论文标题
伯克哈特四分之一的曲折三倍
Twists of the Burkhardt Quartic Threefold
论文作者
论文摘要
我们研究了伯克哈特四分之一的曲折，这是与2,3,5不同特征的非广场封闭基地的曲折。我们表明他们都在投影四个空间中承认四分之一的模型。我们确定了一个GALOIS-循环障碍，该障碍物是指给定的扭曲对Abelian品种的模量空间的含义。这种障碍对这些品种的理性点具有影响。结果，我们看到所有可能的3级结构都可以通过Abelian表面实现，而Kummer 3级结构可以理论上可以接受，在某些基础场上可能无法实现。我们在双变量函数字段上举例说明了伯克哈特四分之一的示例，该函数根本没有理性点。 我们的方法基于SP（4,3），Galois的共同学和Burkhardt四重奏的经典代数几何形状的表示理论。
We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2,3,5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological obstruction that measures if a given twist is birational to a moduli space of abelian varieties. This obstruction has implications for the rational points on these varieties. As a result, we see that all possible 3-level structures can be realized by abelian surfaces, whereas Kummer 3-level structures that group-theoretically may be admissible, may not be realizable over certain base fields. We give an example of a Burkhardt quartic over a bivariate function field whose desingularization has no rational points at all. Our methods are based on the representation theory of Sp(4,3), Galois cohomology, and the classical algebraic geometry of the Burkhardt quartic.
