论文标题
亚洲品种中原始点的划分
Division of primitive Points in an abelian Variety
论文作者
论文摘要
让$ a $是在数字字段$ k $上定义的阿贝里安品种。我们说，如果在a（\ overline {\ overline {\ mathbb {q}}）中，在$ p $ a的定义上定义了$ p $ a的定义$ k $ [q $ k $ po = p $ for $ k $ n] q $ [n] q $ n Inte $ n $ n \ ge 2 $ N，则在（\ overline {\ mathbb {q}}）中的一个点$ p \是原始的。对于任何原始点$ p \ in（\ overline {\ mathbb {q}}）$，正整数$ n $和point $ q \ in（\ overline {\ mathbb {q}}）$，使得$ [n] q = p $ a $ n $ n $ n $ n $ k y n $ [n] q = p $ a n $ k y n $ k y n $ k y n $ k y n $ n $ n $ n n $ n $ n $ $ a，k $和$ p $ $ k $的定义领域程度。证明基于Masser对扭转点程度的估计。 我们将此结果与Manin-Mumford的均匀版本相结合，以证明一个不太可能的相互作用类型结果：如果$ p \在A（\ overline {\ Mathbb {q}}）$中是原始的，则定义为$ d $ a $ a $ a $ a $ a $ a $ a $ a $ x \ cap [$ x $ a $ x $ a $ x \ cap cap [ $ x $的弱特殊部分，前提是$ n $大于$ d $的合适功率。 作为一种应用，我们研究了一个椭圆形的逆向方程，类似于由pila处理的模块化费摩特方程。
Let $A$ be an abelian variety defined over a number field $K$. We say that a point $P \in A(\overline{\mathbb{Q}})$ is primitive if there is no $Q \in A(\overline{\mathbb{Q}})$ defined on the field of definition of $P$ over $K$ such that $[N]Q=P$ for some positive integer $N \ge 2$. For any primitive point $P \in A(\overline{\mathbb{Q}})$, positive integer $N$ and point $Q \in A(\overline{\mathbb{Q}})$ such that $[N]Q=P$, we prove an effective lower bound on the degree of the field of definition of $Q$ over $K$ of the form $N^δ$ that depends only on $A,K$ and the degree of the field of definition of $P$ over $K$. The proof is based on the estimates of the degree of torsion points by Masser. We combine this result with a uniform version of Manin-Mumford to prove an effective Unlikely Intersections-type result: if $P \in A(\overline{\mathbb{Q}})$ is primitive, defined over a field of degree $d$ over $K$, and $X$ is a subvariety of $A$, then $X \cap [N]^{-1}P$ is contained in the weakly special part of $X$, provided $N$ is bigger than a suitable power of $d$. As an application, we study an inverse elliptic Fermat equation, analogous to a modular Fermat equation treated by Pila.
