论文标题
有限田的Markoff型K3表面上的大轨道
Large orbits on Markoff-type K3 surfaces over finite fields
论文作者
论文摘要
我们研究表面$ \ MATHCAL {W} _K：X^2 + Y^2 + Z^2 + X^2 y^2 Z^2 Z^2 = K X Y Z $ in $（\ Mathbb {p}^1）^3 $，三键式k3（tik3）表面。我们解释了Fuchs，Litman，Silverman和Tran所注意到的一种现象：在有限的订单领域$ \ equiv 1 $ mod $ 8 $，$ \ Mathcal {w} _4 $的点不形成单个大牛仔大小相等的子集。该现象可追溯到表面的显式双层盖。
We study the surface $\mathcal{W}_k : x^2 + y^2 + z^2 + x^2 y^2 z^2 = k x y z$ in $(\mathbb{P}^1)^3$, a tri-involutive K3 (TIK3) surface. We explain a phenomenon noticed by Fuchs, Litman, Silverman, and Tran: over a finite field of order $\equiv 1$ mod $8$, the points of $\mathcal{W}_4$ do not form a single large orbit under the group $Γ$ generated by the three involutions fixing two variables and a few other obvious symmetries, but rather admit a partition into two $Γ$-invariant subsets of roughly equal size. The phenomenon is traced to an explicit double cover of the surface.
