## On the configurations of even unimodular lattices of rank 48.(English)Zbl 0571.10020

Let $$\Gamma_{48}$$ be the genus consisting of all isomorphic classes of positive definite even unimodular lattices of rank 48. Let L be an element of $$\Gamma_{48}$$. An element x of L is called a 2m-vector if it satisfies $$(x,x)=2m$$, where (, ) is the metric attached to L. We let $${\mathcal S}_{2m}(L)$$ denote the sublattice of L generated by all 2m- vectors and $${\mathcal S}_{2m_ 1+2m_ 2}(L)$$ the sublattice of L generated by all $$2m_ 1$$-vectors and $$2m_ 2$$-vectors in L. Let a(2t,L) be the number of 2t-vectors in L for positive integer t. $$A_ n$$, $$D_ n$$, $$E_ n$$ is used to denote the root lattices.
The main purpose of this paper is to prove Theorem 1. Let L be an element of $$\Gamma_{48}$$. If $$a(2,L)=a(4,L)=0$$, then $${\mathcal S}_ 6(L)=L.$$
Theorem 2. Let L be an element of $$\Gamma_{48}$$. Assume that either $$a(2,L)>0$$ or $$a(4,L)>0$$ hold. Then we have,
(i) if rank $${\mathcal S}_ 2(L)\geq 3$$, then rank $${\mathcal S}_ 4(L)=48,$$
(ii) if rank $${\mathcal S}_ 2(L)=2$$ and $${\mathcal S}_ 2(L)\cong A_ 2$$ (isomorphic), then either rank $${\mathcal S}_ 4(L)=48$$ or rank $${\mathcal S}_ 6(L)=48,$$
(iii) if rank $${\mathcal S}_ 2(L)=2$$ and $${\mathcal S}_ 2(L)\cong A_ 1\oplus A_ 1$$, then rank $${\mathcal S}_ 4(L)=48,$$
(iv) if rank $${\mathcal S}_ 2(L)=1$$, then either rank $${\mathcal S}_ 4(L)=48$$ or rank $${\mathcal S}_ 6(L)=48,$$
(v) if rank $${\mathcal S}_ 2(L)=0$$, then rank $${\mathcal S}_ 4(L)=48$$.

### MSC:

 11E12 Quadratic forms over global rings and fields 11F27 Theta series; Weil representation; theta correspondences 11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)

### Keywords:

sublattice; 2m-vectors
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### References:

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