Arithmetic dynamics on smooth cubic surfaces

TL;DR

This paper investigates the dynamics of birational automorphisms on smooth cubic surfaces over number fields, focusing on periodic points, local-global properties, and connections to the Mordell--Weil problem.

Contribution

It provides new characterizations of periodic points, conditions for strong residual periodicity, and links to the Mordell--Weil problem in the context of cubic surface automorphisms.

Findings

Characterization of $K$ and $ar{K}$-periodic points

Necessary and sufficient conditions for strong residual periodicity

A dynamical result related to the Mordell--Weil problem

Abstract

We study dynamical systems induced by birational automorphisms on smooth cubic surfaces defined over a number field $K$. In particular we are interested in the product of non-commuting birational Geiser involutions of the cubic surface. We present results describing the sets of $K$ and $\bar{K}$-periodic points of the system, and give a necessary and sufficient condition for a dynamical local-global property called strong residual periodicity. Finally, we give a dynamical result relating to the Mordell--Weil problem on cubic surfaces.