Almost strong approximations for definite quadratic spaces

Abstract

A fundamental result in the arithmetic of integral quadratic forms is the strong approximation theorem for the spin group Spin of a non-degenerate inde®nite quadratic form in three or more variables. (In practice, one invokes the corresponding version for the kernel O0 of the spinor norm function.) This result is due to the works of Eichler and Kneser [Ei], [Kn] in the 1950s, and has ever since played a central role in the proofs of a number of important theorems for both inde®nite as well as de®nite quadratic forms. On the other hand, it is well-known that such a strong approximation cannot possibly hold for i† the group SO of a inde®nite quadratic form as it is not simply connected, and ii† for the group Spin of a de®nite quadratic form as Spin1 is compact. The purpose of this paper is to show that there exists a version of an ``almost'' strong approximation theorem for de®nite quadratic forms, one forOm and one forO ‡ m m 3† and show that they too have serious applications in the arithmetic theory; e.g., in the asymptotic distributions of classes in a genus and in the representation theory of de®nite forms. It is anticipated that such almost strong approximations would also extend to other classical groups and will be treated in a future article. We give a brief outline of the contents of this paper. Terminology and notations are generally those from [ OM]. For simplicity as well as convenience we shall consider here only positive de®nite integral Z-lattices. A key step in the proof of the strong approximation theorem for O0 is the following result on inde®nite representations with approximation property: Invent. math. 129, 471±487 (1997)