Odd order obstructions to the Hasse principle on general K3 surfaces
Odd order obstructions to the Hasse principle on general K3 surfaces
复制标题
一般 K3 表面上哈斯原理的奇数阶障碍
DOI：
--
复制
发表时间：
2018
影响因子：
2
通讯作者：
Anthony Várilly
中科院分区：
文献类型：
--
作者：
J. Berg;Anthony Várilly
我们证明了K3曲面的Brauer群的奇数阶超越元可以阻碍Hasse原理。我们展示了一个一般的K3曲面$Y$的程度2 $\mathbb{Q}$连同一个三扭Brauer类$\alpha$是unramified在所有的素数，除了3，但分歧在所有的3-adic点的$Y$。受霍奇理论的启发，对$（Y，\alpha）$是从一个三次四重的判别式18双有理的X$到一个纤维化到六次del Pezzo曲面在投影平面上构造的。值得注意的是，我们的构造并不依赖于$\alpha$的中心简单代数代表的存在。相反，我们证明了这样一个布劳尔类阻碍哈塞原理的充分条件是不溶性的四重$X$（因此纤维）超过$\mathbb{Q}_3$和本地溶解度在所有其他素数。
We show that odd order transcendental elements of the Brauer group of a K3 surface can obstruct the Hasse principle. We exhibit a general K3 surface $Y$ of degree 2 over $\mathbb{Q}$ together with a three torsion Brauer class $\alpha$ that is unramified at all primes except for 3, but ramifies at all 3-adic points of $Y$. Motivated by Hodge theory, the pair $(Y, \alpha)$ is constructed from a cubic fourfold $X$ of discriminant 18 birational to a fibration into sextic del Pezzo surfaces over the projective plane. Notably, our construction does not rely on the presence of a central simple algebra representative for $\alpha$. Instead, we prove that a sufficient condition for such a Brauer class to obstruct the Hasse principle is insolubility of the fourfold $X$ (and hence the fibers) over $\mathbb{Q}_3$ and local solubility at all other primes.
影响因子：
1.8
作者：
D. Huybrechts
通讯作者：
D. Huybrechts