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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
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1. "il arrive que" means "sometimes" not "it follows that". 2. The isomorphism needs to be compatible with the action of Galois (with Galois acting trivially on $(\mathbf{Z}/n)^2$.) In other words, for a level $n$ structure to exist, all the $n$ torsion needs to be defined over $\mathbf{C}((T))$, which needn't be the case. –  user18237 Jul 2 '12 at 5:47
    
Buy Online Access to this Chapter Individual Book Chapter (Electronic Only) EUR 24.95 –  Chandan Singh Dalawat Jul 3 '12 at 5:42
    
Professor Dalawat - Could you please post a link to where I can find a translation of this paper? thanks –  oxeimon Jul 14 '12 at 9:50
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1 Answer

up vote 12 down vote accepted
  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.

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So, to clarify, $E_\eta$ admits a level $n$ structure over a field $K$ if and only if the full $n$-torsion subgroup is $K$-rational? Re: gb's comment - I don't understand what the isomorphism being compatible with Galois means when the image is just a group, not a scheme. ...or are we thinking of $(\mathbb{Z}/n\mathbb{Z})^2$ as a group scheme? If so, how exactly are we doing this? –  oxeimon Jul 2 '12 at 9:38
    
Yes, we're thinking of $(\mathbf{Z}/n\mathbf{Z})^2$ as a group scheme. Over any base scheme $S$, the scheme structure on $(\mathbf{Z}/n\mathbf{Z})_S^2$ is just $n^2$ disjoint copies of $S$. –  Rebecca Bellovin Jul 2 '12 at 19:40
    
Ahh, interesting. Thanks! –  oxeimon Jul 3 '12 at 2:00
    
@Rebecca Sorry for reviving this old post, but I was just thinking about this, and Deligne remarks in the next paragraph that "this discussion suggests that the points at infinity of $\mathcal{H}^*/\Gamma$ correspond to these subgroups $E_0$ with level structure. Why would the discussion suggest this? What do elliptic curves $\mathbb{C}((T))$ have to do with the point at infinity? And what does the Neron model have to do with anything? Is he saying that the elliptic curve over the un-compactified modular curve always has bad reduction at the cusps? –  oxeimon Aug 15 '12 at 1:20
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