Failures of weak approximation in families

TL;DR

This paper investigates the prevalence of Brauer-Manin obstructions to weak approximation in families of varieties over number fields, showing that such obstructions occur for almost all fibers that are locally soluble.

Contribution

It establishes conditions under which a Brauer-Manin obstruction explains failures of weak approximation for nearly all fibers in a family of varieties.

Findings

Brauer-Manin obstruction accounts for failures of weak approximation in most fibers.

Conditions are identified that predict when the obstruction occurs.

The result applies to a broad class of families over number fields.

Abstract

Given a family of varieties $X\to \mathbb{P}^n$ over a number field $k$, we determine conditions under which there is a Brauer-Manin obstruction to weak approximation for $100\%$ of the fibres which are everywhere locally soluble.