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?

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    $\begingroup$ 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/… . $\endgroup$ – Vesselin Dimitrov Oct 29 '17 at 0:18

This may be more suitable as a long comment. I remember someone asking Soulé current open problems in Arakelov theory during a walk at the summer school (2017) in Grenoble. The adelic intersection theory is one of the topics he mentioned. The other topic is the questions left out in Arakelov's ICM talk. He said many of these are still open problems.

However, I am not familiar with the recent literature on this topic (to be specific, work by Yuan and Zhang, ACL, Gubler, etc). Berkovich spaces showed up a lot in recent literature so that we may work with archimedean and non-archimedean spaces in the same setup. Since the main difficulty faced in Arakelov theory is from the places at infinity, I do not know what is the essentially new contribution to the theory from the adelic point of view. But I am also quite ignorant.

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.


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