Integer-valued quadratic forms and quadratic diophantine equations

Abstract

We investigate several topics on a quadratic form Φ over an algebraic number field including the following three: (A) an equation ξΦ · tξ = ψ for another form ψ of a smaller size; (B) classification of Φ over the ring of algebraic integers; (C) ternary forms. In (A) we show that the "class" of such a ξ determines a "class" in the orthogonal group of a form Θ such that Φ ≈ ψ ⊕ Θ. Such was done in [S3] when Φ is a scalar. We will treat the case of nonscalar Φ, and prove a class number formula and a mass formula, both of new types. In [S5] we classified all genera of Z-valued Φ. We generalize this to the case of an arbitrary number field, which is topic (B). Topic (C) concerns some explicit forms of the formulas in (A) when Φ is of size 3 and ψ is a scalar.