Strong approximation of quadrics and representations of quadratic forms
Let \(F\) be a number field, \(S\) a finite set of primes of \(F\) containing all the archimedean primes of \(F\), and \(\mathfrak o\) the ring of \(S\)-integers of \(F\). Let \(V\) be a nondegenerate quadratic space over \(F\) of dimension \(m\) such that \(V_{\mathfrak p}\) is isotropic for some prime \(\mathfrak p \in S\), and let \(U\), \(M\) and \(N\) be a nondegenerate subspace of \(V\) of dimension \(n\), an \(\mathfrak o\)-lattice on \(V\), and an \(\mathfrak o\)-lattice on \(U\) that is represented everywhere locally by \(M\), respectively. The main result of this paper gives necessary and sufficient conditions, when \(m-n \geq 2\), for the existence of a representation of \(N\) by \(M\) which approximates prescribed representations given on the completions with respect to a finite set of primes disjoint from \(S\). To state the result more precisely, let \(T\) be a finite set of primes disjoint from \(S\). Then \(M\) is said to represent \(N\) with approximation at \(T\) if, for any given integer \(k \geq 1\) and any family of representations \(\phi_{\mathfrak p}:N_{\mathfrak p}\rightarrow M_{\mathfrak p}\) for \(\mathfrak p \in T\), there exists a representation \(f:N\rightarrow M\) such that \(f \equiv \phi_{\mathfrak p} \text{ mod } \mathfrak p^kM_{\mathfrak p}\) for all \(\mathfrak p \in T\). Under the assumptions stated above, it is proven that if \(W\) is isotropic or \(m-n\geq 3\), then \(M\) represents \(N\) with approximation at any finite set of primes disjoint from \(S\). When \(W\) is anisotropic and \(m-n = 2\), the conditions under which \(M\) represents \(N\) with approximation at any finite set of primes disjoint from \(S\) are more subtle, and the result depends on the theory of spinor exceptional representations. This result on representation with approximation is applied to provide necessary and sufficient conditions stated in terms of the arithmetic of the underlying quadratic spaces for the variety \(Y\) of representations of \(U\) by \(V\) to have strong approximation with respect to \(S\). The connection is that the variety \(Y\) has strong approximation with respect to \(S\) if and only if, for all lattices \(M\) on \(V\) and \(N\) on \(U\) such that gen(\(M\)) represents \(N\), \(M\) represents \(N\) with approximation at any finite set of primes disjoint from \(S\). The strong approximation result for \(Y\) takes the following form: Suppose that \(\prod_{\mathfrak p \in S} Y_{\mathfrak p}\) is noncompact. If \(m-n\geq 3\), then \(Y\) has strong approximation with respect to \(S\). If \(m-n =2\), then \(Y\) has strong approximation with respect to \(S\) if and only if either \(W\) is isotropic, or \(W\) is anisotropic and there exists \(\mathfrak p \in S\) for which \(W_{\mathfrak p}\) is anisotropic and additionally \(V_{\mathfrak p}\) is isotropic if \(\mathfrak p\) is real. The special case of this strong approximation result when \(n=1\) appears in a paper of \textit{I. V. Erovenko} and \textit{A. S. Rapinchuk} [J. Number Theory 119, No. 1, 28--48 (2006; Zbl 1168.20020)], where it is noted that the \(m\geq 4\) case follows from Theorem 104:3 of \textit{O. T. O'Meara}'s book [Introduction to quadratic forms. Reprint of the 1973 edition. Berlin: Springer (2000; Zbl 1034.11003)], and the \(m=3\) case is proven using general criteria involving Galois cohomology for a homogeneous space of simply connected algebraic groups to have strong approximation with respect to a given set.
representation with approximation property
scientific article; zbMATH DE number 5379237