Let $ S = \operatorname{Spec}A $ be an affine scheme, $ f : E \to S $ an elliptic curve and $\mathscr{I}$ the ideal sheaf of the $0$-section.
(This is invertible since the section defines the effective relative Cartier divisor.)  

Assume that $f\_\* \Omega\_{E/S}, f\_\*\mathscr{I}^n$ are free over $\mathscr{O}\_S$.
($n = 1, \cdots , 6$)  

I want to show $\hat{E} \cong \operatorname{Spf} A[[T]]$.
And I don't understand the formal expansion, of a basis $\omega$ of $f\_\* \Omega\_{E/S}$ and a basis of $f\_\*\mathscr{I}^n$, along the $0$-section.


Here is what I have tried:
Since the $0$-section is a regular immersion, for any $x \in S$, there exists affine opens $ x \in V \subseteq S$, $ 0(x) \in U \subseteq E$ s.t. $0(V) \subseteq U$ and the diagram


$$\require{AMScd}
\begin{CD}
S @>{0}>> E \\
@VV{1}V @VV{f}V \\
S @>{1}>> S
\end{CD}$$


corresponds to


$$\require{AMScd}
\begin{CD}
C @<{0}<< B \\
@A{\text{localization by one element}}AA @AAA \\
A @<{1}<< A,
\end{CD}$$


where the kernel $I$ of $B \to C$ is generated by $t \in B$, a regular element.


I showed that $\hat{B}$ (the completion of $B$ along the kernel $I$) $\cong C[[t]]$.
And $\Omega\_{B/A} \otimes\_B \hat{B} = dt \hat{B} = dt C[[t]].$  

That is, I can show $ \hat{E} \cong \operatorname{Spf}A[[t]]$ locally, and I can expand $\omega$ locally.


How can I extend these operation globally?  

I also showed that the isomorphism $\hat{B} \cong C[[t]]$ is compatible with localization.
So I think intuitively that these isomorphisms (at any points) are glued together, and we have $ \hat{E} \cong \operatorname{Spf}A[[t]]$.


Any help will be much appreciated!



Formal completion of an elliptic curve along the $0$ sectioin and the formal expansion of functions
Yes, what you are saying is true, at least over an algebraically closed base $k$ of characteristic $0$. In fact, all you need is that $S$ is affine and that the normal bundle $I/I^2$ is trivial (as a bundle over $S$), which in your case is equivalent to $f\_\*\Omega(E/S)$ being free. The rest follows essentially from deformation theory.


There is a standard way to reduce questions about formal thickenings to questions in deformation theory, and this is called **deformation to the normal cone**. Here's the idea. Let $R = O(\widehat{E})$ be the algebra of function on the formal thickening, viewed as a formal ring over $A$. Then this algebra is filtered by $R\_n : = I^n,$ with associated graded ring $$R\_{gr}:=\bigoplus\_n \frac{I^n}{I^{n-1}}\cong A[I/I^2].$$ 
Since $I/I^2$ is trivial and one-dimensional over $A$, we have $R\_{gr}\cong A[[t]].$ Now the *deformation to the normal cone* is an algebra over $\text{Spf} (k[[s]])$ interpolating between $R$ and $R\_{gr}$, given by the Rees construction $$R\_{\text{rees}} : = R[s^{-1}\cdot I]^{\wedge} = \bigoplus\_n^\wedge s^{-n}I^n,$$ where the completion is taken along (positive) powers of $s$. The key thing to observe here is that the fiber over $s=0$ of $R\_{\text{rees}}$ is $\bigoplus s^{-n}(I^n/I^{n+1})$ (not completed), which is $R\_{gr}$ and the generic fiber over $k((s))$ is isomorphic to $R\otimes k((s))$. 


Now a standard result in the deformation theory of schemes tells us that flat deformations of a smooth scheme $X$ up to isomorphism (as well as of a smooth formal scheme) are classified by (what is noncanonically isomorphic to) a subset of $H^1(X,T)^\infty,$ i.e. infinite sequences of vectors in the first homology of the tangent bundle (deformations of order $n$ would be certain sequences of size $n$). In particular, if $X$ is affine (in your case — $X = \text{Spf}A[[t]],$ the special fiber of the Rees construction), the relevant $H^1$ is trivial, so there is only one deformation — the trivial one. Equivalently, $R\_{\text{rees}}\cong R\_{gr}[[s]]$.


Now you're done! Basechanging to the generic point gives $R\otimes k((s)) \cong A[[t]]\otimes k((s)).$ But since you are over an algebraically closed base of characteristic zero, such an isomorphism implies an isomorphism $R\cong A[[t]]$ over $k$.


