Euclidean quadratic forms and ADC forms: I

Pete L. Clark
2012 Acta Arithmetica  
A classical result of Aubry, Davenport and Cassels gives conditions for an integral quadratic form to integrally represent every integer that it rationally represents. We give a generalization which allows one to pass from rational to integral representations for suitable quadratic forms over a normed ring. The Cassels-Pfister theorem follows as another special case. This motivates a closer study of the classes of forms which satisfy the hypothesis of our theorem ("Euclidean forms") and its
more » ... forms") and its conclusion ("ADC forms"). Normed rings. Let R be a commutative, unital ring. We write R A normed ring is a pair (R, | |) where | | is a norm on R. A ring admitting a norm is necessarily an integral domain. We denote the fraction field by K. The norm extends uniquely to a homomorphism of groups (K × , ·) → (Q >0 , ·). Example 1: The usual absolute value | | ∞ (inherited from R) is a norm on Z. Example 2: Let k be a field, R = k[t], and let a ≥ 2 be an integer. Then the map f ∈ k[t] • → a deg f is a non-Archimedean norm | | a on R. When k is finite, it is most natural to take a = #k (see below). Otherwise, we may as well take a = 2. Example 3: An abstract number ring is an infinite integral domain R such that for every nonzero ideal I of R, R/I is finite. (In particular, R may be an order in a number field, the ring of regular functions on an irreducible affine algebraic curve over a finite field, or any localization or completion thereof.) The map x ∈ R • → #R/(x) gives a norm on R [Cl10, Prop. 5], which we will call the canonical norm. The standard norm | | ∞ on Z is canonical, as is the norm | | q on the polynomial ring F q [t]. Example 4: Let R be a discrete valuation ring (DVR) with valuation v : K × → Z
doi:10.4064/aa154-2-3 fatcat:zfifatcpcbakjeend7sawomwb4