The Construction of Definite Indecomposable Hermitian Forms

Abstract

Introduction. This research is concerned primarily with the construction of positive definite indecomposable hermitian forms over the integers of algebraic number fields. The interest in indecomposables stems from the fact that every definite hermitian form over the integers of a number field splits uniquely into the orthogonal sum of indecomposable components. Indecomposables have also been used to obtain a lower bound for the class number of a definite hermitian form in terms of the rank (see Gerstein [2]). The analogous problem for positive definite quadratic forms has been studied by O'Meara [13]. Combining his work with that of Erdos and Ko [1], O'Meara shows the existence of positive definite integral indecomposable quadratic forms of rank n and discriminant d over 7/ when n > 10 and d > 0, with exactly five exceptions. That no indecomposables exist in the exceptional cases had already been shown by Kneser [10]. In this paper we will first show that every indecomposable quadratic form over 7/ lifts (by tensor product) to an indecomposable hermitian form over the integers ? of any imaginary quadratic number field E. Hence the quadratic forms mentioned above provide indecomposable hermitian forms with the same invariants. The principle aim of this research is to give an explicit global construction of positive definite integral indecomposable hermitian forms over ? of arbitrary rank and discriminant. While this duplicates some of the results obtained by lifting indecomposable quadratic forms, many of the quadratic forms constructed by O'Meara were obtained by exhibiting their localizations. Further, our constructions will produce indecomposable hermitian forms which do not come from lifting quadratic forms. We will adopt the terminology of O'Meara's book [12] and refer to lattices instead of forms. Our results on indecomposable hermitian lattices are obtained by adapting some of the methods of O'Meara's paper [13] to the hermitian setting and by using a neighbor lattice technique over the integers of E. The neighbor lattice idea is due to Kneser [10], who developed it for quadratic lattices over Z. lyanaga [5] subsequently developed a neighbor lattice