On the solutions of quadratic Diophantine equations

Abstract

We determine a finite set of representatives of the set of local solutions in a maximal lattice modulo the stabilizer of the lattice in question for a quadratic Diophantine equation. Our study is based on the works of Shimura on quadratic forms, especially citeSh3 and citeSh4. Indeed, as an application of the result, we present a criterion (in both global and local cases) of the maximality of the lattice of $(11.6\textrm{a} )$ in citeSh3. This gives an answer to the question $(11.6\textrm{a} )$. As one more global application, we investigate primitive solutions contained in a maximal lattice for the sums of squares on each vector space of dimension $4, 6, 8$, or $10$ over the field of rational numbers.