I understand that in the number field / function field analogy, the ideles $ \mathbb I_K$ of a number field $ K$ are supposed to be analogous to the Picard group of a function field.
Question: Is this more than an analogy? Is there an actual geometric setting in which the ideles parameterize “line bundles” over a geometric object attached to $ K$ ?
My first inclination was to see if this was the case in Berkovich geometry, in the case $ K = \mathbb Q$ . It’s encouraging to note that ideles correspond to Cech 1-cocycles for reasonable coverings of the Berkovich space $ \mathcal M(\mathbb Z)$ with values in $ \mathcal O_\mathbb Z^\times$ . But nevertheless, every such cocycle is a coboundary, so it seems to not literally be the case that the ideles parameterize line bundles on $ \mathcal M(\mathbb Z)$ .