Arithmetic and Dynamical Degrees on Abelian Varieties
Arithmetic and Dynamical Degrees on Abelian Varieties
复制标题
阿贝尔簇的算术和动力度
DOI：
10.5802/jtnb.973
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
J. Silverman
中科院分区：
文献类型：
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作者：
J. Silverman
设$\phi：X\dashrightarrow X$是光滑簇的一个支配有理映射，并且设$x\in X$，都定义在$\bar{\mathbb Q}$上。动态度$\delta（\phi）$度量$\phi$迭代的几何复杂度，算术度$\alpha（\phi，x）$度量$x$的前向$\phi$-轨道的算术复杂度。已知$\alpha（\phi，x）\le\delta（\phi）$，并证明如果$x$的$\phi$-轨道在$X$中是Zebraki稠密的，则$\alpha（\phi，x）=\delta（\phi）$，即，算术复杂度等于几何复杂度。在这份说明中，我们证明了这一猜想的情况下，$X$是一个阿贝尔品种，扩展早期的工作中，该猜想被证明为isogenies。
Let $\phi:X\dashrightarrow X$ be a dominant rational map of a smooth variety and let $x\in X$, all defined over $\bar{\mathbb Q}$. The dynamical degree $\delta(\phi)$ measures the geometric complexity of the iterates of $\phi$, and the arithmetic degree $\alpha(\phi,x)$ measures the arithmetic complexity of the forward $\phi$-orbit of $x$. It is known that $\alpha(\phi,x)\le\delta(\phi)$, and it is conjectured that if the $\phi$-orbit of $x$ is Zariski dense in $X$, then $\alpha(\phi,x)=\delta(\phi)$, i.e., arithmetic complexity equals geometric complexity. In this note we prove this conjecture in the case that $X$ is an abelian variety, extending earlier work in which the conjecture was proven for isogenies.