2 Strongly modular lattices with long shadow . Gabriele Nebe

To an integral lattice L in the euclidean space (R, (, )), one associates the set of characteristic vectors v ∈ R with (v, x) ≡ (x, x) mod 2Z for all x ∈ L. They form a coset modulo 2L, where L = {v ∈ R | (v, x) ∈ Z ∀x ∈ L} is the dual lattice of L. Recall that L is called integral, if L ⊂ L and unimodular, if L = L. For a unimodular lattice, the square length of a characteristic vector is congruent to n modulo 8 and there is always a characteristic vector of square length ≤ n. In [1] Elkies characterized the standard lattice Z as the unique unimodular lattice of dimension n, for which all characteristic vectors have square length ≥ n. [2] gives the short list of unimodular lattices L with min(L) ≥ 2 such that all characteristic vectors of L have length ≥ n− 8. The largest dimension n is 23 and in dimension 23 this lattice is the shorter Leech lattice O23 of minimum 3. In this paper, these theorems are generalized to certain strongly modular lattices. Following [7] and [8], an integral lattice L is called N -modular, if L is isometric to its rescaled dual lattice √ NL. A N -modular lattice L is called strongly N modular, if L is isometric to all rescaled partial dual lattices √ mL, for all exact divisors m of N , where L := L ∩ 1 m L.