PlanetMath: criterion of Néron-Ogg-Shafarevich

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criterion of Néron-Ogg-Shafarevich

(Theorem)

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

Definition 1 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 1 (Criterion of N ${\bf {\acute{e}}}$ ron-Ogg-Shafarevich) Let be an elliptic curve defined over . The following are equivalent:

has good reduction over ;
is unramified at $\nu$ for all $m\geq1$ , $\gcd(m,\operatorname{char}(\mathbb{F}))=1$ ;
The Tate module is unramified at $\nu$ for some (all) l, $l\neq \operatorname{char}(\mathbb{F})$ ;
is unramified at $\nu$ for infinitely many integers $m\geq 1$ , $\gcd(m,\operatorname{char}(\mathbb{F}))=1$ .

Corollary 1 Let be an elliptic curve. Then has potential good reduction if and only if the inertia group $I_{\nu}$ acts on through a finite quotient for some prime $l\neq \operatorname{char}(\mathbb{F})$ .

Proof. [Proof of Corollary] ( $\Rightarrow$ ) Assume that

has potential good reduction. By definition, there exists a finite extension of

, call it

, such that

has good reduction. We can extend

(if necessary) so

is a Galois finite extension.

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

( $\Leftarrow$ ) 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 , then ${\overline{K}}^J/{\overline{K}}^{I_{\nu}}$ is a finite extension, so we can find a finite extension so that ${\overline{K}}^J={K'}{\overline{K}}^{I_{\nu}}$ . So the inertia group of is equal to , and acts trivially on . Hence the criterion ( (3) $\Rightarrow$ (1) ) implies that has good reduction over , and since is finite, has potential good reduction. $\qedsymbol$

Proposition 1 Let be an elliptic curve. Then has potential good reduction if and only if its -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

to a finite extension

so that

has a Weierstrass equation:

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

(1)

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

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

(2)

then $\lambda\in R$ and $\lambda\neq 0,1 \mod \mathcal{M}'$ ( $\Rightarrow$ $\Delta '\in (R')^*$ ). Hence

has good reduction, i.e.

has potential good reduction.

( $\Rightarrow$ ) Assume that has potential good reduction, so there exists so that has good reduction. Let $\Delta'$ , the usual quantities associated to the Weierstrass equation over . Since has good reduction, $\Delta '\in (R')^*$ , and so $j(E)={{({c_4}')^3}\over {\Delta '}}\in R'$ . But since is defined over , $j(E)\in K$ , so $j(E)\in K\bigcap{R'}=R$ . $\qedsymbol$