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_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, Zbl 0132.41403 
• Ogg, A. P. (1967), "Elliptic curves and wild ramification", American Journal of Mathematics 89: 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: 492–517, doi:10.2307/1970722, ISSN 0003-486X, JSTOR 1970722, MR 0236190, Zbl 0172.46101