Papers › A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold
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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In this paper we investigate the 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 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 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 ℂ.
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