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
Full Text:

References:

 [1] E. Hecke, Analytische Arithmetik der positiven quadratischen Formen. Konigh. Danske Videnskabernes Selskab. Math.-Fys. Medd.17, 12 (1940). · Zbl 0024.00902 [2] J. Leech andN. J. A. Sloane, Sphere packings and error-correcting codes. Canad. J. Math.23, 718-745 (1971). · Zbl 0218.52008 [3] M. Ozeki, Note on the positive definite integral quadratic lattice. J. Math. Soc. Japan28, 421-446 (1976). · Zbl 0324.10013 [4] M.Ozeki, On even unimodular lattices of rank 32. To appear, in Math. Z. · Zbl 0564.10016 [5] M. Peters, Definite unimodular 48-dimensional quadratic forms. Bull. London Math. Soc.15, 18-20 (1983). · Zbl 0506.10016 [6] B. B. Venkov, On the classification of integral even unimodular 24-dimensional quadratic forms. Proc. Steklov Inst. Math.148, 63-74 (1980). · Zbl 0443.10021 [7] B. B. Venkov, Even unimodular Euclidean lattices of dimension 32 (Russian). Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)116, 44-55 (1982). · Zbl 0506.10015 [8] B. B. Venkov, On even unimodular extremal lattices (Russian). Trudy Mat. Inst. AN SSSR165, 43-48 (1984). · Zbl 0544.10017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.