# Neron models and ramification

I encountered this result while reading a few things, and it was stated without reference. I am having a hard time finding a reference for it (or a simple proof), so maybe you can help me:

let $E$ be an elliptic curve over $\mathbb{Q}$, $\ell$ a prime number, and consider $\rho=\rho_{E,\ell}$ the Galois representation of $Gal(\bar{\mathbb{Q}}/\mathbb{Q})$ on the $\ell$-adic Tate module of $E$. Let $p\neq\ell$ be a finite prime number where $\rho$ is ramified. Let $\ell^n$ be the highest power of $\ell$ such that $\rho\mod\ell^n$ is unramified at $p$.

Then $(\ell)^n=(c_p)$ in $\mathbb{Z}_{\ell}$, where $c_p$ is the Tamagawa factor of $E$ at $p$ (the order of the component group of the special fiber at $p$ of the N\'eron model of $E$ over $\mathbb{Z}$).

The proof is not hard but it also far from being obvious (to me, anyway). Let me recall it briefly. A separate, quasi-finite scheme $X$ over $\mathbb Z_{p}$ decomposes as a disjoint union of a finite part $X^{f}$ which has same special fiber as $X$ with a scheme $X'$ with zero special fiber. Denote by $\mathcal E$ the Néron model of $E$. The composition of $\mathcal E(\bar{\mathbb Q}_{p})^{f}\rightarrow (\mathcal E\times_{\mathbb Z_{p}}\mathbb F_{p})(\bar{\mathbb F}_{p})\rightarrow \Phi$ is surjective and its kernel is $\mathcal E(\bar{\mathbb Q}_{p})^{f,\circ}$. As multiplication by $\ell^n$ is surjective on $\mathcal E^{\circ}(\bar{\mathbb Q}_{p})^{f}$, the Snake Lemma gives an isomorphism $\Phi[\ell^{n}]\simeq\mathcal E[\ell^n]^{f}/\mathcal E[\ell^n]^{f,\circ}$. The image of $(T_\ell E)^f\rightarrow E[\ell^n]^f$ is $\ell$-divisible so in $E[\ell^n]^{f,\circ}$ as $\Phi$ is finite and coincides with this subspace by surjectivity of multiplication by $\ell^n$. Hence $(T_\ell E)^{f}\otimes\mathbb Z/\ell^{n}\mathbb Z\simeq \mathcal E[\ell^n]^{f,\circ}$.
Finally, the finite part of $\mathcal E[\ell^n]$ is given by the invariants under the inertia group $I_p$. Putting all this together, we get the following explicit description of the $\ell^n$-torsion of the component group and the result $$\Phi[\ell^n]\simeq\frac{(T_{\ell}E\otimes\mathbb Z/\ell^{n}\mathbb Z)^{I_p}}{(T_{\ell}E)^{I_p}\otimes\mathbb Z/\ell^{n}\mathbb Z}.$$
• Does my question really follows from this result? It seems that this is using $\Phi(\bar{\mathbb{F}_p})$, whereas the Tamagawa factor is the cardinality of $\Phi(\mathbb{F}_p)$, right? May 17, 2016 at 14:21
If $E$ has (split) multiplicative reduction, then you can probably get this by a direct computation on the Tate model. And for the additive reduction cases, one could possibly use explicit models for the various reduction types, a la Tate's algortithm, and do a direct computation. I expect this is fairly easy if $p\ge5$, reasonable if $p=3$, and quite painful (if even possible) for $p=2$. So the SGA reference that Olivier gives is undoubtedly the right approach, but if you want to avoid all the machinery, this might give a way to do it.