We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some analogues over number fields.[-]

We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some analogues over number fields.[-]

We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some analogues over number fields.[-]

We will explain some results about the arithmetic structure of algebraic points over a variety defined over a function fields in one variable. In particular we will introduce the weak and strong Vojta conjectures and explain some consequences of them. We will expose some recent developments on the subject : Curves, Varieties with ample cotangent bundle, curves in positive characteirstic, hypersurfaces.... If there is time we will explain some analogues over number fields.[-]

Any algebraic (resp. linear) relation over the field of rational functions with algebraic coefficients between given analytic functions leads by specialization to algebraic (resp. linear) relations over the field of algebraic numbers between the values of these functions. Number theorists have long been interested in proving results going in the other direction. Though the converse result is known to be false in general, Mahler's method provides one of the few known instances where it essentially holds true. After the works of Nishioka, and more recently of Philippon, Faverjon and the speaker, the theory of Mahler functions in one variable is now rather well understood. In contrast, and despite the contributions of Mahler, Loxton and van der Poorten, Kubota, Masser, and Nishioka among others, the theory of Mahler functions in several variables remains much less developed. In this talk, I will discuss recent progresses concerning the case of regular singular systems, as well as possible applications of this theory. This is a joint work with Colin Faverjon.[-]

As is well known, simultaneous rational approximations to the values of smooth functions of real variables involve counting and/or understanding the distribution of rational points lying near the manifold parameterised by these functions. I will discuss recent results in this area regarding lower bounds for the Hausdorff dimension of $\tau$-approximable values, where $\tau\geq \geq 1/n$ is the exponent of approximations. In particular, I will describe a very recent development for non-degenerate maps as well as a recently introduced simple technique based on the so-called Mass Transference Principle that surprisingly requires no conditions on the functions except them being $C^2$.[-]

It goes back to Lagrange that a real quadratic irrational has always a periodic continued fraction. Starting from decades ago, several authors proposed different definitions of a $p$-adic continued fraction, and the definition depends on the chosen system of residues mod $p$. It turns out that the theory of p-adic continued fractions has many differences with respect to the real case; in particular, no analogue of Lagranges theorem holds, and the problem of deciding whether the continued fraction is periodic or not seemed to be not known until now. In recent work with F. Veneziano and U. Zannier we investigated the expansion of quadratic irrationals, for the $p$-adic continued fractions introduced by Ruban, giving an effective criterion to establish the possible periodicity of the expansion. This criterion, somewhat surprisingly, depends on the ‘real' value of the $p$-adic continued fraction.[-]

We will present work in progress, joint with Hexi Ye, towards a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds for common torsion points of nonisomorphic elliptic curves. We introduce a general approach towards uniform unlikely intersection bounds based on an adelic height pairing, and discuss the utilization of this approach for uniform bounds on common preperiodic points of dynamical systems, including torsion points of elliptic curves.[-]