Related papers: Minoration effective de la hauteur des points d'un…

In this short note, we give a method for computing a non-torsion point of smallest canonical height on a given elliptic curve $E/\mathbb{Q}$ over all number fields of a fixed degree. We then describe data collected using this method, and…

We obtain a lower bound for the normalised height of a non-torsion hypersurface $V$ of a C.M. abelian variety $A$ which is a refinement of a precedent result. This lower bound is optimal in terms of the geometric degree of $V$, up to an…

In this note we obtain effective lower bounds for the canonical heights of non-torsion points on $E(\mathbb{Q})$ by making use of suitable elliptic curve ideal class pairings $$\Psi_{E,-D}: E(\mathbb{Q})\times E_{-D}(\mathbb{Q})\mapsto…

We tackle the "relative" Lehmer problem on algebraic subvarieties of a multiplicative torus. Generalizing a theorem of F. Amoroso and U. Zannier, we give a lower bound for the normalized height of a non torsion hypersurface in terms of its…

We obtain a lower bound for the normalised height of a non-torsion subvariety $V$ of a C.M. abelian variety. This lower bound is optimal in terms of the geometric degree of $V$, up to a power of a ``log''. We thus extend the results of F.…

Let $E$ be an elliptic curve over the rationals. We will consider the infinite extension $\mathbb{Q}(E_{\text{tor}})$ of the rationals where we adjoin all coordinates of torsion points of $E$. In this paper we will prove an explicit lower…

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:…

Let $A/K$ be an abelian variety over a number field $K$. We prove in this article that a good lower bound (in terms of the degree $[K(P):K]$) for the N\'eron-Tate height of the points $P$ of infinite order modulo every strict abelian…

We give examples of points with particularly low height on elliptic curves over quadratic fields, recovered by a search over elliptic divisibility sequences. The smallest example identified satisfies dh(P)=0.0077127...: improving on the…

Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of non-torsion points on E defined over the maximal abelian extension K^ab of…

We establish an explicit lower bound for the N\'eron-Tate height on elliptic curves with complex multiplication, for nontorsion points defined over the maximal abelian extension of a number field. Building on a strategy developed by…

Let E/K be an ellptic curve defined over a number field, let h be the canonical height on E, and let K^ab be the maximal abelian extension of K. Extending work of M. Baker, we prove that there is a positive constant C(E/K) so that every…

Let E be an elliptic curve over the rationals. Let L be an infinite Galois extension of the rationals with uniformly bounded local degrees at almost all primes. We will consider the infinite extension L(E_tor) of the rationals where we…

Currently, the best upper bounds on the number of rational points on an absolutely irreducible, smooth, projective algebraic curve of genus g defined over a finite field F_q come either from Serre's refinement of the Weil bound if the genus…

Let $K$ be a number field, $S$ a finite set of places. For $\mathbb{G}_m$ or an elliptic curve $E$ defined over $K$, we establish uniformity results on the number of $S$-integral torsion points relative to a non-torsion point $\beta$, as…

Let $E$ be an elliptic curve over a number field $K$. Descent calculations on $E$ can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this…

Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(\bar{K})$. More precisely,…

Let $E/\mathbb{Q}$ be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if $E$ has full rational $2$-torsion, the number $N_E(B)$ of rational points with Weil height bounded by $B$ is…

Consider a component of the Hilbert scheme whose general point corresponds to a degree d genus g smooth irreducible and nondegenerate curve in a projective variety X. We give lower bounds for the dimension of such a component when X is P^3,…

A point (x1, x2) with coordinates in a subfield of R of transcendence degree one over Q, with 1, x1, x2 linearly independent over Q, may have a uniform exponent of approximation by elements of Q^2 that is strictly larger than the lower…