Inseparable field extensions associated to $p$-divisible groups
(Un exposé de Richard Pink au séminaire de
théorie des nombres de Chevaleret le 14 février 2005)
Résumé :
Let $X$ be a $p$-divisible group over a field $K$ of
characteristic $p>0$. Then the field of definition $K(x)$ of any point
$x\in X$ is a finite extension of~$K$, which may be separable or
inseparable. How do the separable and inseparable degrees of $K(x)/K$
vary with $x$? The separable degree is determined by the action of the
absolute Galois group of $K$ on the Tate module of~$X$, and hence by
group theory. In a joint project by Frans Oort and the speaker, a
partial answer is given for the inseparable degree. One of the tools
used is deformation theory of $p$-divisible groups. Another is the
theory of multilinear morphisms for commutative group schemes and
$p$-divisible groups, which incorporates tensor product and inner hom,
and which allows the extension of the results to a variety of similar
questions.