Brauer–Manin obstructions on degree 2 K3 surfaces
Brauer–Manin obstructions on degree 2 K3 surfaces
复制标题
2 级 K3 表面上的布劳尔-马宁障碍物
DOI：
10.1007/s40993-018-0126-x
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发表时间：
2017
影响因子：
0.8
通讯作者：
M. Nakahara
中科院分区：
文献类型：
--
作者：
Patrick Corn;M. Nakahara
研究了$${\martbb {P}}^{2}}$$P2在对角六次曲线上分歧的双覆盖对$${{\martbb {Q}}$$Q上K3曲面上有理点的Brauer-Manin阻塞.在找到一个明确的一组发电机的几何皮卡德群这样的表面，我们发现两种类型的无限家庭的反例哈塞原则解释的代数Brauer-Manin障碍。第一种类型的障碍来自一个四元数代数，第二种类型来自一个3-挠元素的Brauer组，这给了一个肯定的答案所提出的问题Ieronymou和Skorobogatov。
We analyze the Brauer–Manin obstruction to rational points on the K3 surfaces over $${{\mathbb {Q}}}$$Q given by double covers of $${{\mathbb {P}}^{2}}$$P2 ramified over a diagonal sextic. After finding an explicit set of generators for the geometric Picard group of such a surface, we find two types of infinite families of counterexamples to the Hasse principle explained by the algebraic Brauer–Manin obstruction. The first type of obstruction comes from a quaternion algebra, and the second type comes from a 3-torsion element of the Brauer group, which gives an affirmative answer to a question asked by Ieronymou and Skorobogatov.