Singular curves and the etale Brauer-Manin obstruction for surfaces

Yonatan Harpaz (1) , Alexei N. Skorobogatov (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 Department of Mathematics [Imperial College London]

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.

Domaines: Géométrie algébrique [math.AG]

Yonatan Harpaz, Alexei N. Skorobogatov. Singular curves and the etale Brauer-Manin obstruction for surfaces. Annales Scientifiques de l'École Normale Supérieure, 2014. ⟨hal-02409239⟩