Abstract

Let W⊂P1×P1×P1$\mathcal{W}\subset {\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$$\mathcal{W}\subset {\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$ be a surface given by the vanishing of a (2, 2, 2)-form. These surfaces admit three involutions coming from the three projections W→P1×P1$\mathcal{W}\to {\mathbb{P}}^1\times {\mathbb{P}}^1$$\mathcal{W}\to {\mathbb{P}}^1\times {\mathbb{P}}^1$, so we call them tri-involutive K3 (TIK3) surfaces. By analogy with the classical Markoff equation, we say that W$\mathcal{W}$$\mathcal{W}$ is of Markoff type (MK3) if it is symmetric in its three coordinates and invariant under double sign changes. An MK3 surface admits a group of automorphisms G$\mathcal{G}$$\mathcal{G}$ generated by the three involutions, coordinate permutations, and sign changes. In this paper we study the G$\mathcal{G}$$\mathcal{G}$-orbit structure of points on TIK3 and MK3 surfaces. Over finite fields, we study fibral connectivity and the existence of large orbits, analogous to work of Bourgain, Gamburd, Sarnak and others for the classical Markoff equation. For a particular 1-parameter family of MK3 surfaces Wk${\mathcal{W}}_k$, we compute the full $\mathcal{G}$-orbit structure of ${\mathcal{W}}_k({\mathbb{F}}_p)$ for all primes $p\le 113$, and we use this data as a guide to find many finite $\mathcal{G}$-orbits in ${\mathcal{W}}_k(\mathbb{C})$, including a family of orbits of size 288 parameterized by a curve of genus 9.

摘要

让瓦⊂磷1×磷1×磷1$\mathcal{瓦}\subset {\mathbb{磷}}^1\times {\mathbb{磷}}^1\times {\mathbb{磷}}^1$$\mathcal{W}\subset {\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$是由 (2, 2, 2) 形式的消失给出的曲面。这些表面允许来自三个投影的三个对合瓦→磷1×磷1$\mathcal{瓦}\to {\mathbb{磷}}^1\times {\mathbb{磷}}^1$$\mathcal{W}\to {\mathbb{P}}^1\times {\mathbb{P}}^1$，所以我们称它们为三重渐开 K3 (TIK3)曲面。与经典马尔可夫方程类比，我们可以说瓦$\mathcal{瓦}$$\mathcal{W}$如果它在三个坐标上对称并且在双符号变化下不变，则它是马尔可夫类型（MK3）。MK3 曲面承认一组自同构G$\mathcal{G}$$\mathcal{G}$由三次对合、坐标排列和符号变化生成。在本文中，我们研究G$\mathcal{G}$$\mathcal{G}$-TIK3和MK3表面上点的轨道结构。在有限域上，我们研究纤维连通性和大轨道的存在，类似于 Bourgain、Gamburd、Sarnak 等人对经典马尔可夫方程的工作。对于 MK3 曲面的特定 1 参数族瓦k${\mathcal{W}}_k$，我们计算完整的$\mathcal{G}$- 轨道结构${\mathcal{瓦}}_k（{\mathbb{F}}_p）$对于所有素数$p\le 113$，我们使用这些数据作为指导来找到许多有限的$\mathcal{G}$- 轨道在${\mathcal{瓦}}_k（\mathbb{C}）$，包括由 9 号曲线参数化的大小为 288 的轨道族。