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 citeSh3 and citeSh4. Indeed, as an applicat...

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Bibliographic Details
Published inDocumenta mathematica Journal der Deutschen Mathematiker-Vereinigung. Vol. 15; pp. 347 - 385
Main Author Yoshinaga, Takashi
Format Journal Article
LanguageEnglish
Published 2010
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Summary: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.
ISSN:1431-0635
1431-0643
DOI:10.4171/dm/300