Date:
Mon, 10/08/2015 - 16:30 - 17:30
Given a fibration of spaces there exists an obstruction theory for the existence of sections,
With values in certain cohomology groups. We will explain how to create an analog theory In arithmetic geometry based on e'tale homotopy. The Brauer-Manin obstruction is probably the best known obstruction to the existence of points on an algebraic variety. The BM obstruction can also be used to obstruct the existence of zero cycles (Galois invariant formal sums of geometric points). For rational points stronger obstruction exists. In 99' Skorobogatov defined the more refined etale-Brauer-Manin obstruction, which is a finer obstruction to the existence of such points. However, this obstruction cannot be applied to zero cycles . The theory of e'tale homotopy gives us a way to reinterpret these obstructions and to give a homotopical explanation for this fact. The difference between BM and e'tale-BM lies in the difference between homotopy and homology, and it is homology's abelian nature that allows to extend the obstruction.
The homotopy theorist knows that taking homology is a very violent abelinization, a finer process is to work with infinite loop spaces. This suggest that a better obstruction should exist in stable e'tale homotopy We'll present such an obstruction and explain how to compute it.
