On quadratic Diophantine equations in four variables and orders associated with lattices (Q2439234)

The paper is based on a ``principle of Shimura'' [\textit{G. Shimura}, Arithmetic and analytic theories of quadratic forms and Clifford groups. Providence, RI: AMS (2004; Zbl 1135.11018), Theorem 11.6]. It is shown that the number of classes of primitive solutions of a quadratic Diophantine equation in four variables coincides with the type number of a certain order in a quaternion algebra under suitable conditions. This result is illustrated by several explicit examples.