Arithmetic of Unitary Groups

The purpose of this paper is to develop the theory of elementary divisors, to prove the approximation theorem, and to determine the class number for the following two types of algebraic groups: (i) the unitary group of a hermitian form over an algebraic number field; (ii) the unitary group of a hermitian form over a quaternion algebra, having an algebraic number field as center, with respect to the canonical involution. The latter includes the symplectic group as a special case, since we admit the total matric algebra of degree two as a quaternion algebra. Of course, the approximation theorem is obtained only in the case where the hermitian form is indefinite. A problem of the same kind was investigated for the group of regular elements of a simple algebra by Eichler [5], and for the orthogonal group by Eichler [6, 7] and Kneser [9]. In [11], we treated the groups of type (ii) with the same intent and obtained a somewhat weaker result than in the present paper. We now explain our result by giving a summary of each section. Let F be an algebraic number field of finite degree and K a quadratic extension of F. Let V be a vector space over K of dimension n and ?F(x, y) a non-degenerate hermitian form on V with respect to the non-trivial automorphism of K over F. We denote by U( V, F) and G( V, F) respectively the unitary group of iF and the group of similitudes of iF (cf. 1.1 below). Further, we denote by SU(V, (D) the subgroup of U(V, iF) consisting of the elements with determinant 1. If n is even, we can consider the group of direct similitudes H( V, iF) (cf. 2.2). In the first two sections, we give preliminaries for these four groups, and study elementary properties of maximal lattices, whose definition is as follows. Let g and x be the ring of integers in F and in K, respectively. Let L be an x-lattice in V. We denote by Me(L) the ideal in F generated by the elements iF(x, x) for all x e L, and call it the norm of L. We say that L is maximal if L is a maximal one among the x-lattices with the same norm. Let p be a prime ideal of F, and F, the completion of F with respect to P; and let K, = K ?F Fp and V, = V OF Fi. In ?? 3 and 4, we treat the local theory of elementary divisors for the lattices in V,. If p decomposes in F, KP is isomorphic to F, x F,, and therefore things are much easier. Section 3 is concerned with this case. In ? 4, we discuss the case where K, is a field. 369