 2011 at 3:42 | comment | added | Will Jagy | My understanding is that an even lattice has all inner products integral and all norms even, so that includes the hyperbolic plane above, which I would prefer to write as $h(x,y) = 2 x y.$ Now, as far as I can make out, the sum of $k$ squares would be an odd lattice, as all inner products are integral but many vectors have odd norm (such as 1). So, while I would write a perfectly good ternary forms as $f(x,y,z) = x^2 + y^2 + z^2 + y z + z x + x y,$ in order to get integral inner products we need to double it, giving an even lattice. Annoying to me. | |
| Jul 19, 2011 at 3:32 | comment | added | S. Carnahan♦ | If $f$ and $g$ correspond to odd lattices, are you writing them as quadratic forms with half-integer coefficients? | |
| Jul 18, 2011 at 20:50 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Removed para indents
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| Jul 18, 2011 at 20:21 | history | asked | Will Jagy | CC BY-SA 3.0 |