标题： 拓扑阿代尔曲线：代数覆盖、数的几何和闭点的高度 显示英文标题

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

摘要： 在本文中，我们介绍了拓扑阿代尔曲线。粗略地说，拓扑阿代尔曲线是一个满足乘积公式的给定域上的（广义）绝对值的拓扑空间。拓扑阿代尔曲线是Chen和Moriwaki引入的阿代尔曲线的拓扑对应物。它们旨在处理可能不可数域上的Arakelov几何，并在Diophantine逼近与Nevanlinna理论之间的类比形式化中提供进一步的想法。利用之前预印本中开发的伪绝对值概念，我们证明了拓扑阿代尔曲线的几个基本性质：代数覆盖、Harder-Narasimhan形式主义、体积函数的存在性。我们还定义了闭点的高度，并在此框架下给出了Nevanlinna第一基本定理的推广。 显示英文摘要

摘要： 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.