Computational Arithmetic Geometry Seminar

Let X be a variety defined over Q. Even if X has points in every local field, it still may not have any global points: the Hasse Principle may fail. The Brauer-Manin obstruction attempts to account for this failure. These ideas can be generalized to the case of 0-cycles, and Colliot-Thelene has conjectured that the Brauer-Manin obstruction should be the only obstruction to the Hasse Principle for 0-cycles of degree 1. We prove that a slightly weaker statement is true, assuming that the Tate-Shafarevich group of the Albanese of X is finite.