TITLE: Are the ideles literally a Picard group?
QUESTION [10 upvotes]: 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 $\mathcal M(\mathbb Z)$.

REPLY [3 votes]: It seems it would be helpful to discuss idealic/adelic view on line/vector bundles, which is quite simple and intuitive. At the end we might arrive at a kind of asnwer to the question. 
Step 1:
 Bundles are described by transition functions from one chart of the covering to another - in case of line bundles we have invertible elements living on intersection , in vector bundles case - $GL(O_{intersection})$. (Looking forward idels are exactly these transition functions for specific covering). 
Do not forget about factorization $GL(O_{U1})$ \ $GL(O_{intersection})$ / $GL(O_{U2})$.
Step 2: Consider 1d-scheme - i.e. curve (over ANY(!) field ) or Spec(Z).
Consider quite specific covering - we take all geometric points $x$ of the curve and "general point" (i.e. Spec(FractionField)). 
And consider "infinitesemal neighbourhood" of every point $U_x$ - it is $Spec( K_x [[t]] )$ . Fact 1. Infinitesemal neighbourhoods of different points do not intersect. Fact 2. Intersection of $U_x$ and "generic point" is 
$Spec( K_x ((t))  )$. 
Comibining step 1 and 2 we arrive to idelic/adelic picture of bundles:
Line bundles =  K(x) \ $ \prod_x  K_x ((t))^*$ / $\prod_x  K_x [[t]]^*$
You can recognize idels at the middle: $ \prod_x  K_x ((t))^*$ 
Vector bundles =  GL( FractionField) \ $ \prod_x  GL( K_x ((t)) ) $ / $\prod_x GL( K_x [[t]] )$
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I am not sure other steps are necessary, but let me discuss.
If I undestand you question correctly you want to exclude factorization over 
/ $\prod_x  K_x [[t]]^*$ from the factor above - just to have idels.
That can be done.  
Claim:   K(x) \ $ \prod_x  K_x ((t))^*$  describes line bundles over geometiric object - each point $x$ is glued to all points in its infinitesemal neighbourhood.
Step 3: Gluing points A,B means we consider functions: f(A)=f(B).
Now take $A = B+ \epsilon$ for $\epsilon^n=0$. If we consider f(A)=f(B) we glue A to its n-neighbourhood. The function field will be $f^{(k)}(A)=0$ for all k <= n,
that means series of the form $C+ c_{n+1}t^{n+1}+c_{n+2}t^{n+2}$. 
Taking limit n to infinity we kill all the functions.
However the fraction field does not change - i.e. intersection with generic point is still the same. 
In that way idels are preserved but the factor is $\prod_x  K_x [[t]]^*$ is killed.
PS
Curves with $f^{(k)}(A)=0$ considered in Serre book "Algebraic Groups and Class Fields", despite being singular, one can consider their Jacobians
and it is related to ramified covering in the same way as usual Jacobion related to  unramified coverings.