Question

http://www.springerlink.com/content/04x54gr171v556m4/fulltext.pdf

On page 149 (DeRa-7), in the middle of the page, I can translate the middle paragraph that starts "3. La surface de Riemann ..." as follows:

3.The Riemann surface $X/\Gamma$ is not compact. Geometrically, this fact is reflected as follows: If $E_\eta$ is an elliptic curve equipped with a level $n$ structure over $\mathbb{C}((T))$, it follows that the minimal model of $E_\eta$ over $\mathbb{C}[[T]]$ has bad reduction. In this case, the special fiber $E_0'$ of the Neron model $E'$ of $E_\eta$ over $\mathbb{C}[[T]]$ is isomorphic to $\mathbb{C}^*\times\mathbb{Z}/kn\mathbb{Z}$ for some suitable $k$. Let $E_0$ be the subgroup of $E_0'$ consisting of the components of $E_0'$ which have order dividing $n$ in $\pi_0(E_0')$. This subgroup is isomorphic to $\mathbb{C}^*\times\mathbb{Z}/n\mathbb{Z}$...

Assuming I've translated it correctly, I've got three questions:

1. The first sentence seems to be saying that any elliptic curve over $\mathbb{C}((T))$ has bad reduction, which (unless the minimal model isn't what I think it is) is obviously not true, since any elliptic curve over $\mathbb{C}$ is also an elliptic curve over $\mathbb{C}((T))$. Where am I going wrong here?

2. Earlier the author defined a level $n$ structure as just an isomorphism from the $n$-torsion of the elliptic curve to $(\mathbb{Z}/n\mathbb{Z})^2$. Thus, any such elliptic curve $E_\eta$ has level $n$ structure for every $n$, which would seem to imply that the special fiber of the Neron model is isomorphic to $\mathbb{C}^*\times\mathbb{Z}/kn\mathbb{Z}$, for every $n$, which obviously makes no sense. (I thought the Neron model depends only on the base ring and the elliptic curve?)

3. I'd appreciate a quick description of what $\pi_0$ of a scheme is (or a good reference for learning about it), and specifically how it's a group (I understand it should intuitively represent the connected components). I've just this week learned about the etale fundamental group $\pi_1$, though I'm not entirely sure how the definition generalizes to higher/lower fundamental groups.

(Also, is there a good translation of this somewhere?)

thanks

• will

Answer 1

1. "Il arrive que..." means "sometimes". So the paragraph says that sometimes the minimal model of $E_\eta$ over $\mathbf{C}[[t]]$ has bad reduction, which is true.

2. You're starting with an elliptic curve over $\mathbf{C}((t))$, not over $\mathbf{C}$. There's no reason that $E_\eta$ admits a level $n$ structure over $\mathbf{C}((t))$ (as opposed to over some finite extension). Geometrically what's going on (morally, in complex analytic geometry) is that you have a family of elliptic curves over the punctured complex unit disk, and you want to choose a level $n$ structure on the elliptic curve at each point so that the level structures vary nicely. You can do this locally, but not globally (in general). The algebro-geometric analogue of this fact is that for any fixed $n$, $E_\eta$ admits a level $n$ structure after some finite extension of the base (= passing to some cover of the punctured unit disk) but not necessarily over $\mathbf{C}((t))$ itself. You can see this concretely and algebraically by thinking aboutWeierstrass equations for an elliptic curves over $\mathbf{C}((t))$, thinking about the equations defining the coordinates of the $n$-torsion points, and seeing that there's no reason for those equations to be solvable if the ground field is not algebraically closed.

3. $\pi_0(E_0')$ refers to the group of connected components of $E_0'$. Normally $\pi_0$ is just a set (literally the set of connected components, not just intuitively), but since $E_0'$ has a group structure, you can add two connected components by picking a point on each one, adding the points, and taking the component this sum lands on (you should check that this is well-defined).

Finally, you should note that this paragraph doesn't really make sense, except as motivation: $X/\Gamma$ is a complex analytic object, whereas the rest of the paragraph takes place in the category of schemes.