The representations of the automorphism groups and the Frobenius invariants of K3 surfaces
The representations of the automorphism groups and the Frobenius invariants of K3 surfaces
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K3曲面自同构群和Frobenius不变量的表示
DOI：
10.1307/mmj/1457101815
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Junmyeong Jang
中科院分区：
文献类型：
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作者：
Junmyeong Jang
对于奇特征代数闭域上的K3曲面，其自同构群在整体两种形式上的表示是有限的。如果K3曲面是超奇异的，则它同构于Neron-Severi群的判别群上的表示。如果K3曲面的高度有限，则（etale或crystallized）超越循环上的表示也是有限的，并且存在从超越循环上的表示到两种形式上的表示的典范投影。我们证明了，如果基域是一个有限域的代数闭包，这个投影是同构和超越循环的秩是一个倍数的顺序表示的两种形式。它们类似于复K3曲面上的经典结果。从这些结果，我们推出的高度和Artin不变量的一个K3曲面的纯非辛自同构的高阶由一个合同类的基特征。
For a K3 surface over an algebraically closed field of odd characteristic, the representation of the automorphism group on the global two forms is finite. If the K3 surface is supersingular, it is isomorphic to the representation on the discriminant group of the Neron-Severi group. If the K3 surface is of finite height, the representation on the (etale or crystalline) transcendental cycles is also finite and there is a canonical projection from the representation on the transcendental cycles to the representation on the two forms. We prove that, if the base field is an algebraic closure of a finite field, this projection is an isomorphism and the rank of the transcendental cycles is a multiple of the order of the representation on the two forms. They are analogous to the classical results on complex K3 surfaces. From these results, we deduce that the height and the Artin invariant of a K3 surface with a purely non-symplectic automorphism of higher order are determined by a congruence class of the base characteristic.