Rational points on smooth projective curves of genus $g \ge 2$ over number fields are in finite number thanks to a theorem of Faltings from 1983. The same result was known over function fields of positive characteristic since 1966 thanks to a theorem of Samuel. The aim of the talk is to give a bound as uniform as possible on this number for curves defined over such fields. In a first part we will report on a result by Rémond concerning the number field case and on a way to strengthen it assuming a height conjecture. During the second part we will focus on function fields of positive characteristic and describe a new result obtained in a joined work with Pacheco.[-]

We will present some of the original definitions, results, and proof techniques about Pfaffian functions on the reals by Khovanskii.
A simple example of a Pfaffian function is an analytic function $f$ in one variable $x$ satisfying a differential equation $f^\prime = P(x,f)$ where $P$ is a polynomial in two variables. Khovanskii gives a notion of complexity of Pfaffian functions which in the example is just the degree of $P$. Using this complexity, he proves analogues of Bézout's theorem for Pfaffian curves (say, zero loci of Pfaffian functions in two variables), with explicit upper bounds in terms of the ocurring complexities.
We explain a recent application by J. Pila and others to a low-dimensional case of Wilkie's conjecture on rational points of bounded height on restricted Pfaffian curves. The result says that the number of rational points of height bounded by $T$, on a transcendental restricted Pfaffian curve, grows at most as a power of log$(T)$ as $T$ grows. This improves the typical upper bound $T^\epsilon$ in Pila-Wilkie's results in general o-minimal structures, the improvement being due to extra geometric Bézout-like control.
In the non-archimedean setting, I will explain analogues of some of these results and techniques, most of which are (emerging) work in progress with L. Lipshitz, F. Martin and A. Smeets. Some ideas in this case come from work by Denef and Lipshitz on variants of Artin approximation in the context of power series solution.[-]

Given two algebraic curves $X$, $Y$ over a finite field we might want to know if there is a rational map from $Y$ to $X$. This has been looked at from a number of perspectives and we will look at it from the point of view of diophantine geometry by viewing the set of maps as $X(K)$ where $K$ is the function field of $Y$. We will review some of the known obstructions to the existence of rational points on curves over global fields, apply them to this situation and present some results and conjectures that arise.[-]

The Green–Griffiths–Lang–Vojta conjectures relate the hyperbolicity of an algebraic variety to the finiteness of sets of “rational points”. For instance, it suggests a striking answer to the fundamental question “Why do some polynomial equations with integer coefficients have only finitely many solutions in the integers?”. Namely, if the zeroes of such a system define a hyperbolic variety, then this system should have only finitely many integer solutions.
In this talk I will explain how to verify some of the algebraic, analytic, and arithmetic predictions this conjecture makes. I will present results that are joint work with Ljudmila Kamenova.[-]

Curves over finite fields of large genus with many rational points have been of interest for both theoretical reasons and for applications. In the past, various methods have been employed for the construction of such curves. One such method is by means of explicit recursive equations and will be the emphasis of this talk.The first explicit examples were found by Garcia–Stichtenoth over quadratic finite fields in 1995. Afterwards followed the discovery of good towers over cubic finite fields and finally all nonprime finite fields in 2013 (B.–Beelen–Garcia–Stichtenoth). The recursive nature of these towers makes them very special and in fact all good examples have been shown to have a modular interpretation of some sort. The questions of finding good recursive towers over prime fields resisted all attempts for several decades and lead to the common belief that such towers might not exist. In this talk I will try to give an overview of the landscape of explicit recursive towers and present a recently discovered tower over all finite fields including prime fields, except $F_{2}$ and $F_{3}$.
This is joint work with Christophe Ritzenthaler.[-]