# criterion of Néron-Ogg-Shafarevich

In this entry, we use the following notation. $K$ is a local field, complete with respect to a discrete valuation $\nu$, $R$ is the ring of integers of $K$, $\mathcal{M}$ is the maximal ideal of $R$ and $\mathbb{F}$ is the residue field of $R$.

###### Definition.

Let $\Xi$ be a set on which $\operatorname{Gal}(\overline{K}/K)$ acts. We say that $\Xi$ is unramified at $\nu$ if the action of the inertia group ${I_{\nu}}$ on $\Xi$ is trivial, i.e. ${\zeta}^{\sigma}=\zeta$ for all $\sigma\in I_{\nu}$ and for all $\zeta\in\Xi$.

###### Theorem (Criterion of N${\bf{\acute{e}}}$ron-Ogg-Shafarevich).

Let $E/K$ be an elliptic curve defined over $K$. The following are equivalent:

1. 1.

$E$ has good reduction over $K$;

2. 2.

$E[m]$ is unramified at $\nu$ for all $m\geq 1$, $\gcd(m,\operatorname{char}(\mathbb{F}))=1$;

3. 3.

The Tate module $T_{l}(E)$ is unramified at $\nu$ for some (all) l, $l\neq\operatorname{char}(\mathbb{F})$;

4. 4.

$E[m]$ is unramified at $\nu$ for infinitely many integers $m\geq 1$, $\gcd(m,\operatorname{char}(\mathbb{F}))=1$.

###### Corollary.

Let $E/K$ be an elliptic curve. Then $E$ has potential good reduction if and only if the inertia group $I_{\nu}$ acts on $T_{l}(E)$ through a finite quotient for some prime $l\neq\operatorname{char}(\mathbb{F})$.

###### Proof of Corollary.

($\Rightarrow$) Assume that $E$ has potential good reduction. By definition, there exists a finite extension of $K$, call it $K^{\prime}$, such that $E/K^{\prime}$ has good reduction. We can extend $K^{\prime}$ (if necessary) so $K^{\prime}/K$ is a Galois finite extension.

Let $\nu^{\prime}$ and $I_{\nu^{\prime}}$ be the corresponding valuation and inertia group for $K^{\prime}$. Then the theorem above ( (1)$\Rightarrow$(3) ) implies that $T_{l}(E)$ is unramified at $\nu^{\prime}$ for all $l$, $l\neq\operatorname{char}(\mathbb{F})=\operatorname{char}(\mathbb{F}^{\prime})$ (since $\mathbb{F}^{\prime}$ is a finite extension of $\mathbb{F}$). So $I_{\nu^{\prime}}$ acts trivially on $T_{l}(E)$ for all $l\neq\operatorname{char}(\mathbb{F}^{\prime})$. Thus $I_{\nu}\hookrightarrow T_{l}(E)$ factors through the finite quotient $I_{\nu}/I_{{\nu}^{\prime}}$.

($\Leftarrow$) Let $l\neq\operatorname{char}(\mathbb{F})$, and assume $I_{\nu}\hookrightarrow T_{l}(E)$ factors through a finite quotient, say $I_{\nu}/J$. Let ${\overline{K}}^{J}$ be the fixed field of $J$, then ${\overline{K}}^{J}/{\overline{K}}^{I_{\nu}}$ is a finite extension, so we can find a finite extension $K^{\prime}/K$ so that ${\overline{K}}^{J}={K^{\prime}}{\overline{K}}^{I_{\nu}}$. So the inertia group of $K^{\prime}$ is equal to $J$, and $J$ acts trivially on $T_{l}(E)$. Hence the criterion ( (3)$\Rightarrow$(1) ) implies that $E$ has good reduction over $K^{\prime}$, and since $K^{\prime}/K$ is finite, $E$ has potential good reduction. ∎

###### Proposition.

Let $E/K$ be an elliptic curve. Then $E$ has potential good reduction if and only if its $j$-invariant is integral ( i.e. $j(E)\in R$ ).

###### Proof.

($\Leftarrow$) Assume $\operatorname{char}(\mathbb{F})\neq 2$, it is easy to prove that we can extend $K$ to a finite extension $K^{\prime}$ so that $E$ has a Weierstrass equation:

 $E:y^{2}=x(x-1)(x-\lambda)\quad\lambda\neq 0,1$ (1)

Since we are assuming $j(E)\in R$, and:

 $(1-\lambda(1-\lambda))^{3}-j{\lambda}^{2}(1-\lambda)^{2}=0$ (2)

then $\lambda\in R$ and $\lambda\neq 0,1\mod\mathcal{M}^{\prime}$ ( $\Rightarrow$ $\Delta^{\prime}\in(R^{\prime})^{*}$ ). Hence $E/K^{\prime}$ has good reduction, i.e. $E$ has potential good reduction.

($\Rightarrow$) Assume that $E$ has potential good reduction, so there exists $K^{\prime}$ so that $E/K^{\prime}$ has good reduction. Let $\Delta^{\prime}$, $c_{4}^{\prime}$ the usual quantities associated to the Weierstrass equation over $K^{\prime}$. Since $E/K^{\prime}$ has good reduction, $\Delta^{\prime}\in(R^{\prime})^{*}$, and so $j(E)={{({c_{4}}^{\prime})^{3}}\over{\Delta^{\prime}}}\in R^{\prime}$. But since $E$ is defined over $K$, $j(E)\in K$, so $j(E)\in K\bigcap{R^{\prime}}=R$. ∎

Title criterion of Néron-Ogg-Shafarevich CriterionOfNeronOggShafarevich 2013-03-22 17:14:58 2013-03-22 17:14:58 alozano (2414) alozano (2414) 4 alozano (2414) Theorem msc 14H52 criterion of Neron-Ogg-Shafarevich EllipticCurve ArithmeticOfEllipticCurves