Geodesic
On quadratic Diophantine equations in four variables and orders associated with lattices
Documenta mathematica, Tome 19 (2014), pp. 247-284
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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.
DOI : 10.4171/dm/446
Classification : 11D09, 11E12, 11E20 
@article{10_4171_dm_446,
     author = {Manabu Murata},
     title = {On quadratic {Diophantine} equations in four variables and orders associated with lattices},
     journal = {Documenta mathematica},
     pages = {247--284},
     year = {2014},
     volume = {19},
     doi = {10.4171/dm/446},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/446/}
}
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Manabu Murata. On quadratic Diophantine equations in four variables and orders associated with lattices. Documenta mathematica, Tome 19 (2014), pp. 247-284. doi: 10.4171/dm/446

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