  We consider p-divisible groups (also called Barsotti-Tate groups) in
characteristic p, their deformations, and we draw some conclusions.
  For such a group we can define its Newton polygon (abbreviated NP). This is
invariant under isogeny. For an abelian variety (in characteristic p) the
Newton polygon of its p-divisible group is ``symmetric''.
  In 1963 Manin conjectured that conversely any symmetric Newton polygon is
``algebroid''; i.e., it is the Newton polygon of an abelian variety. This
conjecture was shown to be true and was proved with the help of the
``Honda-Serre-Tate theory''. We give another proof.
  Grothendieck showed that Newton polygons ``go up'' under specialization: no
point of the Newton polygon of a closed fiber in a family is below the Newton
polygon of the generic fiber. In 1970 Grothendieck conjectured the converse:
any pair of comparable Newton polygons appear for the generic and special fiber
of a family. This was extended by Koblitz in 1975 to a conjecture about a
sequence of comparable Newton polygons. We prove these conjectures.
