# Neron models of elliptic curves with level N structure?

In the Deligne-Rapoport paper entitled "Les schemas de modules de courbes elliptiques" the following is written (I translated in english):

Let $E$ be an elliptic curve with $\Gamma(N)$-level structure defined over $\mathbb{C}((T))$. Let $E'$ be the minimal model of $E$ over $\mathbb{C}[[T]]$. It may happen that $E'$ has bad reduction (i.e. when one reduces modulo $T$). So let $A$ be the neron model of $E'$ over the d.v.r. $\mathbb{C}[[T]]$. Then it seems that the special fiber of $A$ (i.e. when $T=0$) is always isomorphic to $\mathbb{C}^{\times}\times\mathbb{Z}/kN$ for some suitable integer $k$.

Q: How come this $N$ shows up in the special fiber of $A$ ?

This is a little bit strange since in the definition of a Neron model no such $N$ appears.

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## 1 Answer

Roughly speaking, the N-torsion defined over the base injects into the Neron model (in characteristic 0), so the special fiber of the Neron model needs to have a subgroup isomorphic to Z/NZ x Z/NZ, since by assumption there is full level N structure. The special fiber (since there's bad reduction) has the form C^* x F, where F is a finite cycle group. So F contains an N-torsion element, which means F = Z/kN.

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So the fact that you need characteristic 0 means that it does not follow formally from the universal property of the Neron model. So how does the characteristic 0 assumption intervene? Is related somehow to the fact that in characteristic zero finite flat group schemes are etale? –  Hugo Chapdelaine Jan 6 '11 at 0:51
I wonder if you need to assume $N \ge 3$ to exclude $I_0^*$-reduction. In that case the component group is $\mathbb Z/2 \times \mathbb Z/2$. Or do you assume multiplicative Reduction right away? In that case characteristic zero is not needed. In charakteristic $p$, you have to be careful because additive reduction gives you a lot of $p$-Torsion. –  Holger Partsch Jan 6 '11 at 13:52
Hugo, your question was phrased over C, so I put in the characterstic 0 assumption to avoid saying something false, since I didn't really think it through in finite characteristic. If you assume a priori that you have multiplicative reduction, then I think you're fine in all characteristics, including p=2. If you simply assume bad reduction, then there are probably problems with p=2 and p=3 for various reduction types. But I haven't thought it through carefully. Note that if p divides N, then there is some p-torsion in the formal group, but none in the multiplicative group. –  Joe Silverman Jan 8 '11 at 15:41