In mathematics, the NĂ©ronâOggâShafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and â is a prime not dividing the characteristic of the residue field of K then A has good reduction if and only if the â-adic Tate module Tâ of A is unramified. Andrew Ogg ( 1967) introduced the criterion for elliptic curves. Serre and Tate ( 1968) used the results of AndrĂ© NĂ©ron ( 1964) to extend it to abelian varieties, and named the criterion after Ogg, NĂ©ron and Igor Shafarevich (commenting that Ogg's result seems to have been known to Shafarevich).
References
- NĂ©ron, AndrĂ© (1964), "ModĂšles minimaux des variĂ©tĂ©s abĂ©liennes sur les corps locaux et globaux", Publications MathĂ©matiques de l'IHĂS (in French), 21: 5â128, 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): 1â21, 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): 492â517, doi: 10.2307/1970722, ISSN 0003-486X, JSTOR 1970722, MR 0236190, Zbl 0172.46101
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