# Integer-valued quadratic forms and quadratic Diophantine equations.

### Goro Shimura

Department of Mathematics Princeton University Princeton, New Jersey 08544-1000 U.S.A

## Abstract

We investigate several topics on a quadratic form $Φ$ over an algebraic number field including the following three: (A) an equation $ξΦ⋅_{t}ξ=Ψ$ 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.

## Cite this article

Goro Shimura, Integer-valued quadratic forms and quadratic Diophantine equations.. Doc. Math. 11 (2006), pp. 333–367

DOI 10.4171/DM/213