Singular curves and the etale Brauer-Manin obstruction for surfaces

TL;DR

This paper constructs specific algebraic surfaces over number fields that serve as counterexamples to the Hasse principle while having an infinite etale Brauer-Manin set, highlighting new arithmetic phenomena involving singular curves.

Contribution

It introduces novel constructions of surfaces using the arithmetic of singular projective curves, providing counterexamples to the Hasse principle with infinite etale Brauer-Manin sets.

Findings

Constructed a surface violating the Hasse principle with infinite etale Brauer-Manin set.

Built a surface with a unique rational point and infinite etale Brauer-Manin set.

Demonstrated the role of singular curves in arithmetic obstructions.

Abstract

We construct a smooth and projective surface over an arbitrary number field that is a counterexample to the Hasse principle but has the infinite etale Brauer-Manin set. We also construct a surface with a unique rational point and the infinite etale Brauer-Manin set. The key new ingredient is the arithmetic of singular projective curves.