Does exist a "product formula" for arithmetic surfaces?

Question

Let $K$ be a number field, then the Arakelov geometry of $\operatorname{Spec }O_K$ can be interpreted by means of the adelic ring $\mathbb A_K$. In particular, a key ingredient is the product formula which says that $\prod |a|_v=1$ for any $a\in K^\ast$ and any valuation $v$ attached to the compactification $\widehat{\operatorname{Spec }O_K}$. In particular, the product formula is a very interesting relation between the archimedian e non-archimedian data of $\operatorname{Spec }O_K$.

Now consider a regular arithmetic surface $X\to \operatorname{Spec }O_K$. Here the Arakelov geometry of $\widehat X$ is "more artificial", ideed archimedian fibers and non-archimedian fibers are treated in a complete different way.

Do we have an analogue concept of a "product formula" for $\widehat X$? In other words, is there any "equation" which relates archimedian and non-archimedian data?

Answer 1

Yes. This is the whole purpose of Arakelov geometry: To treat arithmetic schemes $X/\mathcal{O}_K$ + their archimedean bits ``at infinity'' as a proper (compact) arithmetic object on which one can do intersection theory. Without the compactification, i.e., adjoint the bits at infinity, the information you would want from intersection theory could `leak out at infinity'. For example, consider a proper variety over a nice field $k$. If you pluck points, divisors, etc., out of it then the intersection theory won't do anything nice for you---important parts of the intersections could be taking place at the missing parts. This is the idea of Arakelov geometry: those missing parts are the archimedean bits. We glue those in to get a meaningful intersection theory which ties all of the archimedean and non-archimedean information together.