A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation

TL;DR

This paper investigates cohomological obstructions to the Hasse principle and weak approximation, focusing on Tate--Shafarevich groups over p-adic function fields, and introduces new unramified obstructions.

Contribution

It provides a novel analysis of Tate--Shafarevich groups' unramified nature, leading to new obstructions to the Hasse principle for torsors under tori.

Findings

Tate--Shafarevich groups are unramified in certain cases

New obstructions to the Hasse principle are identified

Applications to torsors under tori over p-adic fields

Abstract

We show that certain Tate--Shafarevich groups are unramified which enables us to give an obstruction to the Hasse principle for torsors under tori over p-adic function fields.