Local -global problems and the Brauer -Manin obstruction.

Abstract

This thesis is concerned with local to global (Hasse) principles in algebraic number theory. We consider two such principles in detail. The Hasse Norm Principle states that for a cyclic extension <italic>L</italic>/<italic>K</italic> of number fields, an element of <italic>K</italic> is a global norm if and only if it is a local norm over each completion. Leep and Wadsworth asked if a similar theorem holds for elements which are a product of a global (or local) norm with a global (or local) power. We give necessary and sufficient cohomological conditions for a local to global principle to hold in this situation. Our results have application to the theory of quadratic forms. The classical Hasse Principle holds for a class V of varieties if for each <italic>V</italic> in V the existence of local points on implies the existence of global points. In general this is far from true, and so various authors have introduced obstructions to the validity of the Hasse Principle. We give a new obstruction for curves in this thesis, and then relate it to the well known but computationally intractable Brauer-Manin obstruction. This enables us both to produce new examples of the failure of the Hasse Principle, and to calculate the Brauer-Manin obstruction in these cases.