Euclidean quadratic forms and ADC forms. I (Q2893038)

Let \((R,|\cdot |)\) be a normed integral domain of characteristic different from \(2\) and let \(K\) be the fraction field of \(R\). A quadratic form \(q\) over \(R\) (i.e., a homogeneous degree 2 polynomial in \(R[x_1,\dots,x_n]\)) is defined to be Euclidean if for all \(x \in K^n\setminus R^n\), there exists \(y \in R^n\) such that \(0<|q(x-y)|<1\). It is proved that every Euclidean quadratic form has the following property: for \(d\in R\), there exists \(y \in R^n\) such that \(q(y)=d\) whenever there exists \(x \in K^n\) such that \(q(x)=d\). Forms satisfying this latter property are referred to here as ADC forms.NEWLINENEWLINEIn this paper, the author initiates a systematic study of Euclidean forms and ADC forms over normed rings, particularly those of arithmetic interest such as complete discrete valuation rings (CDVRs) and Hasse domains. It is shown that these properties behave nicely under localization and completion, under some additional restrictions on \(R\). In general, the quadratic \(R\)-lattice associated to a Euclidean form is maximal, and the converse is also true over a CDVR. Over a Hasse domain, the ADC forms in at least four variables are precisely the quadratic forms that are sign-universal, in the sense that they represent all elements of \(R\) that are represented at all real places of \(K\). The positive definite Euclidean forms over \(\mathbb Z\) correspond to the integral lattices in Euclidean space with covering radius strictly less than \(1\). The lattices with this property are known due to a result of \textit{G. Nebe} [Beitr. Algebra Geom. 44, No. 1, 229--234 (2003; Zbl 1142.11340)], and all have class number one. It is conjectured in the present paper that in fact any anisotropic Euclidean quadratic form over a Hasse domain has class number one.