Néron–Ogg–Shafarevich criterion

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

Andrew Ogg introduced the criterion for elliptic curves.^[1] Jean-Pierre Serre and John Tate^[2] used the results of André Néron^[3] 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).