Division of primitive Points in an abelian Variety

Abstract

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 $N^{\delta}$ 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.