On the solutions of quadratic Diophantine equations

  • Takashi Yoshinaga

    Department of Mathematics Ritsumeikan University Kusatsu Shiga 525-8577 Japan
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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 (11.6a)(11.6\textrm{a} ) in citeSh3. This gives an answer to the question (11.6a)(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 4,6,84, 6, 8, or 1010 over the field of rational numbers.

Cite this article

Takashi Yoshinaga, On the solutions of quadratic Diophantine equations. Doc. Math. 15 (2010), pp. 347–385

DOI 10.4171/DM/300