ICTS

The famous Bogomolov conjecture states that, given a curve C of genus bigger or equal than 2 and defined over a number field, then there exists a number ϵ > 0 such that the set of algebraic points of C whose Néron–Tate height is smaller that ε is finite. This conjecture was proved by Ullmo and Zhang at the end of last century, based on a pioneering work of Szpiro, Ullmo and Zhang on equidistribution of points of small height.

Arakelov theory brings geometric intuition to arithmetic and is a natural framework where to study heights and many questions related to equidistribution.

In this course we will give an introduction to heights using Arakelov theory, we will discuss classical equidistribution results and their applications, putting on equal foot the arithmetic and the geometric setting. Finally we will explain the recent developments on heights on quasi-projective varieties due to Yuan and Zhang and their application to algebraic dynamics.

We discuss holomorphic forms on  the moduli space M_g of smooth projective complex curves of genus g and on its unramified coverings. Under some hypothesis on the monodromy, we prove the vanishing of holomorphic 1-forms on the preimage of the smooth locus of M_g . This applies to several moduli spaces, as the moduli space of curves with level structures, of spin curves and of Prym curves. It is a joint work with F.Favale ,J.C. Naranjo and S. Torelli.

The famous Bogomolov conjecture states that, given a curve C of genus bigger or equal than 2 and defined over a number field, then there exists a number ϵ > 0 such that the set of algebraic points of C whose Néron–Tate height is smaller that ε is finite. This conjecture was proved by Ullmo and Zhang at the end of last century, based on a pioneering work of Szpiro, Ullmo and Zhang on equidistribution of points of small height.

Arakelov theory brings geometric intuition to arithmetic and is a natural framework where to study heights and many questions related to equidistribution.

In this course we will give an introduction to heights using Arakelov theory, we will discuss classical equidistribution results and their applications, putting on equal foot the arithmetic and the geometric setting. Finally we will explain the recent developments on heights on quasi-projective varieties due to Yuan and Zhang and their application to algebraic dynamics.

We will explain the strategy of two recent articles which construct sl(2,C)-holomorphic differential systems on Riemann surfaces with Fuchsian monodromy and other with cocompact monodromy in SL(2,C). As a consequence, this construction produces holomorphic maps from compact Riemann surfaces of genus g >1 into compact quotients of SL(2,C) which does not factor through elliptic curves. This answers positively a question raised by Huckleberry and Winkelmann and also by Ghys. This work is joint with Indranil Biswas, Lynn Heller and Sebastian Heller.

The famous Bogomolov conjecture states that, given a curve C of genus bigger or equal than 2 and defined over a number field, then there exists a number ϵ > 0 such that the set of algebraic points of C whose Néron–Tate height is smaller that ε is finite. This conjecture was proved by Ullmo and Zhang at the end of last century, based on a pioneering work of Szpiro, Ullmo and Zhang on equidistribution of points of small height.

Arakelov theory brings geometric intuition to arithmetic and is a natural framework where to study heights and many questions related to equidistribution.

In this course we will give an introduction to heights using Arakelov theory, we will discuss classical equidistribution results and their applications, putting on equal foot the arithmetic and the geometric setting. Finally we will explain the recent developments on heights on quasi-projective varieties due to Yuan and Zhang and their application to algebraic dynamics.

As for any symmetric space, the tangent space to Siegel's upper-half space is endowed with an operation B, coming from the Lie bracket on the Lie algebra. The object of the seminar is the pull-back of this operation to the moduli space of curves via the Torelli map j: M_g—>A_g. We first motivate how this is related to studying the geometry of M_g inside A_g. Thus, we recall the definition (due to Kobayashi) and some properties of the Bergman kernel associated with an algebraic curve. Finally, we characterize j*B, in a moduli point [C], in terms of the geometry of the curve, using the Bergman kernel.

The famous Bogomolov conjecture states that, given a curve C of genus bigger or equal than 2 and defined over a number field, then there exists a number ϵ > 0 such that the set of algebraic points of C whose Néron–Tate height is smaller that ε is finite. This conjecture was proved by Ullmo and Zhang at the end of last century, based on a pioneering work of Szpiro, Ullmo and Zhang on equidistribution of points of small height.

Arakelov theory brings geometric intuition to arithmetic and is a natural framework where to study heights and many questions related to equidistribution.

In this course we will give an introduction to heights using Arakelov theory, we will discuss classical equidistribution results and their applications, putting on equal foot the arithmetic and the geometric setting. Finally we will explain the recent developments on heights on quasi-projective varieties due to Yuan and Zhang and their application to algebraic dynamics.