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

## TL;DR

This paper develops an idelic and adelic framework for interpreting the Deligne pairing on arithmetic surfaces, advancing the understanding of Arakelov intersection theory through new algebraic and cohomological methods.

## Contribution

It introduces the first idelic and adelic interpretations of the Deligne pairing, linking it to idelic groups and cohomology for the first time.

## Key findings

- Deligne pairing lifted to a pairing on a kernel of a differential map in idelic groups
- Idelic interpretation involves a pairing on a subspace of the two-dimensional idelic group
- Adelic interpretation achieved through a purely cohomological approach

## Abstract

For an arithmetic surface $X\to B=\operatorname{Spec} O_K$ the Deligne pairing $\left <\,,\,\right > \colon \operatorname{Pic}(X) \times \operatorname{Pic}(X) \to \operatorname{Pic}(B)$ 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 $\left<\,,\,\right>_i:\ker(d^1_\times)\times \ker(d^1_\times)\to\operatorname{Pic}(B) $, where $\ker(d^1_\times)$ 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.