Division of primitive Points in an abelian Variety

TL;DR

This paper establishes effective lower bounds on the degree of points preimage under multiplication in abelian varieties, leading to results on unlikely intersections and applications to an inverse elliptic Fermat equation.

Contribution

It introduces explicit degree bounds for preimages of primitive points in abelian varieties, combining Masser's estimates with a uniform Manin-Mumford to address unlikely intersections.

Findings

Effective lower bounds on degrees of preimages of primitive points.

An unlikely intersections result for subvarieties intersecting preimages.

Application to inverse elliptic Fermat equations.

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.