Title: Local-Global Principles for Diophantine Equations and Topology

Abstract:
If a variety has no p-adic points for some prime p, then it has no rational points, and this is a standard way to prove non-existence of rational points. The Hasse or Local-Global Principle asks the converse, whether a variety with p-adic points for every p must have a rational point. Lindt, Reichard, and later Selmer found counterexamples to the Hasse principle in the 1940's, which they showed using reciprocity laws. In 1971, Manin unified their methods into a single "obstruction" to the local-global principle using class field theory, via the Brauer group, and Skorobogatov extended this further in the 90's into what is known as the etale-Brauer obstruction to the local-global principle. In 2009, Poonen constructed a smooth proper variety with no rational points, whose lack of rational points is not explained even by the etale-Brauer obstruction, and since then, Harpaz-Schlank reinterpreted the etale-Brauer obstruction in terms of etale homotopy theory. In this talk, we review the above and then discuss recent work of Schlank and the speaker using the topological perspective of Harpaz-Schlank to better understand Poonen's counterexample.