Topological adelic curves: algebraic coverings, geometry of numbers and heights of closed points

Abstract: In this article, we introduce topological adelic curves. Roughly speaking, a topological adelic curve is a topological space of (generalised) absolute values on a given field satisfying a product formula. Topological adelic curves are topological counterparts to adelic curves introduced by Chen and Moriwaki. They aim at handling Arakelov geometry over possibly uncountable fields and give further ideas in the formalisation of the analogy between Diophantine approximation and Nevanlinna theory. Using the notion of pseudo-absolute values developed in a previous preprint, we prove several fundamental properties of topological adelic curves: algebraic coverings, Harder-Narasimhan formalism, existence of volume functions. We also define height of closed points and give a generalisation of Nevanlinna's first main theorem in this framework.