Abstract for the talk on 23.06.2022 (14:30 h)ICM 2022 preparatory lecture
Sachi Hashimoto (MPI MiS, Leipzig)
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.