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 ${\mathrm{PSL}}_2\left(\mathbf{Z}\right)\setminus \mathbf{H}$.

Given a rational function $\varphi \left(T\right)$ on ${\mathbb{P}}^1$ of degree at least 2 with coefficients in a number field $k$, we show that for each place $v$ of $k$, there is a unique probability measure ${\mu }_{\varphi ,v}$ on the Berkovich space ${\mathbb{P}}_{\mathrm{Berk},\mathrm{v}}^1/{ℂ}_v$ such that if $\left\{z_n\right\}$ is a sequence of points in ${\mathbb{P}}^1\left(\overline{k}\right)$ whose $\varphi$-canonical heights tend to zero, then the $z_n$’s and their $\mathrm{Gal}(\overline{k}/k)$-conjugates are equidistributed with respect to ${\mu }_{\varphi ,v}$.The proof uses a polynomial lift $F(x,y)=(F_1(x,y),F_2(x,y))$ of $\varphi$ to construct a two-variable Arakelov-Green’s function $g_{\varphi ,v}(x,y)$ for each $v$. The measure ${\mu }_{\varphi ,v}$ is obtained by taking the...