Failure of the Hasse principle for Enriques surfaces

TL;DR

This paper constructs specific Enriques and K3 surfaces over Q demonstrating failures of the Hasse principle without algebraic Brauer-Manin obstructions, highlighting transcendental obstructions' role.

Contribution

It provides explicit examples of Enriques and K3 surfaces where the Hasse principle fails due to transcendental obstructions, not algebraic ones.

Findings

Enriques surface with empty étale-Brauer set and no rational points

Existence of K3 surface with transcendental obstruction to Hasse principle

Demonstrates transcendental obstructions can cause failure of the Hasse principle

Abstract

We construct an Enriques surface X over Q with empty \'etale-Brauer set (and hence no rational points) for which there is no algebraic Brauer-Manin obstruction to the Hasse principle. In addition, if there is a transcendental obstruction on X, then we obtain a K3 surface that has a transcendental obstruction to the Hasse principle.