In this talk I will report on joint work (in progress) with Ronno Das and Sean Howe, where we propose a suitable axiomatic set-up, based on the theory of pre-lambda-rings, in which an "Euler product" operation may be defined. In the context of the Grothendieck ring of varieties this construction recovers the notion of motivic Euler product, which appears in the expression of the limit densities of many natural sequences of moduli spaces. This point of view sheds a new light on some of the properties (and non-properties) of motivic Euler products and on their connection with classical Euler products.

These lectures will discuss recent concrete results in diophantine geometry, such as the holonomy bounds of Calegari-Dimitrov-Tang used in their irrationality breakthrough, or sparsity results for integral points on general affine schemes. We will describe how their proofs use both classical methods from Arakelov geometry, as well as infinite-dimensional objects from geometry of numbers.

Over the past ten or twenty years, a number of works—by Bilu & Browning, Browning & Sawin, Browning & Vishe, Bourqui, Chambert-Loir & Loeser, Glas & Hase-Liu, Peyre, among others—have demonstrated how number theory, particularly analytic number theory, can provide new insights into the study of moduli spaces of curves. Notably, the recent development of motivic versions of number-theoretic tools (harmonic analysis on adèles, the circle method, lifting to universal torsors) has paved the way for a motivic version of Manin’s program: the dictionary between number fields and function fields allows one to move from the fine study of the distribution of rational points on Fano varieties to predictions about the class, in a certain Grothendieck ring of algebraic varieties, of the moduli space of morphisms from a given curve to a Fano variety over the complex numbers. The purpose of this talk is to present the results of a collaboration with Margaret Bilu (CNRS/École Polytechnique), in which we develop a multiplicative version of the motivic Poisson formula. This allows us to demonstrate a motivic stabilisation phenomenon concerning certain moduli spaces of curves on split projective toric variety.

In this course, we introduce Campana’s notion of special varieties, motivated by Campana's conjectures on the potential density of rational and integral points.

Campana's Orbifold Mordell Conjecture can be viewed as a generalization of Faltings’s theorem for curves of genus at least two to Campana’s C-pairs. This conjecture leads naturally to the notions of weakly special varieties and the Weakly Special Conjecture. Some examples of weakly special but non-special varieties will be described, based on joint work with Finn Bartsch, Frédéric Campana, and Olivier Wittenberg. Recent joint work with Finn Bartsch using stacks to help clarify aspects of Campana's theory will also be discussed. If time permits, some results on higher dimensional generalizations of Orbifold Mordell over function fields will be considered, from joint work with Guoquan Gao and Erwan Rousseau.

Determining the image of the set of rational points under a morphism of varieties is a natural and subtle question. Abramovich introduced the notion of firm points in order to study this problem. In this project, in collaboration with L. Herr, M. Pieropan, and T. Poiret, we address the same question and reformulate the notion of firm points in the language of logarithmic geometry. This approach allows us to establish a tropical criterion for lifting rational points along log smooth morphisms.

K3 surfaces that arise as a double cover of the projective plane branched on a sextic curve admit natural fibrations on hyperelliptic curves, namely the pull-backs of linear systems of lines in the plane. In this talk, I will discuss geometric constructions of multisections for such fibrations and their role in the variation of the Mordell-Weil ranks of the associated family of Jacobians. This will be based on joint work with Ander Arriola (U.Groningen).

The study of heights over generalised global fields is a problem that has gained a lot of importance in recent years. For instance, it turns out that a lot of classical results in Diophantine geometry over number fields have a generalisation over arithmetic function fields, namely, finitely generated extensions of Q (cf. Lang, Moriwaki, Vojta).  

More recently, two frameworks aiming at handling heights over more general classes of global fields have been proposed. On the one hand, the adelic curve framework, introduced by Chen and Moriwaki. It stems from the general principle that heights can be constructed from the data of a field together with a set of absolute values satisfying a product formula. In this case, the set of absolute values involved is a measure space, and the product formula is an integral. On the other hand, the \emph{globally valued fields} framework, introduced by Ben Yaacov and Hrushovski. Roughly speaking, it is a model theory of heights. It starts with the abstract data of a height function on a field. It fits in the world of unbounded continuous logic. Although the definition of these two frameworks seems to be "dual" at first glance, these two formalisms are equivalent in the case where the base field is countable. 

In this talk, we will describe the counterpart of this result in the uncountable case. We will first motivate the study of height over uncountable fields by two examples. The first one is related to ultraproducts, a fundamental tool of model theory, which is a very convenient way to handle uniformity questions in Arakelov geometry. The second one is related to the analogy between Diophantine approximation and Nevanlinna theory (cf. Vojta). Although the definition of adelic curves also makes sense over uncountable base fields, it turns out that absolute values are too restrictive in this context. To address this issue we introduce the notion of pseudo-absolute value. We will see that the correct counterpart of adelic curves is roughly speaking a measure on the space of pseudo-absolute values, which is a compact Hausdorff locally ringed space (which we will interpret as the Berkovich analytification of a Zariski-Riemann space), and we will describe the construction of heights in this case. Finally, we will describe how these constructions can be adapted to include the Nevanlinna characteristic functions in our framework, formalising the first part of Vojta's dictionary.

We will discuss the (algebraic) exceptional set appearing in conjectures of Lang and Vojta on the distribution of integral points on log surfaces (S,D), where D has at least three irreducible components. We prove that the exceptional set is finite or empty in most cases. Our results yield an explicit description even in situations where the conjectures are not known to hold, and they produce new examples of Brody hyperbolic varieties. This is joint work with Lucia Caporaso.

I will present a new intersection theory for norms on rings. This purely algebraic theory both generalizes and provides a unified framework for geometric and arithmetic intersection theory of line bundles. I will also discuss possible applications to other areas of mathematics, such as discrete geometry and the theory of modular forms.