Splitting Quadratic Forms Over Integers of Global Fields

Abstract

We are interested in decomposition of quadratic forms over Hasse domains of global fields. By a global fleld F we mean either an algebraic number field or an algebraic function field in one variable over a finite constant field. Throughout assume that characteristic of F is not 2. A Hasse domain o of F is a Dedekind domain which can be obtained as intersection of almost all valuation rings on F. So o is ring of integers corresponding to a set S of almost all discrete spots on F. This is accepted generalization of situation in which F is field Q of rational numbers and o is ring Z of rational integers, or where F is an algebraic number field and o is usual ring of integers contained therein. See O'Meara 2 [6], p. 79, and [5], Chapter X, or Weiss [9], pp. 189 ff. Now consider a regular quadratic space V over global field F. With o a Hasse domain in F, we consider an o-lattice L on V; that is, L is a finitely generated o-module which spans V. We ask following question: Is there some number no, depending only on o, such that L has a non-trivial orthogonal splitting L L,1 IL2 whenever rank L nO? This problem has been investigated in case in which F== Q with Hasse domain o =Z. In fact, Erd6s and Ko [2] proved that in this situation for every integer n > 5 there exists an indecomposable definite quadratic form of rank n. This means that there does not exist a number no in definite case! But indefinite case was later investigated by G. L. Watson in his book Integral Quadratic Forms [8], where he proved that every indefinite Z-form of rank n ? 12 splits non-trivially, and in fact that with a slight restriction on behavior at archimedean spot, splitting occurs when n ? 8. Watson goes on to remark that the general problem of deconlposition has not received attention it deserves. It is our purpose in this