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The well known isomorphism: $$Cl(K) \cong Ker\\ \lgroup\\ H^1(G_K, U) \rightarrow \prod_{\mathfrak{p}} H^1(G_{K_{\mathfrak{p}}},U_p) \rgroup$$

is great. ("Visibility of Ideal Classes", Schoof and Washington)

So ideal classes are like locally trivial cocycles. But there usually aren't many ideal classes - so this can't be the whole picture.

What results of similar nature exist for the Arakelov class group? Are divisor classes like certain cocycles on some larger space?

I am guessing the question can also be asked regarding abelian varieties, where the Arakelov divisor class group has an analogue as some kind of extension of the Shafarevich-Tate group by the Mordell-Weil group of the variety (over $K$). If so, then I am asking that question as well!

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Sorry but what is $U$ here? –  Daniel Loughran Oct 10 '10 at 22:56
    
Algebraic units (in a closure). –  Dror Speiser Oct 10 '10 at 22:59
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