References on "relative Lie groups"

Question

I'm trying to read Deligne's Formes modulaires et représentations $\ell$-adiques. In section 2, he briefly goes over a number of facts about (complex-analytic) elliptic curves in a relative setting. I know very little analysis, so I find myself struggling to understand some basic facts like the following. If $f:E\to S$ is an elliptic curve over the base manifold $S$, then one is supposed to end up with the following "exponential exact sequence": $$ 0 \to \mathsf R^1 f_\ast \mathbb Z^\vee \to \underline{\operatorname{Lie}}_S(E) \to E \to 0 $$ Here $\underline{\operatorname{Lie}}_S(E)$ is the dual to the pullback $\omega=e^\ast \Omega_{E/S}^1$ of the sheaf of $1$-forms on $E$ by the identity section $e:S\to E$. In the case that $S$ is a point, this sequence is just $$ 0 \to \operatorname{H}_1(E,\mathbb Z) \to \Omega^1(E)^\vee \to E \to 0 $$ where the first map comes from Poincare duality and the second is the exponential map. I guess I just don't understand where $\mathsf R^1 f_\ast \mathbb Z^\vee \to \omega^{-1}$ comes from when $S$ is not just a point. Any explanation would be appreciated, and I would also be grateful if someone could point a decent reference for "Lie groups in a relative setting."

Answer 1

It looks like there aren't any readily available references. Section 1.3.4 of Brian Conrad's (hard to find) draft of Modular forms, cohomology and the Ramanujan conjecture gives a very nice proof that there is a canonical way of defining a sequence $$ 0 \to \left(\mathsf R^1 f_\ast \mathbb Z\right)^\vee \to \left(e^\ast \Omega_{A/S}^1\right)^\vee \to A \to 0 $$ whenever $A/S$ is a relative complex torus. However, the construction of this map is a bit involved.