On the structure of even unimodular extremal lattices of rank 40
Michio Ozeki · Rocky Mountain Journal of Mathematics · 1989
In memory of the late Professor Hei Braun 1. Introduction.Let Tgk(k > 1) be the genus consisting of all equivalence classes of positive definite even unimodular quadratic lattices of rank 8k.In an element L of Tg*:, a vector x in L is called a 2ra-vector if x satisfies (#, x) = 2ra, where (, ) is the inner product of L and 2ra is an even integer.In obtaining a complete picture of the configurations of all equivalence classes in T%k{k > 4), the classes of lattices without 2-vectors would be a main obstacle.In this paper, we study the subfamily T^o of r 40 consisting of all equivalence classes of lattices without 2-vector.As in [7], we use £2m(L) (respectively, C2m l +2m 2 {L)) to denote the sublattice of L generated by all 2m-vectors (respectively 2mi-vectors and 2m2-vectors) in L. In §2, we prove THEOREM 1.Let L be a lattice in T^o-Then we have L = £4+6 (L).In §3,we shall introduce the notion of the c-sublattice of a lattice in T4o,o-We expect this notion would play a role in the study of the structures of lattices in T4o,o, and also in r 32 ,o.However our present study of the e sublattice is merely a beginning of exploration.We collect some standard notations used throughout the paper: Q is the field of rational numbers, Z is the ring of rational integers, M(l, k) (respectively S(l, k)) is the linear space of modular (respectively cusp) forms of degree 1 and weight &, Efc(z) is Eisenstein series of degree 1 and weight fc, Ai2(z) is the normalized cusp form of degree 1 and weight 12. Special notations are explained in the appropriate places if necessary.