Weak approximation for tori over $p$-adic function fields

TL;DR

This paper investigates weak approximation for tori over p-adic function fields, establishing exact sequences in Galois cohomology to analyze the failure of weak approximation and linking it to unramified cohomology.

Contribution

It constructs explicit Poitou--Tate type exact sequences for tori over p-adic function fields, extending classical number field results to this setting.

Findings

Established a 9-term Poitou--Tate sequence for tori over p-adic function fields.

Linked the defect of weak approximation to a subgroup of the third unramified cohomology group.

Provided tools to analyze local-global principles for tori in this context.

Abstract

This is the companion piece to "Local-global questions for tori over p-adic function fields" by the first and third authors. We study local-global questions for Galois cohomology over the function field of a curve defined over a p-adic field, the main focus here being weak approximation of rational points. We construct a 9-term Poitou--Tate type exact sequence for tori over a field as above (and also a 12-term sequence for finite modules). Like in the number field case, part of the sequence can then be used to analyze the defect of weak approximation for a torus. We also show that the defect of weak approximation is controlled by a certain subgroup of the third unramified cohomology group of the torus.