Representations of positive definite quadratic forms with congruence and primitive conditions
Let \(R=\mathbb Z\), \(K=\mathbb Q\) or \(R= \mathbb Z_ p\), \(K=\mathbb Q_ p\) (= \(p\)-adic completion of \(\mathbb Z\) resp. \(\mathbb Q\)). Let \(V\) be a quadratic space over \(K\). A quadratic \(R\)-module \(M\) means a finitely generated \(R\)-module in \(V\). We call \(M\) a lattice if \(KM=V\). A representation of the quadratic \(R\)-module \(N\) by the quadratic \(R\)-module \(M\) is an isometry \(f\) so that \(f(N)\subseteq M\). The main theorem of the paper under review is as follows. Let \(M\) be a positive definite quadratic \(\mathbb Z\)-module of rank \(m\). Let \(n\), \(e\), \(s\) be positive integers with \(m\geq 2n+3\). Let \(T\) be a finite set of primes including the prime 2 and all primes \(p\) for which the corresponding local \(\mathbb Z_ p\)-modules \(M_ p\) are not unimodular. Let \(q\not\in T\) be a prime. Then there are constants \(C_ 1, C_ 2>0\) depending on \(m\), \(n\), \(M\), \(T\), \(s\), \(q\), \(e\) with the following property: Let \(N\) be a quadratic \(\mathbb Z\)-module of rank \(n\) such that there exists local representations of \(N_ p\) by \(M_ p\) for all primes \(p\) and for the archimedean valuation of \(\mathbb Q\). Let \(f_ p: N_ p\to M_ p\) for \(p\in T\) be local representations of \(N_ p\) by \(M_ p\) and let \(\min(N)> C_ 1\). Then there exists a representation \(f: N\to M\) so that (1) \(f(x)\equiv f_ p(x)\bmod p^ s M_ p\) for \(x\in N_ p\) and for all \(p\in T\), (2) For every \(p\not\in T\cup\{ q\}\) there is a \(\mathbb Z_ p\)-module \(N_ p'\) of \(M_ p\) so that \(f(N_ p)\oplus N_ p'= M_ p\), (3) \([\mathbb Q_ p f(N)\cap M_ q: f(N_ q)] <C_ 2\). If in addition \(N\) satisfies \(\text{ord}_ q (\text{vol} (N_ q)) \leq e\) then there is a representation \(f\) of \(N\) by \(M\) such that condition (1) holds and condition (2) holds for all \(p\not\in T\). The proof of this result is based on several technical lemmas and some of the authors' earlier results.
positive definite quadratic \(\mathbb Z\)-module
scientific article; zbMATH DE number 638548