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).


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 $E_0$

http://abel.math.harvard.edu/~jay/writings/deligne-l-adic.pdf (Page 3)


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.