Duality theorems for curves over local fields

We prove duality theorems for the étale cohomology of split tori on smooth curves over a local field of positive characteristic. In particular, we show that the classical Brauer–Manin pairing between the Brauer and Picard groups of smooth projective curves over such a field extends to arbitrary smooth curves over the field. As another consequence, we obtain a description of the Brauer group of the function fields of curves over local fields in terms of the characters of the idele class groups.