Jacques Martinet

Perfect lattices in Euclidean spaces

1 General Properties of Lattices.- 2 Geometric Inequalities.- 3 Perfection and Eutaxy.- 4 Root Lattices.- 5 Lattices Related to Root Lattices.- 6 Low-Dimensional Perfect Lattices.- 7 The Voronoi Algorithm.- 8 Hermitian Lattices.- 9 The Configurations of Minimal Vectors.- 10 Extremal Properties of Families of Lattices.- 11 Group Actions.- 12 Cross-Sections.- 13 Extensions of the Voronoi Algorithm.- 14 Numerical Data.- 15 Appendix 1: Semi-Simple Algebras and Quaternions.- 16 Appendix 2: Strongly Perfect Lattices.- References.- List of Symbols.

Class Groups of Number Fields: Numerical Heuristics

Extending previous work of H. W. Lenstra, Jr. and the first author, we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four (given the signature and the Galois group of the Galois closure). The theoretical justifications for these conjectures will appear elsewhere, but the agreement with the existing tables is quite good. 1. Introduction and Notations. In (3), H. W. Lenstra, Jr., and the first author developed a method for conjecturing quantitative results on class groups of quadratic fields and cyclic extensions of prime degree. In a forthcoming paper (4) we shall show that this technique can be extended to a much wider class of number fields, and also to relative extensions. The aim of the present paper is to rapidly make available the numerical conjec- tures obtained, for people not really interested in our heuristic reasoning or not wanting to wait for (4) to appear. Hence, apart from a total lack of justifications for the conjectures that we present, this paper is essentially self-contained. The plan is as follows. In the rest of this section we present the notations used in the sequel. Some of them being nonstandard (and in general differing from the notations of (3)), we urge the reader to read the notations carefully before applying the conjectures. In the next section we present templates for the subsequent conjectures, and then the conjectures themselves, illustrated by numerical examples, first for their own sake, and second as a double check for the reader to understand the templates. These conjectures are given for all types of fields of degree less than or equal to four. In the final section we comment on the consistency of the conjectures with existing tables (which is quite good). Combinatorial Notations: * If X is a set, {XI denotes its cardinality.

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.

Heuristics on class groups: some good primes are not too good

We correct some too optimistic predictions in an earlier paper of ours.

The minimum discriminant of totally real octic fields

Abstract The minimum discriminant of totally real octic algebraic number fields is determined. It is 282,300,416 and belongs to the ray class field over Q (√2) of conductor (7 + 2 √2): F = Q (√α) for α = (7 + 2 √2 + (1 + √2) √7 + 2 √2)/2. There is—up to isomorphy—only one field of that discriminant. The next two smallest discriminant values are 309,593,125 and 324,000,000. For each field we present a full system of fundamental units and its class number.

BASES OF MINIMAL VECTORS IN LATTICES, III

We prove that all Euclidean lattices of dimension n ≤ 9 which are generated by their minimal vectors, also possess a basis of minimal vectors. By providing a new counterexample, we show that this is not the case for all dimensions n ≥ 10.

On integral lattices having an odd minimum

The kissing number of integral lattices of odd minimum is studied, with special emphasis on the case of minimum 3.

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

The Configurations of Minimal Vectors

We say that two lattices Λ and Λ′ with respective sets of minimal vectors S and S′ are minimal-equivalent if there exists u ∈ GL(E) which maps Λ onto Λ′ and S(Λ) onto S (Λ′).

Appendix 2: Strongly Perfect Lattices

This chapter is intended to give a survey of some recent results which have been discovered after the French version of this book was written, and which are of great importance in the theory of extreme lattices: on the one hand, they allow us to prove that some lattices having a comparatively high dimension are extreme without having any precise description of their minimal vectors (example: 80-dimensional even unimodular lattices of minimum 8, for which two lattices are known from [Bac-Ne2]); on the other hand, they connect the theory of extreme lattices to that of spherical designs, which properly belongs to combinatorics. The basic reference is Venkov’s [Ven3] (English title: “Lattices and Spherical Designs”). We account for some of the results proved in [Ven3] or in some other articles which appeared in the same issue of “L’Enseignement Mathematique” (Bachoc, Martinet, Venkov), and also for results involving group theory, due to Dummigan, Lempken, Schroder and Tiep ([D-T],[L-S-T]).