Newton polygons and formal groups: Conjectures by Manin and Grothendieck
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 ``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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- June 2000
- DOI:
- 10.48550/arXiv.math/0007201
- arXiv:
- arXiv:math/0007201
- Bibcode:
- 2000math......7201O
- Keywords:
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- Mathematics - Algebraic Geometry
- E-Print:
- 24 pages, published version, abstract added in migration