Yet another proof of the uniqueness of the E_{8} lattice

Noam D. Elkies August, 2004

Theorem (Mordell 1938 [2]). Let L be an even unimodular positive-definite lattice of rank 8. Then L ≅ E_{8}.

Yet another proof : We construct an odd “2-neighbor” L′ of L, identify L′ with Z^{8} using our characterization of Z^{n} by its minimal characteristic norm [1], and use this to identify L with E_{8}.

A “2-neighbor” of L is a self-dual lattice L′ ⊂ L ⊗ Q such that L_{0} = L ∩ L′ is a sublattice of index 2 in each of L and L′. To obtain L_{0} and L′, first fix a lattice vector v_{0} ∉ 2L such that 4|(v_{0}, v_{0}). (Such a vector exists in every even lattice of rank at least 3.) We may assume that (v_{0}, v_{0}) ≡ 4 mod 8, since if 8|(v_{0}, v_{0}) we may replace v_{0} by v_{0} + 2w with (v_{0}, w) odd. Set

L_{0} = \{v ∈ L : 2|(v, v_{0})\}, L′ = L_{0} ∪ (L_{0} + \frac{v_{0}}{2} ).

We claim that L′ is an odd self-dual lattice. That it is a lattice is clear since certainly v_{0} ∈ L_{0}. It is contained in its dual by construction. Since [L′ : L_{0}] = [L : L_{0}] = 2, we have disc(L′) = disc(L) = 1, so L′ ⊆ L′^{∗} implies L′ = L′^{∗}. Finally L′ is odd because it contains the vector v_{0}/2 of odd norm.

Recall (see for instance [1]) that every unimodular lattice M has characteristic vectors: lattice vectors c ∈ M such that (c, v) ≡ (v, v) mod 2 for all v ∈ M. These constitute a coset of 2M in M, and if M is positive definite of rank n then (c, c) ≡ n mod 8 for all characteristic c ∈ M. Since L′ is odd, 0 is not a characteristic vector of L′. Hence for every characteristic vector c of L′ we have (c, c) ≥ 8 = rank(L′). By [1] it follows that L′ ≅ Z^{8}.

Therefore L_{0}, being an even lattice contained with index 2 in L′, is a D_{8} lattice. Thus L is one of the three lattices intermediate between D_{8} and D_{8}^{∗}. One of these lattices is L′ = Z^{8}. The other two are isomorphic to E_{8}, Q.E.D.

Acknowledgements. This work was supported in part by the the National Science Foundation (grant DMS-0200687), and took place during a workshop at the American Institute of Mathematics.

References

[1] Elkies, N.D.: A characterization of the Z^{n} lattice. Math. Res. Letters 2

     #3 (5/1995), 321–326. math.NT/9906019 at arXiv.org.

[2] Mordell, L.J.: The definite quadratic forms in eight variables with determinant unity, J. Math. Pures Appl. 17 (1938), 41–46.