Quelques notes privées qui ne sont pas destinée à la publication (Some private notes not to be published).

1. Formule d'Ogg d'après Saito (Ogg's formula after Saito) (March 1998)

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 donnée 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.

2. Néron models of elliptic curves (with Dino Lorenzini, March 2002, minor changes on Sept. 2003)

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.

3. Algebraic subgroups of abelian varieties. (July 2011)

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.

4. Discriminant and smoothness of a hyperelliptic equation.