论文标题
小算术学点的非密度
Non-density of points of small arithmetic degrees
论文作者
论文摘要
如果在一个数字字段上的投影品种上有滤波的内态$ f：x \ to x $，可以在$ x $中以$ x $ $ x $ $ x $定义算术学位$α_f（x）$ $ f $。 Kawaguchi -Silverman猜想（KSC）预测，$ x $的$ x $的任何前向$ f $ oRBIT，算法$α_f（x）$严格小于第一个动力学$δ_f$ $ f $ $ f $的$Δ_F$不是zariski浓密。我们将KSC扩展到沙子（=小算术非密度）的猜想，即所有小算术程度的所有点的基因座$ z_f（d）$不是zariski浓密，并验证这种构想的构态性，以实现投影性的内态性，包括表面，hyperkähler的变化，莫名其妙的变化，莫名其妙的变化，莫名其妙的变化，莫名其妙的变化，莫名其妙的变化，使其与众不同。内态性，平滑的三倍，承认内态性内态性和一些纤维空间。我们显示了对小动力学程度的周期性亚变化的沙子猜想的等效性。我们还展示了沙子猜想与莫顿和西尔弗曼在投射空间的内态性的统一界限之间的密切关系，以及在阿贝尔品种中扭转点统一的扭转点上的另一个长期猜想。
Given a surjective endomorphism $f: X \to X$ on a projective variety over a number field, one can define the arithmetic degree $α_f(x)$ 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 $α_f(x)$ is strictly smaller than the first dynamical degree $δ_f$ 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, HyperKähler 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.
