Obstructions to the integral Hasse principle for generalized affine Chatelet surfaces
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This dissertation contains results on the integral Hasse principle and strong approximation for generalized affine Chatelet surfaces defined over a number field k by x^2 −ay^2 =P(t) inside affine 3 space, with P(t) in k[t] a separable polynomial in one variable. The first portion of this dissertation is devoted to enumerating the isomorphism types of the Brauer groups of such surfaces, under certain conditions on the Galois groups of the polynomial P(t). We then provide an approach for constructing explicit Brauer classes for a larger class of varieties known as norm form varieties, and use this approach to compute generators of the Brauer groups of the generalized affine Chatelet surfaces. We provide an explicit method for computing the Brauer-Manin set for the non-cyclic algebras generating these Brauer groups. Finally, we use the results of the previous chapters to provide counterexamples to strong approximation explained by a Brauer-Manin obstruction.