Hermitian Forms Over Local Hereditary Orders

Abstract

Let A be a simple algebra of finite dimension over its center (a field), and assume the center is complete with respect to some discrete valuation. Suppose A is an order in A which is hereditary, i.e. all ideals in A are projective as A-modules. We assume also that there is an involution J on A with AJ = A. We wish to investigate the properties of hermitian or skew hermitian forms H: M X M -+ A where M is a A-lattice. As one might expect, the theory is similar in many respects to that of quadratic forms over local rings of integers (cf. [5], [6]). Thus for example, there is a cancellation theorem for hyperbolic planes (Theorem 21), every non-degenerate form has a Jordan decomposition into modular forms (Corollary 12, Theorem 13), every orthogonally indecomposable form is unary or binary (Proposition 11) and for the sharpest results an arithmetic assumption