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A surface is a two-dimensional topological manifold. The term can also be used to describe a smooth surface, depending on the context.

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Does exist a "product formula" for arithmetic surfaces?

Let $K$ be a number field, then the Arakelov geometry of $\operatorname{Spec }O_K$ can be interpreted by means of the adelic ring $\mathbb A_K$. In particular, a key ingredient is the product formula …