Newton polygons and formal groups: Conjectures by Manin and Grothendieck.

TLDR: In this paper, the authors consider the problem of finding the Newton polygon of an abelian fiber from a matrix defined by a given Dieudonne module, and show that it is invariant under isogeny.

Abstract: 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 "HondaSerre-Tate theory". We give another proof in Section 5. 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. In Section 6 we show these conjectures to be true. These results are obtained by deforming the most special abelian varieties or p-divisible groups we can think of. In describing deformations we use the theory of displays; this was proposed by Mumford, and has been developed in [17], [18], and recently elaborated in [32] and [33]; also see [11], [31]. Having described a deformation we like to read off the Newton polygon of the generic fiber. In most cases it is difficult to determine the Newton polygon from the matrix defined by F on a basis for the (deformed) Dieudonne module. In general I have no procedure to do this (e.g. in case we deform away from a formal group where the Dieudonne module is not generated by one element). However in the special case we consider here, a(Go) = 1, a noncommutative version of the theorem of Cayley-Hamilton ("every matrix satisfies its own