Counting periodic points over finite fields

TL;DR

This paper investigates how group actions and symmetries influence the counting of periodic points of endomorphisms on algebraic varieties over finite fields, revealing relations between counts on quotients and twists.

Contribution

It establishes relations between periodic point counts on quotients of varieties and their twists using group ring idempotents, especially for abelian groups.

Findings

Idempotent relations in group rings relate periodic point counts on quotients.

Periodic point counts on twisted varieties are connected to counts on the original variety.

Results apply to endomorphisms commuting with finite group actions over finite fields.

Abstract

Let $V$ be a quasiprojective variety defined over $\mathbb{F}_q$, and let $\phi:V\rightarrow V$ be an endomorphism of $V$ that is also defined over $\mathbb{F}_q$. Let $G$ be a finite subgroup of $\operatorname{Aut}_{\mathbb{F}_q}(V)$ with the property that $\phi$ commutes with every element of $G$. We show that idempotent relations in the group ring $\mathbb{Q}[G]$ give relations between the periodic point counts for the maps induced by $\phi$ on the quotients of $V$ by the various subgroups of $G$. We also show that if $G$ is abelian, periodic point counts for the endomorphism on $V/G$ induced by $\phi$ are related to periodic point counts on $V$ and all of its twists by $G$.