Intersection de courbes et de sous-groupes, et probl\`emes de minoration de hauteur dans les vari\'et\'es ab\'eliennes C.M

TL;DR

This paper proves a special case of the Zilber-Pink conjecture for curves in abelian varieties of C.M. type, establishing finiteness of intersections with certain subgroup schemes and providing new height bounds.

Contribution

It introduces a novel lower bound for Néron-Tate heights on C.M. abelian varieties, extending Lehmer's problem and generalizing previous results to new cases.

Findings

Finiteness of intersections for curves in C.M. abelian varieties.

New lower bounds for Néron-Tate heights.

Application to bounds on the absolute minimum of subvarieties.

Abstract

We prove a special case of the following conjecture of Zilber-Pink generalising the Manin-Mumford conjecture : let $X$ be a curve inside an Abelian variety $A$ over $\bar{\Q}$, provided $X$ is not contained in a torsion subvariety, the intersection of $X$ with the union of all subgroup schemes of codimension at least 2 is finite ; we settle the case where $A$ is a power of a simple Abelian variety of C.M. type. This generalises the previous known result, due to Viada and R\'emond-Viada (who was able to prove the conjecture for power of an elliptic curve with complex multiplication). The proof is based on the strategy of R\'emond (following Bombieri, Masser and Zannier) with two new ingredients, one of them, being at the heart of this article : it is a lower bound for the N\'eron-Tate height of points on Abelian varieties $A/K$ of C.M. type in the spirit of Lehmer's problem. This lower bound is an analog of the similar result of Amoroso and David \cite{ad2003} on $\G_m^n$ and is a generalisation of the theorem of David and Hindry \cite{davidhindry} on the abelian Lehmer's problem. The proof is an adaptation of \cite{davidhindry} using in our abelian case the new ideas introduced in \cite{ad2003}. Furthermore, as in \cite{ad2003} and adapting in the abelian case their proof, we give another application of our result : a lower bound for the absolute minimum of a subvariety $V$ of $A$. Although lower bounds for this minimum were already known (decreasing multi-exponential function of the degree for Bombieri-Zannier), our methods enable us to prove, up to an $\epsilon$ the optimal result that can be conjectured.