Newton polygons as lattice points

Abstract

Let M be the reduction modulo p of a Shimura variety, and let N ( M ) be the partially ordered set predicted to be the set of all types occurring in the stratification of M by Newton polygons, in the context of F -isocrystals with additional structures. We prove that N ( M ) is ranked (or catenary ), in the sense that any two maximal chains have the same length. We also give a Lie-theoretic formula for the distance between two comparable elements in N ( M ). This formula can be used to give the expected dimension of the Newton strata of M .