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Primitive solutions of nonscalar quadratic Diophantine equations. (English) Zbl 1459.11083

Summary: There is a mapping of the set of primitive integral representations of a quadratic form in \(n - 1\) variables by a quadratic form in \(n\) variables into the set of primitive representations of a rational number by a quadratic form in \(n\) variables. Such a correspondence was introduced by Shimura in 2004. The purpose of this paper is to study the former representations by describing the inverse image of the latter set under that map. As an application we discuss some specific Fourier coefficients of theta series of degree \(n - 1\).

MSC:

11D09 Quadratic and bilinear Diophantine equations
11E12 Quadratic forms over global rings and fields
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11F27 Theta series; Weil representation; theta correspondences
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References:

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