Definite lattices over real algebraic function domains
type: Item
instance of: scholarly article
title: Definite lattices over real algebraic function domains
zbMATH Open document ID: 0556.10012
DOI: 10.1007/BF01455860
author: H.-G. Quebbeman
published in: Mathematische Annalen
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164025
review text: Let K be a formally real field. The problems considered in this paper have arisen from the failure of a quadratic analogue of Serre's conjecture for the polynomial ring \(k[x_ 1,...,x_ m]\), \(m\geq 2\). More generally, let A be a k-domain of algebraic functions having only real poles with respect to some normal projective model of its quotient field K, and let \(q: V\to K\) be a quadratic form which is definite at these infinite prime divisors of A. Then for an A-lattice \(M\subset V\) the vectors \(v\in M\) with q(v) bounded by a given infinite divisor d form a finite-dimensional k-vector space M(d). Starting from this result the paper develops a structure theorem for the orthogonal group O(M), a unique decomposition theorem for the lattice M, and a construction principle for (indecomposable, unimodular) lattices M having a prescribed sublattice AM(d) of maximal rank.
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 11E57
zbMATH DE Number: 3885987
zbMATH Keywords: formally real field
zbMATH Keywords: domain of algebraic functions
zbMATH Keywords: quadratic form
zbMATH Keywords: A- lattice
zbMATH Keywords: structure theorem
zbMATH Keywords: orthogonal group
zbMATH Keywords: unique decomposition theorem
zbMATH Keywords: construction principle
zbMATH Keywords: indecomposable unimodular lattices
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3885987