Local-global principles for rational points

Let Q be a smooth projective quadric of dimension at least 1 over a number field k. Then Q has a k-point if and only if it has points over the completions of k at every place. For example, the conic Q: $x^2+y^2 = -1$ has no real solutions, and therefore no rational points. However, the genus 1 curve $2y^2 = x^4 - 17z^4$ has solutions over the reals and the p-adics for every prime p, but does not have any rational points. Here, there is an obstruction to the local-global principle called the Brauer-Manin obstruction. In this talk, we will give an introduction to the Brauer-Manin obstruction, weak approximation, and local-global principles.

This is a preparatory lecture for the ICM talk of Olivier Wittenberg.