Definite lattices over real algebraic function domains
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.
scientific article; zbMATH DE number 3885987
scientific article