Résume | In 1963 Manin conjectured that every p-divisible group in positive characteristic,which is isogenous with its dual, comes from an abelian variety.This problem was solved in the Honda-Serre-Tate theory (by reducing CMabelian varieties constructed incharacteristic zero). We give another proof of this conjecture viamethods purely in characteristic p. It is part of understanding deformationsof formal groups, which leads to a proof of the conjecture by Grothendieck(Montreal, 1970) about possible deformations of p-divisible groupswhen Newton polygons (of the closed and the generic fibre) are given.In the proof, a non-commutative version of the theorem of Cayley-Hamilton(every matrix satisfies its own characteristic polynomial) plays a centralrole in choosing coordinates on the deformation space of certain formalgroups. |