Várilly-Alvarado, Anthony2019-05-162019-05-162019-052019-04-19May 2019Johnson, Alexis Katherine. "Brauer groups of Kummer surfaces arising from elliptic curves with complex multiplication." (2019) Diss., Rice University. <a href="https://hdl.handle.net/1911/105424">https://hdl.handle.net/1911/105424</a>.https://hdl.handle.net/1911/105424The Brauer group of a variety often captures arithmetic information about the space. In this thesis, we study the Brauer group of a special kind of K3 surface, namely, a Kummer surface associated with a self-product of an elliptic curve over a number field with complex multiplication by a non-maximal order in an imaginary quadratic number field. Skorobogatov has conjectured that the Brauer group controls the existence of rational points on K3 surfaces. In practice, given a K3 surface, one often needs explicit descriptions of Brauer elements in order to study the behavior of rational points. The surfaces studied in this thesis have a geometrically rich structure that enables us to explicitly compute both the algebraic and transcendental Brauer groups over the rational numbers. Furthermore, over arbitrary number fields, we bound the transcendental Brauer group.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.Brauer groupsK3 surfaceselliptic curvesalgebraic number theoryalgebraic geometryThesis2019-05-16