Duality for cohomology of split tori on curves over local fields

TL;DR

This paper establishes duality theorems for étale cohomology of sheaves and split tori on curves over local fields, linking Brauer groups to idele characters and extending classical pairings to broader curve classes.

Contribution

It introduces new duality results for cohomology of logarithmic Hodge-Witt sheaves and split tori on curves over local fields, and generalizes Brauer-Manin pairings.

Findings

Duality theorems for étale cohomology of sheaves and tori.

Description of Brauer groups via idele group characters.

Extensions of Brauer-Manin pairings to various types of curves.

Abstract

We prove duality theorems for the {\'e}tale cohomology of logarithmic Hodge-Witt sheaves and split tori on smooth curves over a local field of positive characteristic. As an application, 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 groups. We also show that the classical Brauer-Manin pairing between the Brauer and Picard groups of smooth projective curves over local fields has analogues for arbitrary smooth curves, smooth projective curves with modulus and singular projective curves over such fields.