On montre comment écrire de grandes familles, avec de hautes multiplicités, de cas d’égalité $A+B=C$ pour l’inégalité de Stothers-Mason (si $A\left(X\right),B\left(X\right),C\left(X\right)$ sont des polynômes premiers entre eux, le nombre exact de racines du produit $ABC$ dépasse de $1$ le plus grand des degrés des composantes $A,B,C)$. On développera pour cela des techniques polynomiales itératives inspirées des décompositions de Dunford-Schwartz et de fonctions de Belyi. Des exemples d’application avec les conjectures $\left(abc\right)$ ou de M. Hall sont développés.

We study some functional equations between Mahler measures of genus-one curves in terms of isogenies between the curves. These equations have the potential to establish relationships between Mahler measure and especial values of $L$-functions. These notes are based on a talk that the author gave at the “Cuartas Jornadas de Teoría de Números”, Bilbao, 2011.

We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.

Let $K$ be a number field. It is well known that the set of recurrencesequences with entries in $K$ is closed under component-wise operations, and so it can be equipped with a ring structure. We try to understand the structure of this ring, in particular to understand which algebraic equations have a solution in the ring. For the case of cyclic equations a conjecture due to Pisot states the following: assume $\left\{a_n\right\}$ is a recurrence sequence and suppose that all the $a_n$ have a $d^{\mathrm{th}}$ root in the field $K$; then (after...

We compute, in a unified way, the equations of all hyperelliptic modular curves. The main tool is provided by a class of modular functions introduced by Newman in 1957. The method uses the action of the hyperelliptic involution on the cusps.

Let be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in dual terms) that every graph P ∈ has a decomposition into factors P₀, P₁, P₂ (indexed by elements of the cyclic group Q = 0,1,2) such that every factor $P_q$ consists of two induced paths of the same length M(q), and K(q) - 1 induced cycles of the same length 2M(q). For q ∈ Q, we define an integer S⁺(q) such that the vector (K(q),M(q),S⁺(q)) determines the graph P (if P is...

We shortly introduce non-archimedean valued fields and discuss the difficulties in the corresponding theory of analytic functions. We motivate the need of $p$-adic cohomology with the Weil Conjectures. We review the two most popular approaches to $p$-adic analytic varieties, namely rigid and Berkovich analytic geometries. We discuss the action of Frobenius in rigid cohomology as similar to the classical action of covering transformations. When rigid cohomology is parametrized by twisting characters,...

C. J. Smyth was among the first to study the spectrum of the Weil height in the field of all totally real numbers, establishing both lower and upper bounds for the limit infimum of the height of all totally real integers, and determining isolated values of the height. Later, Bombieri and Zannier established similar results for totally p-adic numbers and, inspired by work of Ullmo and Zhang, termed this the Bogomolov property. In this paper, we use results on equidistribution of points of low height...

In this paper we demonstrate the relation between uniform distribution modulo 1 of the sequence cpα, p prime, and the zero free regions of the Riemann zeta function.

On montre dans cet article que le théorème d’équidistribution de Szpiro-Ullmo-Zhang concernant les suites de petits points sur les variétés abéliennes s’étend au cas des suites de sous-variétés. On donne également une version quantitative de ce résultat.

We give an introduction to adelic mixing and its applications for mathematicians knowing about the mixing of the geodesic flow on hyperbolic surfaces. We focus on the example of the Hecke trees in the modular surface.

Let $X$ be the quotient group of the $S$-adele ring of an algebraic number field by the discrete group of $S$-integers. Given a probability measure $\mu$ on $X^d$ and an endomorphism $T$ of $X^d$, we consider the relation between uniform distribution of the sequence $T^n𝐱$ for $\mu$-almost all $𝐱\in X^d$ and the behavior of $\mu$ relative to the translations by some rational subgroups of $X^d$. The main result of this note is an extension of the corresponding result for the $d$-dimensional torus ${𝕋}^d$ due to B. Host.

We prove a microlocal version of the equidistribution theorem for Wigner distributions associated to cusp forms on ${\mathrm{PSL}}_2\left(\mathbf{Z}\right)\setminus {\mathrm{PSL}}_2\left(\mathbf{R}\right)$. This generalizes a recent result of W. Luo and P. Sarnak who prove equidistribution on ${\mathrm{PSL}}_2\left(\mathbf{Z}\right)\setminus \mathbf{H}$.