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

Abstract

For an arithmetic surface $X\to B=Spec{O}_K$, the Deligne pairing $⟨,⟩:Pic(X)\times Pic(X)\to Pic(B)$ 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 ${⟨,⟩}_i:ker(d_{\times }^1)\times ker(d_{\times }^1)\to Pic(B)$, where $ker(d_{\times }^1)$ is an important subspace of the two-dimensional idelic group ${\mathbf{A}}_X^{\times }$. On the other hand, the argument for the adelic interpretation is entirely cohomological.