Non-density of points of small arithmetic degrees
Matsuzawa, Yohsuke Meng, Sheng Shibata, Takahiro Zhang, De-Qi
Matsuzawa, Yohsuke
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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 fiber spaces. We also show close
relations between our 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.
Keywords
math.AG, math.AG, math.DS, math.NT, Primary: 37P55, Secondary: 14G05, 37B40
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2020-02-25
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