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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)
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