# Points de petite hauteur sur une vari\'et\'e semi-ab\'elienne de la forme $\mathbb{G}_m^n \times A$

## TL;DR

This paper extends a conjecture on lower bounds of heights to split semi-abelian varieties, connecting existing results and providing new examples where the conjecture is valid.

## Contribution

It generalizes a conjecture on height lower bounds to split semi-abelian varieties, unifying previous results and introducing new verified cases.

## Key findings

- Extension of the conjecture to split semi-abelian varieties.
- Connection of existing results in the literature.
- New examples confirming the conjecture.

## Abstract

Recently, R\'emond stated a very general conjecture on lower bounds of a normalized height on either an abelian variety or a power of the multiplicative group. In this note, we extend a particular case of this conjecture to split semi-abelian varieties of the form $\mathbb{G}_m^n \times A$. This allows us to connect many results already existing in the litterature. Finally, we give new examples for which this conjecture holds.