Arakelov divisor on $\operatorname{Spec } O_F$: places or embeddings?

Question

Let $F$ be a number field such that $[F:\mathbb{Q}]=n$ and with ring of integers $O_F$. Let's put $B=\operatorname{Spec } O_F$, then an Arakelov divisor is an element of: $$Div(X)\times \bigoplus_\sigma \mathbb R[\sigma]$$ namely it can be written as $$\bigg(\sum_{\text{$\mathfrak p$ prime $\neq 0$}}n_{\mathfrak p}[\mathfrak p],\sum_{\sigma}x_{\sigma}[\sigma]\bigg)\quad \text{with $x_\sigma\in\mathbb R$, $n_{\mathfrak p}\in\mathbb Z$}$$

Now: some references say that $\sigma$ runs among the archimedean places of $F$ other say that $\sigma$ runs among the complex embeddings of $F$.

There is a substantial difference in the two cases, because in the former the dimension of the vector space of "divisors at infinity" is $r_1+r_2$, whereas in the latter it is $n=r_1+2r_2$ (as usual $r_1$ and $2r_2$ are the number of real and complex embeddings of $F$ ).

Which is the most natural approach?

In the first case the degree of an Arakelov divisor is $\sum_{\mathfrak p} n_{\mathfrak p}\log(\#(O_F/\mathfrak p))+\sum_\sigma x_\sigma$ whereas in the second case I think that we should put $$\sum_{\mathfrak p} n_{\mathfrak p}\log(\#(O_F/\mathfrak p))+\sum_{\text{$\sigma$ real }} x_\sigma+\frac{1}{2}\sum_{\text{$\sigma$ complex }} x_\sigma$$

Is it correct?

Edit: books/papers using places: Van De Geer,Schoof- Effectivity of Arakelov divisors of a number field,Neukirch ANT.

books/papers using embeddings: Moriwaki's book, De Jong thesis.

Moreover I'd like to understand what happens when you consider Arakelov theory on arithmetic surfaces $X\to B$. Here I think that would be meaningful to consider all the embeddings.

Answer 1

You would usually want the principal Arakelov divisors, i.e. those of the form $(\sum_{\mathfrak{p}}{\rm ord}_{\mathfrak{p}}(a), \sum_\sigma -\log|\sigma(a)|)$ for $a\in F^\times$, to be cocompact in the group of degree $0$ divisors — the volume of the quotient should be the familiar product ${\rm Reg}(F)\cdot{\rm h}(F)$, where ${\rm h}$ denotes the class number. For that, you want $\sigma$ to only run over places, not over all embeddings. Incidentally, you also want to normalise your degree function carefully: the degree of a real place should be $1$, while the degree of a complex place should be $2$.

Answer 2

I think the natural definition is the one using archimedean places (or equivalently, all embeddings, with the pairs of complex conjugates considered as a single embedding), and I'm very curious of the context where you have seen the definition using complex embeddings only.

Consider for example the basic result that the kernel of the mapping $\mathrm{div}:K^*\to \mathcal{Div}(\overline{\mathcal{O}})$ is $\mu(K)$.

This follows directly from the exact sequence

$$1\longrightarrow \mu(K) \longrightarrow \mathcal{O}^*\longrightarrow \Gamma\longrightarrow 0$$

and from the equality (up to a sign) of the composition

$$K^*\longrightarrow\mathcal{Div}(\overline{\mathcal{O}}) \longrightarrow \prod_{\mathfrak{p}|\infty}\mathbb{R}$$

and the logarithmic Minkowski map

$$\lambda: f \mapsto (...,\log |f|_\mathfrak{p},...)$$

This shows quite explicitly why the definition uses necessarily all the embeddings.

For reference, this is also consistent with the use in Moriwaki's and Lang's textbooks on Arakelov theory.