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Integral points on twisted Markoff surfaces. (English) Zbl 1468.11149

Summary: We study the integral Hasse principle for affine varieties of the shape \[ a x^2 + y^2 + z^2 - x y z = m, \] using the Brauer-Manin obstruction, and we produce examples whose Brauer groups include 4-torsion elements. We describe these elements explicitly, and in some cases, we show that there is no Brauer-Manin obstruction to the integral Hasse principle for them.

MSC:

11G35 Varieties over global fields
11D25 Cubic and quartic Diophantine equations
14F22 Brauer groups of schemes
14G12 Hasse principle, weak and strong approximation, Brauer-Manin obstruction
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References:

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