Almost regular quaternary quadratic forms
Let us say that a quadratic form is integral if it is a positive definite nonsingular form with an integer coefficient matrix. An integral quadratic form \(f\) is called \textit{almost regular} (respectively, \textit{regular}) if it represents all sufficiently large (respectively, all) integers which are represented by \(f\) over \(p\)-adic integers \(\mathbb Z_p\) for every prime \(p\). The paper under review investigates the properties of almost regular quaternary quadratic forms. More specifically, a form is called \textit{\(p\)-isotropic} for a given prime \(p\) if it represents zero nontrivially over \(\mathbb Z_p\), and it is called \textit{\(p\)-anisotropic} otherwise. It is an old result contained in many of the classical papers on the subject that every positive definite integral quadratic form which is \(p\)-isotropic for every prime \(p\) is almost regular. An almost regular quadratic form is called \textit{exceptional} if it is \(p\)-anisotropic for at least one prime \(p\). It is known that there are no exceptional forms in more than four variables, and that all almost regular forms in less than four variables are exceptional. In this paper, the authors study exceptional quaternary forms. They show that although there are infinitely many such forms, in a certain sense they are rare. In particular, the authors prove that any exceptional quaternary form can be \(p\)-anisotropic for only one prime \(p\), and this prime has to be no greater than 37. On the other hand, they also show that for \textit{every} prime \(p \leq 37\), there exists an almost regular \(p\)-anisotropic quaternary form. To this end, for each prime \(p \leq 37\) the authors introduce the family \(\mathcal E_p\) of all primitive almost regular \(p\)-anisotropic quaternary quadratic forms, along with its subfamily \(\mathbb R_p\) of regular \(p\)-universal forms (if \(p >2\); \(2\mathbb Z_2\)-universal if \(p=2\)). The authors show that each \(\mathbb R_p\) is finite and define a subfamily \(\mathcal U_p\) of even-universal (i.e. representing all even positive integers) forms in \(\mathbb R_p\). In fact, they prove that for each prime \(17 \leq p \leq 37\), \(\mathbb R_p = \mathcal U_p\), while for each prime \(p \leq 13\), \(\mathcal U_p \subsetneq \mathbb R_p\). Moreover, the authors discover that the families \(\mathbb R_p\) are very small (containing only one or two forms) for \(17 \leq p \leq 37\), while for \(p \leq 13\) they are much larger. The authors carry out an explicit investigation of the structure and properties of the families \(\mathcal E_p\), \(\mathbb R_p\), and \(\mathcal U_p\), using them to investigate certain properties of genera of integral quadratic forms of given type, and finally use these families to prove an \textit{effective} criterion for almost regularity of \(p\)-anisotropic integral quaternary quadratic forms.
scientific article; zbMATH DE number 5315502