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:

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?

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?)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