Dynamics on abelian varieties in positive characteristic

TL;DR

This paper investigates the dynamics of endomorphisms on abelian varieties over fields of positive characteristic, revealing conditions for rationality of the dynamical zeta function, introducing tame dynamics, and analyzing orbit distributions.

Contribution

It introduces the concept of very inseparable endomorphisms, studies the rationality and boundary properties of the dynamical zeta function, and develops a tame zeta function framework for positive characteristic dynamics.

Findings

Dichotomy between rational and transcendental zeta functions based on very inseparability.

Existence of a tame zeta function that is always algebraic.

Orbit distribution patterns differ significantly between very inseparable and not very inseparable cases.

Abstract

We study periodic points for endomorphisms $\sigma$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $\zeta_\sigma$ of $\sigma$ is either rational or transcendental, the first case happening precisely when $\sigma^n-1$ is a separable isogeny for all $n$. We call this condition very inseparability and show it is equivalent to the action of $\sigma$ on the local $p$-torsion group scheme being nilpotent. The "false" zeta function $D_\sigma$, in which the number of fixed points of $\sigma^n$ is replaced by the degree of $\sigma^n-1$, is always a rational function. Let $1/\Lambda$ denote its largest real pole and assume no other pole or zero has the same absolute value. Then, using a general dichotomy result for power series proven by Royals and Ward in the appendix, we find that $\zeta_\sigma(z)$ has a natural boundary at $|z|=1/\Lambda$ when $\sigma$ is not very inseparable. We introduce and study tame dynamics, ignoring orbits whose order is divisible by $p$. We construct a tame zeta function $\zeta^*_{\sigma}$ that is always algebraic, and such that $\zeta_\sigma$ factors into an infinite product of tame zeta functions. We briefly discuss functional equations. Finally, we study the length distribution of orbits and tame orbits. Orbits of very inseparable endomorphisms distribute like those of Axiom A systems with entropy $\log \Lambda$, but the orbit length distribution of not very inseparable endomorphisms is more erratic and similar to $S$-integer dynamical systems. We provide an expression for the prime orbit counting function in which the error term displays a power saving depending on the largest real part of a zero of $D_\sigma(\Lambda^{-s})$.