Integer-valued quadratic forms and quadratic Diophantine equations.

Abstract

We investigate several topics on a quadratic form $\Phi$ over an algebraic number field including the following three: (A) an equation $\xi \Phi \cdot {}^t\xi =\Psi$ for another form $\Psi$ of a smaller size; (B) classification of $\Phi$ over the ring of algebraic integers; (C) ternary forms. In (A) we show that the “class” of such a $\xi$ determines a “class” in the orthogonal group of a form $\Theta$ such that $\Phi \approx \Psi \oplus \Theta$. Such was done in [S3] when $\Psi$ is a scalar. We will treat the case of nonscalar $\Psi$, and prove a class number formula and a mass formula, both of new types. In [S5] we classified all genera of $\mathbf{Z}$-valued $\Phi .$ 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 $\Phi$ is of size 3 and $\Psi$ is a scalar.