Obstructions to the Local-Global Principle Applied to Open Subvarieties

2017/12/22 Fri 13:30 - 14:30
David Corwin
Massachusetts Institute of Technology

In 2008, Bjorn Poonen announced the construction of a variety without rational points but no étale-Brauer obstruction to the existence of rational points. We attempt to create a new obstruction that explains Poonen's example by applying the étale-Brauer obstruction to a Zariski open cover of a variety. On the one hand, we prove a general result that this new obstruction explains every variety without rational points, over quadratic imaginary and totally real number fields. On the other hand, this method is not effective, as the set of adelic points of a non-proper variety is non-compact. Nonetheless, we show how this new obstruction can be applied in Poonen's example. In doing so, we analyze the example from an algebro-topological perspective via the étale homotopy obstruction of Harpaz-Schlank and prove some results of independent interest in this direction. We also mention a related investigation about cuspidal sections of algebraic fundamental groups of higher-dimensional varieties.