On the Solutions of Quadratic Diophantine Equations

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 \cite{Sh3} and \cite{Sh4}. 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 \cite{Sh3}. 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.

2010 Mathematics Subject Classification: 11D09, 11E08, 11E12

Keywords and Phrases: Maximal lattices, Quadratic Diophantine equations

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