I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, please delete it.
In Deligne's article in Séminaire Bourbaki "Formes modulaires et représentation $\ell$-adiques" (http://archive.numdam.org/ARCHIVE/SB/SB_1968-1969__11_/SB_1968-1969__11__139_0/SB_1968-1969__11__139_0.pdf) some passages were not clear to me.
Let $E \xrightarrow{f} S$ be an elliptic curve over an analytic space $S$ and $T_{\mathbb{Z}} (E) = R^1 f_{*} \mathbb{Z}^{\vee} $
Why the map $\alpha$ in $R^1 f_{*} \mathbb{Z}^{\vee} \xrightarrow{\alpha} \text{Lie}_S (E) \xrightarrow{\text{exp}_E} E$ fits into the exact sequence?
Is somehow $T_\mathbb{Z} (E)$ a Tate module for the archimedean place? If yes, how can the analogy be seen?
Why $R^2 f_{*} \mathbb{Z} \cong \mathbb{Z}^{2}$?
In page 143, why $E_X$ exists as a scheme and not just as a stack?
In page 143, how he got the sequence $\omega_{E/S} \cong f^{*}\Omega^{1}_{E/S} \rightarrow R^{1}f_{*}{\mathbb{R}} \otimes_{\mathbb{R}} \mathscr{O}_X \rightarrow R^{1}f_{*} \mathscr{O}_X$? The expected sequence would be $\omega_{E/S} \rightarrow R^{1}f_{*} \Omega^{\bullet}_{dR, E/S} \rightarrow R^1f_{*} \mathscr{O}_X$. So why would $R^{1}f_{*} \Omega^{\bullet}_{dR, E/S} \cong R^{1}f_{*}{\mathbb{R}} \otimes_{\mathbb{R}} \mathscr{O}_X$?
Furthermore,(as in page 143) how to prove that under a suitable trivialization the map $q$ is just $q(ax + by) = a \frac{f(x)}{f(y)} + b$ where $f$?
Thanks in advance.