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arXiv · 2009.05862

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

Sachi Hashimoto·Katrina Honigs·Alicia Lamarche·Isabel Vogt
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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 $\mathbb{C}$.

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Sachi Hashimoto, Katrina Honigs, Alicia Lamarche, Isabel Vogt. 2020-09-12. A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold. https://doi.org/10.1007/s40993-021-00307-4

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