La formule d'Ogg relie le discriminant minimal d'une courbe elliptique à son conducteur. La preuve originelle d'Ogg n'est pas complète sur un corps de valuation discrète de caractéristiques (0, 2). T. Saito a donné une preuve plus conceptuelle.
Ogg's formula relates the minimal discriminant of an elliptic curve defined over a discrete valuation field to its conductor. The original proof of Ogg is a case by case computation and did not treat the case of mixed characteristics (0,2). As a special case of his theorem relating (a kind of) discriminant to the Artin conductor, Takeshi Saito gave in 1988 a complete and conceptual proof of this formula.
We give a new proof of the well-known fact that the Néron model of an elliptic curve is isomorphic to the smooth part of the minimal regular model.
Let X -> X/G be the quotient of a S-scheme X by a finite group G. Let T -> S be any S-scheme. In general, XT/G -> (X/G)T is not an isomorphism. However, we show that it is a homeomorphism. This fact is well-known. The proof here is rather elementary.
Poincaré's complete reducibility theorem is usually proved for abelian varieties over an algebraicailly closed (or perfect) field. Here we give a proof in the general case following hints of Michel Raynaud.