Finding Rational Periodic Points on Wehler K3 Surfaces

TL;DR

This paper develops an algorithm to find rational periodic points on certain K3 surfaces and applies it to identify surfaces with points of various periods, also exploring their zeta functions and Picard numbers.

Contribution

It introduces a novel algorithm for detecting rational periodic points on Wehler K3 surfaces using modular information, expanding understanding of their dynamical properties.

Findings

Identified K3 surfaces with rational periodic points of periods 1 to 16.

Determined the zeta function modulo 3 for a specific K3 surface.

Constructed a family of K3 surfaces with Picard number two.

Abstract

This article examines dynamical systems on a class of K3 surfaces in $\mathbb{P}^{2} \times \mathbb{P}^{2}$ with an infinite automorphism group. In particular, this article develops an algorithm to find $\mathbb{Q}$-rational periodic points using information modulo $p$ for various primes $p$. The algorithm is applied to exhibit K3 surfaces with $\mathbb{Q}$-rational periodic points of primitive period $1,...,16$. A portion of the algorithm is then used to determine the Riemann zeta function modulo 3 of a particular K3 surface and find a family of K3 surfaces with Picard number two.