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 ℓ is a prime not dividing the characteristic of the residue field of K then A has good reduction if and only if the ℓ-adic Tate module T_ℓ 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).

References

Néron, André (1964), "Modèles minimaux des variétés abéliennes sur les corps locaux et globaux", Publications Mathématiques de l'IHÉS (in French), 21: 5–128, doi:10.1007/BF02684271, ISSN 1618-1913, MR 0179172, S2CID 120802890, Zbl 0132.41403
Ogg, A. P. (1967), "Elliptic curves and wild ramification", American Journal of Mathematics, 89 (1): 1–21, doi:10.2307/2373092, ISSN 0002-9327, JSTOR 2373092, MR 0207694, Zbl 0147.39803
Serre, Jean-Pierre; Tate, John (1968), "Good reduction of abelian varieties", Annals of Mathematics, Second Series, 88 (3): 492–517, doi:10.2307/1970722, ISSN 0003-486X, JSTOR 1970722, MR 0236190, Zbl 0172.46101

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