**Introduction:**

Let $M$ be a Riemann surface, then a Green function on $M$ is an element $g\in C^\infty(V)$ where $V=M\setminus\{x_1,\ldots,x_r\}$ and around each point $p\in M$ we have:

$$g=a\log\left|\phi\right|^2+u$$ where $\phi$ is a complex chart centered in $p$, $u$ is a $C^\infty$ function and $a\in \mathbb R$ is independent from the chosen chart. We denote $\operatorname{ord}_p(g):=a$.

The Green functions on $M$ form a vector space $G(M)$and we have a group homomorphism $$\operatorname{div^G}:G(M)\to\operatorname{Div}(M)\otimes_\mathbb Z\mathbb R$$ $$g\mapsto\sum_p\operatorname{ord}_p(g)\{p\}$$

So each Green function defines a real divisor on $M$. The components of $\operatorname{div^G}(g)$ are exactly the points where $g$ is not defined. Now let $K$ be a number field with ring of integers $O_K$ and let $\pi:X\to\operatorname{Spec} O_K$ be a a projective, regular arithmetic surface. For every field embedding $\sigma:K\to \mathbb C$ we obtain a Riemann Surface $X_\sigma:X\times_\sigma\operatorname {Spec}\mathbb C$, and for every divisor $D\subset X$, $D_\sigma$ is the pullback divisor of $D$ through the canonical map $X_\sigma\to X$.

**Question:**

The group of arithmetic divisors of $X$ can be defined in the following way: $$\widehat{\operatorname{Div}(X)}:=\left\{\left(D,\sum_\sigma g_\sigma\sigma\right)\in \operatorname{Div(X)\times\bigoplus_\sigma G(X_\sigma)\sigma}\,:\, \operatorname{div^G}(g_\sigma)=D_\sigma,\;\forall\sigma\right\}$$

The meaning of this is quite obvious: an element $\left(D,\sum_\sigma g_\sigma\sigma\right)$ can be "physically" seen as the divisor $D$ plus the finite set of points where $g_\sigma$ is not defined for each Riemann surface $X_\sigma$.

I don't understand why the arithmetic divisor $F_\sigma:=(0,2\sigma)\in \widehat{\operatorname{Div}(X)}$ is usually identified with the "fiber at infinity" $X_\sigma$. **In particular what is the meaning of the constant $2$**? How can I see $F_\sigma$ as the Riemann surface $X_\sigma$?