On Weak Approximation of Reductive Groups over Higher Dimensional Function Fields

TL;DR

This paper investigates the weak approximation property for reductive groups over higher dimensional function fields, revealing limitations and providing insights through arithmetic dualities.

Contribution

It introduces a defect to weak approximation for reductive groups over such fields using arithmetic dualities, advancing understanding in higher-dimensional arithmetic geometry.

Findings

Identifies a defect to weak approximation in this setting

Uses arithmetic dualities to analyze the problem

Provides new insights into the structure of reductive groups over higher-dimensional fields

Abstract

Let $k$ be a $d$-local field of characteristic 0, and let $K$ be the function field of a nice curve over $k$. We give a defect to weak approximation for reductive groups over $K$ using arithmetic dualities.