Browsing by Subject "Brauer group"
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Item Obstructions to the integral Hasse principle for generalized affine Chatelet surfaces(2016-05) Berg, Jennifer Sara; Voloch, José Felipe; Ciperiani, Mirela; Allcock, Daniel; Keel, Sean; Varilly-Alvarado, AnthonyThis 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.Item Rational embeddings of the Severi Brauer variety(2010-05) Meth, John Charles; Saltman, D. J. (David J.), 1951-; Helm, David; Reid, Alan; Heitmann, Ray; Haile, DarrellIn an attempt to prove Amitsur's Conjecture for cyclic subgroups of the Brauer group, we look at rational embeddings of the Severi Brauer variety of an algebra into its norm hypersurface. We enlarge the collection of such embeddings, and generalize them to embeddings of generalized Severi Brauer varieties into determinantal varieties.