The Brauer-Manin obstruction
Bjorn Poonen

I will define and state some basic properties of Brauer groups of schemes,
and then I will explain the Brauer-Manin obstruction.  As an example,
I will use it to show that the equation y^2 + z^2 = (x^2 - 2)(3 - x^2) 
has no rational solutions.