On primitively universal quadratic forms
If a positive definite integral quadratic form \(Q\) in four or more variables is primitively \(p\)-universal for all primes \(p\) (that is, \(Q\) primitively represents all positive integers over each of the rings \(\mathbb Z_p\) of \(p\)-adic integers), then it is known that \(Q\) primitively represents all sufficiently large positive integers (see \textit{J. W. S. Cassels} [Rational quadratic forms. London, New York, San Francisco: Academic Press (1978; Zbl 0395.10029)]). In this paper, the author studies conditions on a \(p\)-adic Jordan decomposition of \(Q\) that guarantee that \(Q\) is primitively \(p\)-universal. The case of an odd prime \(p\) can be resolved using standard facts about the representation properties of unimodular lattices over \(\mathbb Z_p\). The case \(p=2\) is predictably more complicated, and the author obtains only incomplete results in this case. The local results are applied to obtain the following global result: If \(Q\) is a positive definite classically integral quadratic form of rank at least four with odd square-free determinant, and \(Q\) primitively represents the integers \(1\), \(4\) and \(8\), then \(Q\) primitively represents all sufficiently large positive integers.
almost primitively universal quadratic forms
scientific article; zbMATH DE number 5840703
scientific article