We study series of the form ${\displaystyle \sum _M|{Aut}_R{\left(M\right)|}^{-1}{\left|M\right|}^{-u}}$, where $R$ is a commutative local ring, $u$ is a non-negative integer, and the summation extends over all finite $R$-modules $M$, up to isomorphism. This problem is motivated by Cohen-Lenstra heuristics on class groups of number fields, where sums of this kind occur. If $R$ has additional properties, we will relate the above sum to a limit of zeta functions of the free modules $R^n$, where these zeta functions count $R$-submodules of finite index in $R^n$. In particular we will show that...

The aim of these notes is to generalize Laumon’s construction [20] of automorphic sheaves corresponding to local systems on a smooth, projective curve $C$ to the case of local systems with indecomposable unipotent ramification at a finite set of points. To this end we need an extension of the notion of parabolic structure on vector bundles to coherent sheaves. Once we have defined this, a lot of arguments from the article “ On the geometric Langlands conjecture” by Frenkel, Gaitsgory and Vilonen [11]...

Let $p\ne 2$ be a prime and let $G$ be a $p$-group of matrices in ${\mathrm{SL}}_n\left(\mathbb{Z}\right)$, for some integer $n$. In this paper we show that, when $n\<3(p-1)$, a certain subgroup of the cohomology group $H^1(G,{𝔽}_p^n)$ is trivial. We also show that this statement can be false when $n\ge 3(p-1)$. Together with a result of Dvornicich and Zannier (see [2]), we obtain that any algebraic torus of dimension $n\<3(p-1)$ enjoys a local-global principle on divisibility by $p$.

Le but de cet article est de proposer une nouvelle méthode pour des études dans le cadre de la théorie des “dessins d’enfants” de A. Grothendieck de certaines questions concernant l’action du groupe de Galois absolu sur l’ensemble des arbres planaires.On définit l’application qui associe à chaque arbre planaire à $n$ arêtes, une courbe hyperelliptique avec un point de $n$-division. Cette construction permet d’établir un lien entre la théorie de la torsion des courbes hyperelliptiques et celle des “dessins...

Soit $\theta$ un nombre réel, avec $\theta \>1,$ et soit $A_{\left[\theta \right]}$ l’ensemble des nombres $P\left(\theta \right)$ pour $P$ décrivant les polynômes à coefficients dans $\{0,1,...,[\theta \left]\right\}.$ En utilisant des résultats d’Yves Meyer sur les ensembles harmonieux, on montre que $\theta$ est un nombre de Pisot si et seulement si l’ensemble $A_{\left[\theta \right]}\cup (-A_{\left[\theta \right]})$ est un ensemble de Meyer, et on déduit quelques résultats déjà prouvés par Y. Bugeaud ou P. Erdös et V. Komornik, sur le spectre des nombres de Pisot. Les mêmes outils permettent aussi de montrer que pour $\epsilon \in ]0,1],$ les $\epsilon$-nombres de Pisot appartenant...

Let $L(\theta ,\lambda )$ be the set of limit points of the fractional parts $\left\{\lambda {\theta }^n\right\}$, $n=0,1,2,\cdots$, where $\theta$ is a Pisot number and $\lambda \in \mathbb{Q}\left(\theta \right)$. Using a description of $L(\theta ,\lambda )$, due to Dubickas, we show that there is a sequence ${\left({\lambda }_n\right)}_{n\ge 0}$ of elements of $\mathbb{Q}\left(\theta \right)$ such that $Card\left(L(\theta ,{\lambda }_n)\right)<Card\left(L(\theta ,{\lambda }_{n+1})\right)$, $\forall$$n\ge 0$. Also, we prove that the...

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ℤ of the rational integers such that ℤ[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus...