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

As we said earlier, Arakelov geometry is a

staticgeneralization 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 anadelicvariant 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 anadelic geometryis 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?

Small points and adelic metricsfrom 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/… . $\endgroup$ – Vesselin Dimitrov Oct 29 '17 at 0:18