Let K be a number field, X be a smooth projective curve over it and D be a reduced divisor on X. Let (E,∇) be a vector bundle with connection having meromorphic singularities on D. Let $p_1,...,p_s\in X\left(K\right)$ and $X^o:=X̅\setminus D,p_1,...,p_s$ (the $p_j$’s may be in the support of D). Using tools from Nevanlinna theory and formal geometry, we give the definition of E-section of arithmetic type of the vector bundle E with respect to the points $p_j$; this is the natural generalization of the notion of E-function defined in Siegel-Shidlovskiĭ theory. We prove...

Est-il possible d’utiliser les propriétés de la fonction modulaire $j$ pour avancer un peu en direction de la conjecture des quatre exponentielles ? Sur ce thème, le texte propose plusieurs conjectures équivalentes et quelques résultats partiels.

Dans ce texte, nous déterminons explicitement les idéaux premiers différentiellement stables dans l’anneau des formes quasi-modulaires pour ${\mathbf{SL}}_2\left(\mathbb{Z}\right)$. Les techniques introduites permettent de préciser des résultats de Nesterenko dans [5] et [6].

For $x\in ℂ$, $\left|x\right|\<1$, $s\in \mathbb{N}$, let ${\mathrm{Li}}_s\left(x\right)$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha$ such that $\left|\alpha \right|\<1$, the $\mathbb{Q}\left(\alpha \right)$-vector space spanned by $1,{\mathrm{Li}}_1\left(\alpha \right),{\mathrm{Li}}_2\left(\alpha \right),\cdots$ has infinite dimension. This result extends a previous one by Rivoal for rational $\alpha$. The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.

The goal of this article is to associate a $p$-adic analytic function to the Euler constants ${\gamma }_p(a,F)$, study the properties of these functions in the neighborhood of $s=1$ and introduce a $p$-adic analogue of the infinite sum ${\sum }_{n\ge 1}f\left(n\right)/n$ for an algebraic valued, periodic function $f$. After this, we prove the theorem of Baker, Birch and Wirsing in this setup and discuss irrationality results associated to $p$-adic Euler constants generalising the earlier known results in this direction. Finally, we define and prove certain...

Here we characterise, in a complete and explicit way, the relations of algebraic dependence over $\mathbb{Q}$ of complex values of Hecke-Mahler series taken at algebraic points ${\underline{u}}_1,...,{\underline{u}}_m$ of the multiplicative group ${𝔾}_{\mathrm{m}}^2\left(ℂ\right)$, under a technical hypothesis that a certain sub-module of ${𝔾}_{\mathrm{m}}^2\left(ℂ\right)$ generated by the ${\underline{u}}_i$’s has rank one (rank one hypothesis). This is the first part of a work, announced in [Pel1], whose main objective is completely to solve a general problem on the algebraic independence of values of these series.

On étudie certaines propriétés arithmétiques de fonctions analytique $f$ au voisinage de $0,{\sum }_0^{+\infty }{a}_n{x}^n$ où $a_n\in \mathbb{Q}$ et $f$ satisfaisant une équation fonctionnelle de Poincaré.

On sait (Cobham) qu’une suite $2$- et $3$-automatique est une suite rationnelle. Une question de Loxton et van der Poorten étend ce résultat au cas $2$- et $3$-régulier. On montre dans cet article que, si une suite vérifie une récurrence $2$- et $3$-mahlérienne d’ordre un, elle est rationnelle.