Lattices and spherical designs. (Réseaux et designs sphériques (notes de Jacques Martinet).) (French) Zbl 1139.11320

Martinet, Jacques (ed.), Résaux euclidiens, designs sphériques et formes modulaires. Autour des travaux de Boris Venkov. Genève: L’Enseignement Mathématique (ISBN 2-940264-02-3/pbk). Monogr. Enseign. Math. 37, 10-86 (2001).
Summary: New methods in the study of the classical theory of Korkine and Zolotareff, refined by Voronoï at the beginning of the twentieth century, have arisen in connection with the notion of a spherical design: that the set of minimal vectors of a lattice be a spherical 2- or 4-design is a restrictive form of the properties of eutaxy or extremality. After recalling some basic facts about spherical \(t\)-designs, we study the translation of this theory in terms of lattices, then we solve some classification questions (small dimensional lattices, lattices with a small kissing number, integral lattices with a small minimum). We conclude with some applications to this theory of the theory of modular forms with harmonic (spherical) coefficients as well as some constructions of designs by means of sections of lattices.
For the entire collection see [Zbl 1054.11034].


11H06 Lattices and convex bodies (number-theoretic aspects)
05B30 Other designs, configurations
11H31 Lattice packing and covering (number-theoretic aspects)
11H55 Quadratic forms (reduction theory, extreme forms, etc.)