In Soule's Lectures on Arakelov Geometry, he suggests the following "improvement" of Arakelov geometry:

As we said earlier, Arakelov geometry is a static generalization of infinite descent. For instance, when doing intersection theory on $X$ one is not allowed to move the cycles; no analog of Chow's Moving Lemma is known over $\mathbb{Z}$. A more dynamic approach would be an adelic variant of Arakelov geometry. The main object of study in this theory would be a smooth variety $V$ over $\mathbb{Q}$, and vector bundles on $V$ equipped with metrics at archimedean places, and $p$-adic analogs of these at finite places. Such an adelic geometry is still to be built.

What is the status of this adelic geometry? Do subjects like rigid analytic geometry and Berkovich spaces have anything to say about this?

• It is indeed the 'modern viewpoint' now to consider a variety over $\mathbb{Q}$ and adelically metrized vector bundles over it. This first appears in Shou-wu Zhang's Small points and adelic metrics from 1995. Yes, Berkovich spaces do play a role, since the viewpoint becomes more analytic and less geometric: the full analytic space $V_{/\mathbb{C}_p}^{\mathrm{an}}$ is considered in place of the closed fibre at $p$. For dimension one, see Amaury Thuillier's 2005 thesis at Rennes: tel.archives-ouvertes.fr/file/index/docid/48750/filename/… . Oct 29 '17 at 0:18

My naive understanding is that because there is no "natural metric" on the whole adele ring, the classical Arakelov framework becomes much harder to work with. The points are "thickened" and the $p$-adic analysis analog of objects at infinity may not be available (Zhang has a paper Admissible pairing on a curve exploring this, before the invention of tropical geometry!). For one-dimensional Arakelov theory, there is a theory of matrix divisors by Ichiro Miyada. There is also some possibility of extending it to the functional field. But for higher dimensions I do not know any serious attempt to generalize Faltings-Riemann-Roch to the adelic setting.