Orbits on K3 Surfaces of Markoff Type

TL;DR

This paper investigates the orbit structure of points on special K3 surfaces of Markoff type, analyzing their symmetry groups over finite fields and complex numbers, and identifying large orbits and their geometric properties.

Contribution

It introduces the study of automorphism group orbits on tri-involutive K3 surfaces of Markoff type, extending classical Markoff equation analysis to higher-dimensional algebraic surfaces.

Findings

Computed full orbit structures over finite fields for a family of MK3 surfaces.

Identified large orbits of size 288 parameterized by a genus 9 curve.

Established connections between orbit sizes and geometric properties of the surfaces.

Abstract

Let $\mathcal{W}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be a surface given by the vanishing of a $(2,2,2)$-form. These surfaces admit three involutions coming from the three projections $\mathcal{W}\to\mathbb{P}^1\times\mathbb{P}^1$, so we call them $\textit{tri-involutive K3 (TIK3) surfaces}$. By analogy with the classical Markoff equation, we say that $\mathcal{W}$ is of $\textit{Markoff type (MK3)}$ if it is symmetric in its three coordinates and invariant under double sign changes. An MK3 surface admits a group of automorphisms $\mathcal{G}$ generated by the three involutions, coordinate permutations, and sign changes. In this paper we study the $\mathcal{G}$-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 $\mathcal{W}_k$, we compute the full $\mathcal{G}$-orbit structure of $\mathcal{W}_k(\mathbb{F}_p)$ for all primes $p\le113$, 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 $9$.