Adelic geometry on arithmetic surfaces, I: Idelic and adelic interpretation of the Deligne pairing

For an arithmetic surface X → B = SpecOK, the Deligne pairing 〈, 〉 : Pic(X)×Pic(X)→Pic(B) gives the "schematic contribution"to the Arakelov intersection number. We present an idelic and adelic interpretation of the Deligne pairing; this is the first crucial step for a full idelic and adelic interpretation of the Arakelov intersection number. For the idelic approach, we show that the Deligne pairing can be lifted to a pairing 〈, 〉i : ker(d1×)×ker(d1×)→Pic(B), where ker(d1×) is an important subspace of the two-dimensional idelic groupA× X. On the other hand, the argument for the adelic interpretation is entirely cohomological.