Timeline for Lorentzian characterization of genus

5 events

when toggle format	what		by	license	comment
Jul 20, 2011 at 17:42	vote	accept	Will Jagy
Jul 19, 2011 at 20:01	comment	added	Abhinav Kumar		You're welcome! Good point about classically integral including both even and odd - I forgot about the diagonal :)
Jul 19, 2011 at 18:42	comment	added	Will Jagy		Excellent. Cassels, page 111, for prime $p=2$ he says $x_1^2 + 2 x_2 x_2$ is "properly primitive" (and I believe "odd lattice") while $ 2 x_1^2 + 2 x_1 x_2$ is "improperly primitive" (and I believe "even lattice"). Then we have Lemma 4.2 on page 120 for unary cancellation in $\mathbb Z_2,$ Corollary on page 122 for improperly primitive. Next, odd $p$...
Jul 19, 2011 at 18:11	comment	added	Will Jagy		Thank you so much for a proof out of Cassels, which is approachable. I should say, although it may not matter, that "classically integral" includes both "odd" and "even" lattices, see the first page in chapter 26 of SPLAG for $I_{n,1}$ and $II_{n,1}.$ The odd ones are like $f(x,y) = x^2 + y^2,$ some vectors have odd norm. The even ones are like $g(x,y) = x^2 + x y + y^2,$ to get "integral" we must double to $2 x^2 + 2 x y + 2 y^2,$ and now vector norms are even. So the root lattice $E_8$ and the Leech lattice are "even" in the same sense, as polynomials it is possible to divide through by 2.
Jul 19, 2011 at 15:58	history	answered	Abhinav Kumar	CC BY-SA 3.0