Néron–Ogg–Shafarevich criterion

In mathematics, the Néron–Ogg–Shafarevich criterion states that an elliptic curve or abelian variety A over a local field K has good reduction if, and only if, there is a prime ℓ not dividing the characteristic of the residue field of K (or equivalently, for all such primes) such that the ℓ-adic Tate module T of A is unramified. Andrew Ogg (1967) introduced the criterion for elliptic curves. Serre and Tate (1968) used the results of André Néron (1964) 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).

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