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.