Brauer-Manin obstruction for curves

I gave this talk on the 13th of October of 2026 for the study group on the Brauer-Manin obstruction organised by Tianchen Zhao.

In this talk, I studied the Brauer–Manin obstruction for curves over global fields, and the conjecture that it is the only obstruction to the Hasse principle for curves. I explained the link to the Jacobian and the Tate–Shafarevich group, surveyed the cases where the conjecture is known (genus 0, genus 1 assuming the Tate–Shafarevich group is finite, Jacobians of rank 0, and function fields), and showed how the obstruction can be computed in practice through descent methods. These are the notes I wrote: