Representation of positive definite quadratic forms with congruence and primitive conditions. II
Let \(\text{SR}(m,n)\) be the strong representation property: \(M\) is a positive definite \(\mathbb Z\)-module of \(\text{rank}\leq m\), \(s\in \mathbb N\), \(T\) is a finite set of primes containing 2 and all primes \(p\) for which \(M_p\) is not unimodular. Then there is a constant \(C\) depending on \(m\), \(n\), \(M\), \(T\), \(s\) such that if \(N\) is a quadratic \(\mathbb Z\)-module of rank \(n\) which is locally represented by \(M\), and if \(f_p: N_p\to M_p\) are the representations for \(p\in T\), and if \(C< \min (N)\) then there exists a representation \(f: N\to M\) so that \(f\equiv f_p\bmod p^s M_p\) for all \(p\in T\) and \(f(N_p)\) is primitive in \(M_p\) for all \(p\not\in T\). In [J. Number Theory 48, 88--101 (1994; Zbl 0810.11020)] the author and \textit{Y. Kitaoka} showed that \(\text{SR}(2n+3, n)\) holds true for \(n\geq 1\) if in addition \(\text{ord}_p (\text{Vol} (N))\) is bounded at any fixed prime \(q\not\in T\). The paper under review gives an algebraic proof of \(\text{SR}(7,2)\). It is not known whether \(\text{SR}(2n+ 3,n)\) holds unconditionally for all \(n\geq 1\).
representation of positive definite quadratic forms
scientific article; zbMATH DE number 714115