Almost strong approximations for definite quadratic spaces
The strong approximation theorem for the spin group of an indefinite quadratic space of dimension at least 3 [see, e.g., 104:4 of \textit{O. T. O'Meara}'s book [Introduction to quadratic forms, Grundlehren Math. Wiss. 117. Springer-Verlag, New York (1973; Zbl 0259.10018)], which was proven in the work of Eichler and Kneser in the 1950's, has since that time played a key role in the proofs of fundamental results on the representation and equivalence of indefinite integral quadratic lattices. The purpose of the present paper is to prove ``almost strong approximation theorems'' (one for the spinorial kernel \(O_m'\) and one for the rotation group \(O_m^+\)) for definite quadratic spaces. The statement of the theorem in the case of \(O_m'\) is as follows. Let \(V\) be a regular quadratic space over \(\mathbb{Q}\) of dimension \(m\geq 3\). Then, for any finite set \(T\) of primes, any set \(\{f_p\in O_m' (V_p)\}_{p\in T}\), and any \(\varepsilon> 0\), there exists an extended finite set \(T_e\supset T\) of primes such that for any prime \(q\not\in T_e\) there is a \(\sigma(q)\in O_m' (V)\) satisfying: (i) \(\|\sigma(q)-f_p\|_p< \varepsilon\) for each \(p\in T\), (ii) \(\|\sigma(q)\|_p= 1\) for all \(p\not\in T\cup\{q\}\), and (iii) \(\|\sigma(q)\|_q= q^2\). (The authors show by an example that the exponent 2 appearing in condition (iii) is the best possible; i.e., that this exponent cannot in general be reduced to 1.) The statement of the theorem for the rotation group \(O_m^+\) is somewhat more involved and will not be given here. The proof is correspondingly more difficult, but the applications in some instances yield stronger results than those obtained from the theorem for \(O_m'\). The proofs of both theorems depend upon a refinement and extension of previous results on the primitive representation by a positive definite quaternary quadratic \(\mathbb{Z}\)-lattice of all sufficiently large integers which are primitively represented by its genus. For the present purpose, a version which incorporates the approximation of a given set, of local representations is proven. The general method follows that of a fundamental paper of \textit{J. S. Hsia, Y. Kitaoka} and \textit{M. Kneser} [J. Reine Angew. Math. 301, 132-141 (1978; Zbl 0374.10013)]. For lattices of ranks exceeding 4, the corresponding result has been proven by \textit{M. Jöchner} and \textit{Y. Kitaoka} [J. Number Theory 48, 88-101 (1994; Zbl 0810.11020)]. The strength of these almost strong approximation theorems is demonstrated by a number of interesting applications. For example, it is shown that if \(L\) is a positive definite integral quadratic \(\mathbb{Z}\)-lattice of square-free discriminant, then for all but at most finitely many primes \(q\), representatives of all isometry classes in the genus of \(L\) occur among the \(q\)-neighbors of any vertex in the graph \(\mathbb{Z}(L:q)\) (this is the graph whose vertices are those \(\mathbb{Z}\)-lattices \(K\) on the same ambient space as \(L\) and having \(L_p= K_p\) for all \(p\neq q\) [see \textit{R. Schulze-Pillot}, J. Number Theory 14, 237-250 (1982; Zbl 0483.10020)] for details). In addition, results on the asymptotic representation of forms by forms are obtained (including one which answers a question posed by \textit{Y. Kitaoka} [Arithmetic of quadratic forms, Cambridge Tracts in Mathematics 106, Cambridge University Press (1993; Zbl 0785.11021)], as is an arithmetic proof for a result on the primitive representation of integers by positive ternary quadratic forms.
primitive representation by a positive definite quaternary quadratic \(\mathbb{Z}\)-lattice
primitive representation of integers by positive ternary quadratic forms
scientific article; zbMATH DE number 1054506