Integer-Valued Quadratic Forms and Quadratic Diophantine Equations

We investigate several topics on a quadratic form Φ over an algebraic number field including the following three: (A) an equation ξΦ · ξ = 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 Ψ is a scalar. We will treat the case of nonscalar Ψ, 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. 2000 Mathematics Subject Classification: 11E12 (primary), 11D09, 11E41 (secondary)