论文标题
在阳性特征中有效的各向同性mordell-lang
Effective isotrivial Mordell-Lang in positive characteristic
论文作者
论文摘要
Moosa和Scanlon的各向同性Mordell-lang定理描述了$ x \capγ$当$ x $是有限的字段$ \ mathbb {f} _q $和$ c $的$ g $ g $ g $ g $ q $ q yous quip yus的$ q $ q y的$ x $时，$ \ mathbb {f} _q $ \ mathbb {f} _q $ \ mathbb {f} _q $的子变量。该描述在这里是有效的，并扩展到任意交换代数组$ g $和任意生成的$ \ mathbb {z} [f] $ - $ - subpodules $γ$。该方法是使用有限自动机给出$ x \capγ$的具体描述。这些方法和结果即使在有限字段的$ g $中是一个Abelian品种，$ x \ subseteq g $ a子变量在功能字段$ k $和$γ= g（k）$时定义了$ x \ subseteq g $，即使是新的应用程序。作为自动机理论方法的应用，建立了二分法定理，以增长有限高度的$ x（k）$中的点数。作为$ x \capγ$的有效描述的应用，给出以下三个毒液问题的决策程序：$ x（k）$ nonepty吗？是无限的吗？它包含一个无限的固定垫吗？
The isotrivial Mordell-Lang theorem of Moosa and Scanlon describes the set $X\capΓ$ when $X$ is a subvariety of a semiabelian variety $G$ over a finite field $\mathbb{F}_q$ and $Γ$ is a finitely generated subgroup of $G$ that is invariant under the $q$-power Frobenius endomorphism $F$. That description is here made effective, and extended to arbitrary commutative algebraic groups $G$ and arbitrary finitely generated $\mathbb{Z}[F]$-submodules $Γ$. The approach is to use finite automata to give a concrete description of $X\cap Γ$. These methods and results have new applications even when specialised to the case when $G$ is an abelian variety over a finite field, $X\subseteq G$ a subvariety defined over a function field $K$, and $Γ=G(K)$. As an application of the automata-theoretic approach, a dichotomy theorem is established for the growth of the number of points in $X(K)$ of bounded height. As an application of the effective description of $X\capΓ$, decision procedures are given for the following three diophantine problems: Is $X(K)$ nonempty? Is it infinite? Does it contain an infinite coset?
