Effective computation of Picard groups and Brauer-Manin obstructions of degree two \(K3\) surfaces over number fields. (English) Zbl 1297.14027
The main object of the paper under review is a \(K3\) surface \(X\) of degree 2 defined over a number field \(k\). The authors are interested in the group \(\mathrm{Br}(X)/\mathrm{Br}(k)\), where the Brauer-Manin obstructions to the Hasse principle and weak approximation live. This group is known to be finite A. N. Skorobogatov and Y. G. Zarhin [J. Algebr. Geom. 17, No. 3, 481–502 (2008; Zbl 1157.14008)]. The main result of the paper states that its order can be bounded effectively whenever an explicit equation of \(X\) is given.
On the way, the authors provide several effective assertions which are interesting by their own right, such as an effective Kuga-Satake construction and computation of the Picard module.
On the way, the authors provide several effective assertions which are interesting by their own right, such as an effective Kuga-Satake construction and computation of the Picard module.
Reviewer: Boris Kunyavskii (Ramat Gan)
MSC:
| 14G25 | Global ground fields in algebraic geometry |
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14F22 | Brauer groups of schemes |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
Citations:
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