Anne-Marie Bergé

Sur un problème de dualité lié aux sphères en géométrie des nombres

Abstract We study extremal pairs of polar lattices with respect to the product of the norms of minimal vectors. This notion of a “dual-extreme” lattice is compared with the usual notion of an extreme lattice. This study was suggested by the problem of twin-classes in number fields raised by Zimmert in an analytical context.

A generalization of some lattices of Coxeter

This paper introduces a wide generalization of a family of integral lattices defined by Coxeter, which share with the Coxeter lattices the following properties: they are perfect, with often an odd minimum, and have no nontrivial perfect sections with the same minimum. Let E be an n-dimensional Euclidean space with scalar product x · y and norm N(x) = x · x. A lattice in E is a discrete subgroup Λ of E of rank n. We set m = minΛ = minN(x), x ∈ Λr{0} (the minimum of Λ), S = S(Λ) = {x ∈ Λ | N(x) = m} (the sphere of Λ), and s = 1 2 |S| (its (half ) kissing number). In his 1951 paper [Cox], Coxeter considered the lattices L defined for odd n ≥ 5 by the conditions An ⊂ L ⊂ An and [An : L] = 2, where An stands as usual for the root lattice generated by the root system of type An. He determined their minimum and proved that they are perfect, with s = n(n+1) 2 , the smallest possible kissing number for a perfect lattice. In the sequel, we denote by Coxn their scaled copy to the smallest minimum m which makes them integral (2(n − 1) if n ≡ 1 mod 4, n−1 2 if n ≡ 3 mod 4; see [M], Section 5.2). In her paper [Be], the first author proved that these lattices are hollow for all n ≥ 7 in the sense that they do not have any perfect r-dimensional sections with the same minimum in the range 1 < r < n. (Cox5, which has hexagonal sections of minimum m, is an exception.) Her method relies on the fact that S(An) is of the form {±v0,±v1, . . . ,±vn} with vectors vi which add to zero, and that the minimal vectors of Coxn (up to sign) are then the sums vi + vj for 0 ≤ i < j ≤ n. In relation with his recent joint paper [M-V] with Boris Venkov, the second author tried to construct in various dimensions integral perfect lattices Λ having an odd minimum, using a construction going back to Watson ([W]; see also [M1]): one starts with a lattice Λ′ having a basis (e1, . . . , en) of minimal vectors, and then considers lattices of the form Λ = 〈Λ′, e〉 for a vector e = e ′ d , e ′ ∈ Λ′, d ≥ 2. chosen in such a way that the minimal vectors of Λ lie in the set S0 = {±ei}∪{±(e−ei−ej)}. It appeared that S0 could be viewed as a set {±(vi+vj)} where the vi, i = 0, 1, . . . , n belong to a lattice M containing Λ to index 2 and are related by a single relation c0v0 + c1v1 + · · · + cnvn = 0 with at most one null coefficient ci, an observation which throws new light on Watson’s constructions. Using this device, we are able to prove: Theorem 0.1. For every n ≥ 10, there exist perfect, hollow n-dimensional lattices having an odd minimum. The mere existence of perfect lattices having an odd minimum (hollow or not) for all large enough dimensions is new (the Coxeter family solves the problem only for dimensions congruent to 3 modulo 4). The proof of Theorem 0.1 will be given