We show that the ring of Siegel-Jacobi forms of bounded ratio between weight and index is not finitely generated. Our main tool is the theory of toroidal b-divisors and their relation to convex geometry. As a byproduct of our methods, we prove a conjecture of Kramer about the representation of all Siegel-Jacobi forms as sections of certain line bundles and we recover a formula due to Tai for the asymptotic dimension of the space of Siegel-Jacobi forms of given ratio between weight and index. This is joint work with José Burgos Gil, David Holmes and Robin de Jong.[-]

In this talk we review results on several types of harmonic weak Maass forms that are related to integral even weight newforms. We start with a brief introduction to the theory of harmonic weak Maass forms. These can be related to classical modular forms via a certain differential operator, the so-called $\chi $-operator. Starting with an integral weight newform, we will review different constructions of integral weight harmonic weak Maass forms via (generalized) Weierstrass zeta functions that map to the newform under the $\chi $-operator. A second construction via theta liftings gives a half-integral weight harmonic weak Maass form whose coefficients are given by periods of certain meromorphic modular forms with algebraic coefficients and periods of the integer even weight newform. This is joint work with Jens Funke, Michael Mertens, and Eugenia Rosu resp. Jan Bruinier and Markus Schwagenscheidt.[-]

Berkovich spaces over $\mathbb{Z}$ may be seen as fibrations containing complex analytic spaces as well as $p$-adic analytic spaces, for every prime number $p$. We will give an introduction to those spaces and explain how they may be used in an arithmetic context to prove height inequalities. As an application, following a strategy by DeMarco-Krieger-Ye, we will give a proof of a conjecture of Bogomolov-Fu-Tschinkel on uniform bounds on the number of common images on P1 of torsion points of two elliptic curves.[-]

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.[-]

A pair consisting of a K3 surface and a non-symplectic automorphism of order three is called an Eisenstein K3 surface. We introduce an invariant of Eisenstein K3 surfaces, which we obtain using the equivariant analytic torsion of an Eisenstein K3 surface and the analytic torsion of its fixed locus. Then this invariant gives rise to a function on the moduli space of Eisenstein K3 surfaces, which consists of 24 connected components and each of which is a complex ball quotient depending on the topological type of the automorphism of order three. Our main result is that, for each topological type, the invariant is expressed as the product of the Petersson norms of two kinds of automorphic forms, one is an automorphic form on the complex ball and the other is a Siegel modular form. In many cases, the automorphic form on the complex ball obtained in this way is a so-called reflective modular form. In some cases, this automorphic form is obtained as the restriction of an explicit Borcherds product to the complex ball. This is a joint work with Shu Kawaguchi.[-]