A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold

TL;DR

This paper demonstrates that certain Calabi-Yau threefolds have a transcendental Brauer-Manin obstruction preventing weak approximation, revealing new arithmetic properties linked to their geometric and derived equivalence structures.

Contribution

It establishes the existence of a transcendental Brauer-Manin obstruction on specific Calabi-Yau threefolds and constructs derived equivalences over , linking geometric and arithmetic properties.

Findings

Brauer class causes obstruction to weak approximation

Derived equivalences can be constructed over 

Conditions identified for failure of weak approximation

Abstract

In this paper we investigate the $\mathbb{Q}$-rational points of a class of simply connected Calabi-Yau threefolds, which were originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural $2$-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation. Hosono and Takagi showed that over $\mathbb{C}$ each of these Calabi-Yau threefolds $Y$ is derived equivalent to a Reye congruence Calabi-Yau threefold $X$. We show that these derived equivalences may also be constructed over $\mathbb{Q}$, and we give sufficient conditions for $X$ to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi--Yau variety over $\mathbb{C}$.