#### DOCUMENTA MATHEMATICA, Vol. 11 (2006), 333-367

Goro Shimura

We investigate several topics on a quadratic form $\Phi$ over an algebraic number field including the following three: (A) an equation $\,\xi\Phi\cdot\tr\xi=\Psi$ 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 $\Th$ such that $\Phi \approx\Psiøplus\Th.$ 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 $\Z$-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.