Symmetric p-divisible groups are algebraizable
(Un exposé de F. Oort au séminaire de théorie des
nombres de Jussieu le 14 janvier 1999)
Résumé :
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 CM
abelian varieties constructed in
characteristic zero). We give another proof of this conjecture via
methods purely in characteristic p. It is part of understanding deformations
of formal groups, which leads to a proof of the conjecture by Grothendieck
(Montreal, 1970) about possible deformations of p-divisible groups
when 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 central
role in choosing coordinates on the deformation space of certain formal
groups.