Néron–Ogg–Shafarevich criterion

In mathematics, the Néron–Ogg–Shafarevich criterion states that if $A$ is an elliptic curve or abelian variety over a local field $K$, and $\ell$ is a prime not dividing the characteristic of the residue field of $K$, then $A$ has good reduction if and only if the $\ell$-adic Tate module $T_{\ell }$ of $A$ is unramified.

Andrew Ogg introduced the criterion for elliptic curves.Template:Sfnp Jean-Pierre Serre and John TateTemplate:Sfnp used the results of André NéronTemplate:Sfnp to extend it to abelian varieties, and named the criterion after Ogg, Néron and Igor Shafarevich (commenting that Ogg's result seems to have been known to Shafarevich).

References

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