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!