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.



Arakelov divisor on $\operatorname{Spec } O_F$: places or embeddings?
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 $2$.


