Non-density of points of small arithmetic degrees

TL;DR

This paper extends the Kawaguchi-Silverman Conjecture to a broader non-density context, verifying it for various classes of projective varieties and exploring its connections with other major conjectures in arithmetic dynamics.

Contribution

It introduces the sAND Conjecture for points of small arithmetic degree and proves it for multiple classes of varieties, establishing equivalences and relations with other key conjectures.

Findings

Verification of sAND Conjecture for surfaces, HyperK"ahler, abelian varieties, and more.

Establishment of equivalence between sAND Conjecture and a conjecture on small dynamical degree subvarieties.

Connections drawn between sAND Conjecture, Morton-Silverman Uniform Boundedness, and torsion points conjectures.

Abstract

Given a surjective endomorphism $f: X \to X$ on a projective variety over a number field, one can define the arithmetic degree $\alpha_f(x)$ of $f$ at a point $x$ in $X$. The Kawaguchi - Silverman Conjecture (KSC) predicts that any forward $f$-orbit of a point $x$ in $X$ at which the arithmetic degree $\alpha_f(x)$ is strictly smaller than the first dynamical degree $\delta_f$ of $f$ is not Zariski dense. We extend the KSC to sAND (= small Arithmetic Non-Density) Conjecture that the locus $Z_f(d)$ of all points of small arithmetic degree is not Zariski dense, and verify this sAND Conjecture for endomorphisms on projective varieties including surfaces, HyperK\"ahler varieties, abelian varieties, Mori dream spaces, simply connected smooth varieties admitting int-amplified endomorphisms, smooth threefolds admitting int-amplified endomorphisms, and some fibre spaces. We show the equivalence of the sAND Conjecture and another conjecture on the periodic subvarieties of small dynamical degree; we also show the close relations between the sAND Conjecture and the Uniform Boundedness Conjecture of Morton and Silverman on endomorphisms of projective spaces and another long standing conjecture on Uniform Boundedness of torsion points in abelian varieties.