Basic Arakelov geometry and geometry of numbers

François Charles

In these lectures, we will describe some lattices of infinite rank that occur naturally as spaces of sections of coherent sheaves in arithmetic geometry. We will explain how to attach cohomological invariants to these, and how these invariants allow us to study the geometry of numbers in infinite rank. We will introduce the notion of A-schemes, which are an Arakelov analogue of schemes. We will explain how the geometry of numbers above allows us to study coherent sheaves on A-schemes, and will deduce various results on positivity in arithmetic geometry.