MATH Seminar

Title: A reexamination of the Birch and Swinnerton-Dyer cubic surfaces
Seminar: Algebra and Number Theory
Speaker: Mckenzie West of Emory University
Contact: Michael H. Mertens, michael.mertens@emory.edu
Date: 2015-11-10 at 4:00PM
Venue: W304
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Abstract: The Hasse principle asks whether solutions to an equation in a local field extend to those in a global field. This does not always happen, the Brauer-Manin obstruction being a common explanation. A conjecture of Colliot-Thelene and Sansuc implies that a Brauer-Manin obstruction exists for every cubic surface which fails to satisfy the Hasse principle. In 1975, Birch and Swinnerton-Dyer gave some early examples of cubic surfaces which have a Brauer-Manin obstruction: (cubic norm) = (linear) (quadratic norm). They make a rough number theoretic argument for the Brauer-Manin obstruction in the case that the Hasse principle fails, focusing on the particular fields and constants. We make use of advancements in arithmetic geometry, taking a geometric look at these objects and utilizing the correspondence between the Brauer group and the Picard group of a surface in order to update and generalize their arguments.