Dynamics on abelian varieties in positive characteristic

Abstract

We study periodic points for endomorphisms $σ$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $ζ_σ$ of $σ$ is either rational or transcendental, the first case happening precisely when $σ^n-1$ is a separable isogeny for all $n$. We call this condition very inseparability and show it is equivalent to the action of $σ$ on the local $p$-torsion group scheme being nilpotent. The "false" zeta function $D_σ$, in which the number of fixed points of $σ^n$ is replaced by the degree of $σ^n-1$, is always a rational function. Let $1/Λ$ 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 $ζ_σ(z)$ has a natural boundary at $|z|=1/Λ$ when $σ$ is not very inseparable. We introduce and study tame dynamics, ignoring orbits whose order is divisible by $p$. We construct a tame zeta function $ζ^*_σ$ that is always algebraic, and such that $ζ_σ$ 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 $\log Λ$, 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 $D_σ(Λ^{-s})$.

Appendix by Robert Royals and Thomas Ward. Expanded introduction including a worked example; added argument (Prop. 5.4) to only have to consider zeros instead of zeros and poles; further minor changes; 47 pp