arXiv subjects

Arnaud Plessis

Publications and source records attributed to Arnaud Plessis.

9 recordsLinked to original sources

arXiv2026

Small points in radical extensions of number fields

We study small points in radical extensions of algebraic fields. Given an algebraic extension $\mathbb{F}$ of $\mathbb{Q}$, a finitely generated subgroup $\Gamma\subseteq \mathbb{F}^\times$, and a rational prime $p$, we give a general criterion ensuring that $\mathbb{F}(\Gamma^{p-\mathrm{div}})\setminus \Gamma^{\mathrm{div}}$ has the Bogomolov property. This problem is motivated by a conjecture of R\'emond, formulated when $\mathbb{F}$ is a number field, predicting that such radical extensions contain no unexpected small points outside the divisible hull of the group used to generate them. As applications, we obtain new cases of R\'emond's conjecture for radical extensions generated by division points with respect to a finite set of primes, recovering and extending previous results of Amoroso and the third author. Our argument is based on a recent result on small points in $p$-adic Lie extensions.

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arXiv2025

Lower bounds for heights on some algebraic dynamical systems

Let $v$ be a finite place of a number field $K$ and write $K^{nr,v}$ for the maximal field extension of $K$ in which $v$ is unramified. The purpose of this paper is split up into two parts. The first one generalizes a theorem of Pottmeyer: If $E$ is an elliptic curve defined over $K$ with split multiplicative reduction at $v$, then the N\'eron-Tate height of a non-torsion point $P\in E(\bar{K})$ is bounded from below by $C / e_v(P)^{2 e_v(P)+1}$, where $C>0$ is an absolute constant and $e_v(P)$ is the maximum of all ramification indices $e_w(K(P) \vert K)$ with $w\vert v$. Among other things, we refine this result by showing that given a simple abelian variety $A$ defined over $K$ that is degenerate at $v$, the N\'eron-Tate height of a non-torsion point $P\in A(\bar{K})$ is at least $C / \mathrm{lcm}_{w\vert v} \{e_w(K(P)\vert K)\}^2$, where $C>0$ is an absolute constant. We then give applications towards Lehmer's conjecture. Next, we provide the first examples of polynomials $\phi\in K[X]$ of degree at least $2$ so that the canonical height $\hat{h}_\phi$ of any point in $\bbP^1(K^{nr,v})$ is either $0$ or bounded from below by an absolute positive constant.

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arXiv2023

Equidistribution for sets which are not necessarily Galois stable: On a theorem of Mignotte

An important result of Bilu deals with the equidistribution of the Galois orbits of a sequence $(α_n)_n$ in $\overline{\mathbb{Q}}^*$. Here, we prove a quantitative equidistribution theorem for a sequence of finite subsets in $\overline{\mathbb{Q}}^*$ which are not necessarily stable by Galois action. We follow a method of Mignotte.

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arXiv2023

Bogomolov Property of some infinite nonabelian extensions of a totally $v$-adic field

Let $E$ be an elliptic curve defined over a number field $K$ and let $v$ be a finite place of $K$. Write $K^{tv}$ the maximal extension of $K$ in which $v$ is totally split and $L$ the field generated over $K^{tv}$ by all torsion points of $E$. Under some conditions, we will show that the absolute logarithmic Weil height (resp. Néron-Tate height) of any element of $L^*$ (resp. $E(L)$) is either $0$ or bounded from below by a positive constant depending only on $E, K$ and $v$. This lower bound will be explicit in the toric case when $K=\mathbb{Q}$.

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arXiv2023

Location of small points on an elliptic curve by an equidistribution argument

Let $E$ be an elliptic curve defined over a number field $K$ without complex multiplication. If $Γ\subset E(\overline{K})$ is a subgroup of finite rank, a very special case of a conjecture of Rémond predicts that points of small height in $E(K(Γ))$ lie in the division group of $Γ$. Using an equidistribution argument, we will show that this conjecture is true for groups of rank arbitrarily large.

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arXiv2023

A new way to tackle a conjecture of Rémond

Let $Γ\subset \bar{\mathbb{Q}}^*$ be a finitely generated subgroup. Denote by $Γ_\mathrm{div}$ its division group. A recent conjecture due to Rémond, related to the Zilber-Pink conjecture, predicts that the absolute logarithmic Weil height of an element of $\mathbb{Q}(Γ_\mathrm{div})^*\backslash Γ_{\mathrm{div}}$ is bounded from below by a positive constant depending only on $Γ$. In this paper, we propose a new way to tackle this problem.

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arXiv2022

Points de petite hauteur sur une variété semi-abélienne de la forme $\mathbb{G}_m^n \times A$

Recently, Rémond 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.

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arXiv2019

Minoration de la hauteur de Weil dans un compositum de corps de rayon

The study of Property (B) starts as a special case of Lehmer's conjecture. An algebraic field is said to satisfy Property (B) if there exists a positive constant bounding by below the height of every point of infinite order. In this paper we prove that, under certain uniformity conditions, the compositum of a familly of fields satisfies Property (B).

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arXiv2018

Calcul des groupes de ramification de certaines extensions radicales

For a rational prime $p\neq 2$, we compute the sequence of ramification groups of a Galois, radical and finite extension $L/F$ where $F/\mathbb{Q}_p$ is an unramified finite extension. First, we compute it in the case where the exponent of Gal$(L/F)$ is a power of $p$, by a Hecke's theorem and some inductions,. Finally, we deal with the general case. For the case $p=2$, the same method can work, provided that we make an additional hypothesis.

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