Insufficiency of the \'etale Brauer-Manin obstruction: towards a simply connected example

TL;DR

This paper constructs new examples of algebraic varieties where the étale Brauer-Manin obstruction fails to explain the absence of rational points, including simply connected cases, using Beauville surfaces and assuming the abc conjecture.

Contribution

It provides the first examples of such varieties with trivial Albanese variety and simply connectedness, expanding understanding beyond previous fibrational constructions.

Findings

Constructed examples with trivial Albanese variety.

Proved existence of simply connected examples under the abc conjecture.

Extended the class of known counterexamples to the étale Brauer-Manin obstruction.

Abstract

Since Poonen's construction of a variety $X$ defined over a number field $k$ for which $X(k)$ is empty and the \'etale Brauer--Manin set $X(\mathbf{A}_k)^\text{Br,et}$ is not, several other examples of smooth, projective varieties have been found for which the \'etale Brauer--Manin obstruction does not explain the failure of the Hasse principle. All known examples are constructed using "Poonen's trick", i.e. they have the distinctive feature of being fibrations over a higher genus curve; in particular, their Albanese variety is non-trivial. In this paper, we construct examples for which the Albanese variety is trivial. The new geometric ingredient in our construction is the appearance of Beauville surfaces. Assuming the $abc$ conjecture and using geometric work of Campana on orbifolds, we also prove the existence of an example which is simply connected.