Timeline for Is the square of the covering radius of an integral lattice/quadratic form always rational?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 5, 2011 at 22:22 | vote | accept | Will Jagy | ||
| Jul 4, 2011 at 20:55 | comment | added | Will Jagy | I sent you an email, I'm holding a copy of the first edition of SPLAG. Let's see, they say the deep holes in the Leech lattice have a one-to-one correspondence with the Niemeier lattices, page 413. In Chapter 23 on the covering radius of the Leech lattice, Theorem 2 says there are 23 inequivalent deep holes in the Leech lattice, these correspond with the other 23 Niemeier lattices. | |
| Jul 4, 2011 at 19:48 | comment | added | Noam D. Elkies | Come to think of it, if $L = E_8$ then for each deep hole $L' = D_8$, so that's already a counterexample to $L'=L$ — though here $[L:L']=2$ so we can't get an extra odd factor in the denominator, and as it happens the covering radius is 1 so there's no denominator at all. [If we construct $E_8$ as the union of $D_8$ with its shift by $(\frac12,\frac12,\ldots,\frac12)$ then all deep holes are equivalent to $(1,0,0,\ldots,0)$.] What happens for the Leech lattice? I don't have a copy of SPLAG handy today... | |
| Jul 4, 2011 at 4:07 | comment | added | Noam D. Elkies | The "classes in the genus of the Leech lattice" are the 24 Niemeier lattices, right? | |
| Jul 4, 2011 at 3:30 | comment | added | Will Jagy | Oh, and the part about class number one refers to strict inequality, otherwise the Leech lattice would be included. I don't know the number of classes in the genus of the Leech lattice, however it is not one. | |
| Jul 4, 2011 at 2:33 | comment | added | Will Jagy | Thanks, Noam. All examples are in dimension 10 or less. Also, all Pete's examples have class number one, and it is a result of George Leo Watson that this implies dimension no larger than 10. I have been searching for some time for an a priori proof of class number one, no luck. | |
| Jul 4, 2011 at 2:23 | history | answered | Noam D. Elkies | CC BY-SA 3.0 |