On Some Arithmetic Properties of Linear Algebraic Groups

Abstract

Let Q be the field of rational numbers and k an extension of Q of finite degree. The multiplicative group k* of k, considered as a group of linear transformations of the vector space k over Q, forms an algebraic group, i.e., a Q-torus in the sense of A. Borel. As is well known in the algebraic number theory, the properties of k* and of its related structures, in particular that of the group Jk of id6les of k, play important roles. On the other hand, let f be a quadratic form on a vector space V over Q of finite dimension. The orthogonal group O( V, f ) composed of all linear transformations of V leaving invariant the form f forms an algebraic group. The properties of the group O( V, f) have essential relations to the arithmetic of the quadratic form f, and the study of these relations has been one of the principal themes in M. Eichler's book Quadratische Formen und Orthogonale Gruppen . Recently the theory of algebraic groups of linear transformations has been systematized by C. Chevalley on the basis of fundamental concepts of algebraic geometry, and the classical mechanism of the Lie theory (correspondence between groups and Lie algebras) has been generalized to the case where the basic field K is an arbitrary field of characteristic 0 (cf., Chevalley [2], [3]). By specializing K to Q, we may apply his methods and results to the study of arithmetic properties of algebraic groups. Thus it could be said that the above two theories, i. e., the arithmetic of k* and that of O( V, f) are two profiles of a kind of unified theory which we might call the arithmetic of algebraic groups. In the present paper, we shall formulate some fundamental concepts for algebraic groups from this point of view and prove some results which might possibly give us some suggestions for further developments in this direction. Thus, in Section 1, we shall introduce the notion of rational characters of algebraic groups and determine the structure of the group of rational characters for a special algebraic group, i. e., a Q-torus (Theorem 1). In Section 2, we shall introduce the notion of G-id6les for an algebraic group G which generalizes the usual notion of id6les of an algebraic number field k and define a subgroup J1(G) of the group J(G) of G-id6les in con266