Bounded generation of S-arithmetic subgroups of isotropic orthogonal groups over number fields

Abstract

Let f be a nondegenerate quadratic form in n 5 variables over a number field K and let S be a finite set of valuations of K containing all Archimedean ones. We prove that if the Witt index of f is 2 or it is 1 and S contains a non-Archimedean valuation, then the S-arithmetic subgroups of SO n (f ) have bounded generation. These groups provide a series of examples of boundedly generated S-arithmetic groups in isotropic, but not quasi-split, algebraic groups.

Weakly commensurable arithmetic groups and isospectral locally symmetric spaces

“…Then, of course, the completion K 1v i can be identified with Q p for i = 1, 2. It follows from (12) and the construction of the open sets U (i) (cf. Lemma 3.4) that over K 1v 1 , the torus T 1 is isomorphic to T (1) , hence is split, and over K 1v 2 , it is isomorphic to T (2) , hence is anisotropic.…”

“…Then the set of arithmetically defined locally symmetric spaces X 2 of G 2 , which are length-commensurable to a given arithmetically defined locally symmetric space X 1 of G 1 , is a union of finitely many commensurability classes. 12 It in fact consists of a single commensurability class if G 1 and G 2 have the same type different from A n , D 2n+1 , with n > 1, D 4 and E 6 .…”

Strong approximation of quadrics and representations of quadratic forms

“…Let V be a nondegenerate quadratic space of dimension m over a number field F , and U be a nondegenerate subspace of V of dimension n. Let Y be the variety of representations of U by V . As a variety, Y can be identified as the quotient variety Spin(V )/Spin(W ), where W is the orthogonal complement of U in V (see, for example, [1,Appendix]). A fundamental question concerning the arithmetic of Y is whether Y has strong approximation with respect to a finite set of primes S of F which contains all the archimedean primes.…”

“…A fundamental question concerning the arithmetic of Y is whether Y has strong approximation with respect to a finite set of primes S of F which contains all the archimedean primes. A necessary and sufficient condition, expressed in terms of Galois cohomology, for Y (or more generally for homogeneous spaces of simply connected algebraic groups) to have strong approximation with respect to S is given in [1,Theorem A.4]. For the special case n = 1, that condition can be stated explicitly in terms of the arithmetic of the quadratic spaces.…”

“…For the special case n = 1, that condition can be stated explicitly in terms of the arithmetic of the quadratic spaces. With our notations, it reads as follows: [1,Appendix].) Suppose that n = 1 and that p∈S Y F p is noncompact.…”