Positive definite unimodular lattices with trivial automorphism groups / Etsuko Bannai

It had been unknown so far whether in some n-dimensional Euclidean space there exists a selfdual lattice whose group of isometric automorphisms is trivial in the sense that it consists only of ±identity. In the work under review it is shown now that for any dimension n≥43 (resp. n≥144) there exist odd (resp. even) selfdual lattices with this property. The result is obtained by an upper estimate of the ratio of the Minkowski-Siegel mass of lattices with nontrivial group to the mass of all lattices (in the given genus). It turns out that for n→∞ this ratio tends to zero, so there is, in fact, an abundance of selfdual lattices with trivial group. The improvement over earlier work by J. Biermann [Gitter mit kleiner Automorphismengruppe in Geschlechtern von ℤ-Gittern mit positiv-definiter quadratischer Form (Thesis, Göttingen 1981; Zbl 0486.10018)] comes from the proper treatment of lattices which have an isometry of prime order p whose minimal polynomial is reducible, i.e., equal to x p -1. (Zentralblatt)

Bibliogr. p. 69-70