Transcendental Brauer-Manin obstruction on a pencil of elliptic curves

TL;DR

This paper provides an explicit example of transcendental Brauer-Manin obstruction on an elliptic surface, highlighting its divisibility and its nature as a K3 surface, which were absent in prior examples.

Contribution

It presents the first known example where the Brauer class causing the obstruction is divisible and the variety is an elliptic (K3) surface.

Findings

Demonstrates a divisible Brauer class causing obstruction

Provides an explicit elliptic surface example

Highlights the role of transcendental Brauer classes

Abstract

This note gives an explicit example of transcendental Brauer-Manin obstruction to weak approximation. It has two features which the only previously known example of such obstruction did not have: the class in the Brauer group which is responsible for the obstruction is divisible, and the underlying algebraic variety is an elliptic surface (a K3 surface).