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Positive definite unimodular lattices with trivial automorphism groups / Etsuko Bannai

Auteur principal : Bannai, Etsuko, 1944-, AuteurType de document : MonographieCollection : Memoirs of the American Mathematical Society, 429Langue : anglais.Pays : Etats Unis.Éditeur : Providence : American Mathematical Society, 1990Description : 1 vol. (IV-70 p.) ; 26 cmISBN : 9780821861523; 9780821824917; 0821824910.ISSN : 0065-9266.Bibliographie : Bibliogr. p. 69-70.Sujet MSC : 11H56, Number theory - Geometry of numbers, Automorphism groups of lattices
11E41, Forms and linear algebraic groups, Class numbers of quadratic and Hermitian forms
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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

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