Large orbits on Markoff-type K3 surfaces over finite fields

TL;DR

This paper investigates the orbit structure of a specific K3 surface over finite fields, revealing a surprising partition into two invariant subsets due to an explicit double cover, especially over fields with order congruent to 1 mod 8.

Contribution

It uncovers a novel orbit partition phenomenon on a tri-involutive K3 surface over finite fields and links it to an explicit double cover construction.

Findings

Over finite fields of order ≡ 1 mod 8, points on the surface form two large invariant subsets.

The phenomenon is explained via an explicit double cover of the surface.

The orbit partition is specific to the case when k=4 on the surface.

Abstract

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 $\Gamma$ generated by the three involutions fixing two variables and a few other obvious symmetries, but rather admit a partition into two $\Gamma$-invariant subsets of roughly equal size. The phenomenon is traced to an explicit double cover of the surface.