Strong Approximation for a Family of Quadratic Surfaces

Strong approximation is a property that some schemes have, which relates the geometry of their rational points to the geometry of their p-adic points. For S a subset of the places of the rational numbers, a scheme satisfies strong approximation away from S, if the rational points are dense in some product over all p not in S of the sets of p-adic points. In the last century, a couple of great, general results have been proven which give sufficient or necessary conditions for a scheme or a group scheme to satisfy strong approximation away from some set S. In 2019, Kok and Bright showed, using the Brauer-Manin obstruction, that the scheme representing primitive solutions to the equation X1^2 + 47 X2^2 - 103 X3^2 - 103 * 47 * 17 X4^2 = 0 does not satisfy strong approximation away from infinity. On the other hand, in 2020, Pagano and Bright proved a general result from which it follows that this scheme satisfies strong approximation away from infinity, 17, 47 and 103. This thesis shows...Show moreStrong approximation is a property that some schemes have, which relates the geometry of their rational points to the geometry of their p-adic points. For S a subset of the places of the rational numbers, a scheme satisfies strong approximation away from S, if the rational points are dense in some product over all p not in S of the sets of p-adic points. In the last century, a couple of great, general results have been proven which give sufficient or necessary conditions for a scheme or a group scheme to satisfy strong approximation away from some set S. In 2019, Kok and Bright showed, using the Brauer-Manin obstruction, that the scheme representing primitive solutions to the equation X1^2 + 47 X2^2 - 103 X3^2 - 103 * 47 * 17 X4^2 = 0 does not satisfy strong approximation away from infinity. On the other hand, in 2020, Pagano and Bright proved a general result from which it follows that this scheme satisfies strong approximation away from infinity, 17, 47 and 103. This thesis shows that this scheme satisfies strong approximation away from infinity and 17, and it obtains a more general result for some equations of the form a X1^2 + b X2^2 + c X3^2 + d X4^2 = 0.Show less

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Strong Approximation for a Family of Quadratic Surfaces