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The notions of $H^0(\widehat{ D})$ and $h^0(\widehat{D})$ are not satisfactory
Let $K$ be a number field with ring of integers $O_K$. Moreover consider an Arakelov divisor $\widehat{D}\in\overline{\operatorname{Div }(\operatorname {Spec }O_K)}$, namely
$$D=\sum_{\mathfrak p\;\...
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