Roots of unity of order dividing $n$ equidistribute around the unit circle as $n$ tends to infinity. With some extraeffort the same can be shown when restricting to roots of unity of exact order $n$. Equidistribution is measured by comparing the average of a continuous test function evaluated at these roots of unity with the integral over the complex unit circle. Baker, Ih, and Rumely extended this to test function with logarithmic singularities of the form $\log|P|$ where $P$ is a univariate polynomial in algebraic coefficients. I will discuss joint work with Vesselin Dimitrov where we allow $P$ to come from a class of a multivariate polynomials, extending a result of Lind, Schmidt, and Verbitskiy. Our method draws from earlier work of Duke.[-]

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.[-]

The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows him to deduce his finiteness statements. In this work we prove a new Northcott property for the Faltings height. Namely we show, assuming the Colmez Conjecture and the Artin Conjecture, that there are finitely many CM abelian varieties of a fixed dimension which have bounded Faltings height. The technique developed uses new tools from integral p-adic Hodge theory to study the variation of Faltings height within an isogeny class of CM abelian varieties. In special cases, we are able to use these techniques to moreover develop new Colmez-type formulas for the Faltings height.[-]

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.[-]

I will explain some new connections between the $abc$ conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the $abc$ conjecture, which lies beyond the existing techniques in this context. The proof involves a number of tools such as Shimura curves, CM points, analytic number theory, and Arakelov geometry. It also requires some intermediate results of independent interest, such as bounds for the Manin constant beyond the semi-stable case. If time permits, I will also explain some results towards Szpiro's conjecture over totally real number fields which are compatible with the discriminant term appearing in Vojta's conjecture for algebraic points of bounded degree.[-]

In our talk we will give a panorama of José Burgos' contributions to various generalizations of the classical arithmetic intersection theory developed by Gillet and Soulé. It starts with the extension of Arakelov geometry allowing to incorporate logarithmically singular metrics with applications to Shimura varieties. Further generalizations include toric varieties as well as the most recent results about arithmetic intersections of arithmetic b-divisors with applications to mixed Shimura varieties including the theory of Siegel-Jacobi forms.[-]

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.[-]