On Quadratic Diophantine Equations in Four Variables and Orders Associated with Lattices

This paper treats certain lattices in ternary quadratic spaces, which are obtained from the data of a non-zero element and a maximal lattice in a quaternary space. Each class in the genus of such a lattice with respect to the special orthogonal group corresponds to an isomorphism class in the genus of an order associated with the lattice in a quaternion algebra. Using this result together with the principle of Shimura, we show that the number of classes of the primitive solutions of a quadratic Diophantine equation in four variables coincides with the type number of the order under suitable conditions on the given element. We illustrate this result by explicit examples.

2010 Mathematics Subject Classification: 11E12, 11D09, 11E20

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