What does "completing an elliptic curve $E$ along its identity section $\sigma_0$" mean?

Question

From here, I got to know the method of getting formal group/formal group law from Elliptic from. It says:

Given an elliptic curve $E$ over $R$, $E\to\text{Spec}(R)$, we get a formal group $\hat{E}$ by completing $E$ along its identity section $\sigma_0$, $$E \to \text{Spec}(R) \stackrel{\sigma_0}{\to} E \,$$ we get a ringed space $(\hat E, \hat O_{ E,0}) $, then $$ \hat O_{ E\times E,(0,0)}≃ \hat O_{E,0} \hat \otimes_ k \hat O_{E,0}≃k[[x,y]].$$

I have to understand the line, "we get a formal group $\hat{E}$ by completing $E$ along its identity section $\sigma_0$".

I know definition of section, which is a one way inverse of a morphism. But what does mean by identity section ?

Second, what does mean by $\text{completing $E$ along its identity section $\sigma_0$}$ ?

Please explain those two question.

Thanks

Answer 1

Every fiber of the map $E\to \operatorname{Spec} R$ is a genus one curve; marking a point to be the identity element in the group law on the fiber makes it an honest elliptic curve. The identity section is just this choice made in a coherent way across all fibers. Completion of a scheme along a closed subscheme is a standard construction that you can look up in Hartshorne etc.