Course Description
In a series of lectures I will present a new framework of Arakelov geometry proposed by Atsushi Moriwaki and myself, namely Arakelov geometry over an adelic curve.

In our setting, an adelic curve refers to a field equipped with a family of absolute values (with possible repetitions) parametrised by a measure space. I will begin with an introduction on elementary geometry of numbers and Arakelov geometry of number fields and function fields and then explain why the setting of adelic curves is natural and permits to unify several frameworks in the literature which could be apparently transversal: Arakelov geometry over number fields and function fields, arithmetic geometry over a finitely generated field (Moriwaki), height theory of M-fields (Gubler), R-filtration method (Chen), Siegel fields (Gaudron and Rémond). The construction of arithmetic objects and arithmetic invariants will be discussed, with an emphasis on the geometry of adelic vector bundles and its relationship with the classic geometry of numbers and Arakelov theory.

The lectures will be concluded by a view of further research topics and open problems.