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 ζσ 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 Dσ, 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 ζσ(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 ζσ 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 Dσ(Λ −s ).

Dynamically affine maps in positive characteristic

“…Zeta functions of more general dynamical systems can: − be rational : e.g. for "Axiom A" diffeomorphisms by Manning [40], for a "random" such zeta function by Buzzi [10], for some explicit automorphisms of solenoids by Bell, Miles, and Ward [6], and for most endomorphisms of abelian varieties in characteristic p > 0 by the first two authors [11].…”

“…Because of the analytic nature of the function ζ f (z) revealed by our results, one cannot in general use standard Tauberian methods. We have studied this question via a different route for maps on abelian varieties [11] and for maps on general algebraic groups [12] (which covers the case of dynamically affine maps with trivial Γ, h, and ι, but is more general, since we do not require the group G to be commutative). It would be interesting to extend this to general dynamically affine maps.…”

“…Since Γ is finite and σ is confined, the elements λ j (γ) are roots of unity while λ j (σ) are not. By the discussion in [11,Section 5], the elements λ j (σ) are exactly the roots of the action of σ on T (A). Expanding the expression in (22), one can show that deg(σ sn −γ n ) is given by a linear recurrence, and that the dominant root is unique if and only if |λ j (σ)| = 1 for all j.…”

“…− G is an abelian variety: G is isogenous to a product of simple abelian varieties, and deg(σ n − γ) becomes a product of reduced norms N (σ n i − γ i ) on finitely many simple Q-algebras R i (with τ i ∈ R i and γ i ∈ R × i ) [11,Prop. 2.3].…”

“…Notice that the translation parameter h ∈ G(K) no longer occurs in (11). We now introduce the four hypotheses that we require in order to prove the main theorems.…”