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

Abstract

For an arithmetic surface $X\to B=\mathrm{Spec}{O}_K$ the Deligne pairing $⟨,⟩:\mathrm{Pic}(X)\times \mathrm{Pic}(X)\to \mathrm{Pic}(B)$ 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 ${⟨,⟩}_i:\ker (d_{\times }^1)\times \ker (d_{\times }^1)\to \mathrm{Pic}(B)$, 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.

Some typos have been corrected

References in corpus (2)

• An introduction to higher dimensional local fields and adeles
• An explicit approach to residues on and dualizing sheaves of arithmetic surfaces