# A Universal Elliptic Curve

I'm working through Deligne's "Formes modulaires et representations l-adiques" paper and I find one of his constructions particularily ambigious. I'm hoping someone can give me a bit of clarification so I can work out a careful proof of the proposition that follows it (2.2.i - below) for myself. (The paper is available in English at http://abel.math.harvard.edu/~jay/writings/deligne-l-adic.pdf , page 3)

Easy formalism: Let $e_1, e_2$ and $1, i$ be ordered bases for the $\mathbb{R}$-vectorspaces $\mathbb{R}^2$, and $\mathbb{C}$ respectively. Let $X = \textrm{Hom}^+(\mathbb{R}^2, \mathbb{C}$) denote the set of $\mathbb{R}$-vectorspace isomorphisms that preserve the given orientations (defined by $e_1\wedge e_2 > 0$ and $1\wedge i > 0$). $X$ is naturally given a complex structure.

Deligne says that on this space one arranges a universal exact sequence:

$$0 \rightarrow \mathbb{Z}^2_X \rightarrow \mathbb{G}_a \rightarrow E_0 \rightarrow 0$$

Here, the first term is the constant sheaf with stalk $\mathbb{Z}^2$ on $X$. Although Deligne doesn't explicitly state what $E_0$ is in this sequence, two lines later (Prop 2.2.i) he uses $E_0$ to denote an elliptic curve over $X$, and says that the pair $(X, E_0)$ satisfies a particular universal property (classifying elliptic curves over analytic spaces with some additional data - see the end of post).

$\textbf{Questions:}$

1.) In particular, I'm not sure how to interpret $\mathbb{G}_a$ in this context (all he says is that this is the additive group). What is it? Is it a sheaf for us? The trivial vector bundle on $X$ with fiber $\mathbb{C}$?

2.) What is this an exact sequence of? Perhaps sheaves of abelian groups? In which case, is there a way we interpret the last term $E_0$ as a sheaf of analytic sections of an elliptic curve over $X$ or something? Somehow $E_0$, whatever it is, has to naturally correspond to an elliptic curve over $X$.

3.) What is the map $\mathbb{Z}^2_X \rightarrow \mathbb{G}_a$?

Prop 2.2.(i) The functor which associates to each analytic space $S$ the set of isomorphism classes of elliptic curves $f: E \rightarrow S$ with identity section denoted $e$, equipped with

(A) isomorphisms $e^*(\Omega_{E/S}) \cong \mathbb{G}_a$ (B) isomorphisms $R^1f_*\mathbb{Z}_E \cong \mathbb{Z}^2$ that induce the map $1$ on the second exterior powers

is representable by the analytic space $X$ equipped with a universal elliptic curve $E_0$

Every point of $X$ gives you an embedding of $\mathbb Z^2$ as a lattice in $\mathbb C$ (via the canonical inclusion $\mathbb Z^2 \subset \mathbb R^2$), and hence by quotienting an elliptic curve. This produces a universal diagram $$X \times \mathbb Z^2 \to X \times \mathbb C \to E,$$ where $E$ is a family of elliptic curves over $X$. The analytic sections of this diagram over $X$ give you Deligne's short exact sequence, you can think of it just as a sequence of sheaves of abelian groups.