Hodge-Newton filtration for $p$-divisible groups with ramified endomorphism structure

Abstract

Let O K be a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field. We prove the existence of the Hodge-Newton filtration for p-divisible groups over O K with additional endomorphism structure for the ring of integers of a finite, possibly ramified field extension of Q p . The argument is based on the Harder-Narasimhan theory for finite flat group schemes over O K . In particular, we describe a sufficient condition for the existence of a filtration of p-divisible groups over O K associated to a break point of the Harder-Narasimhan polygon. identical hypothesis on the polygons as above, Katz proves the existence of a "Hodge-Newton" decomposition of the relative F -crystal into two components, whose polygons correspond respectively to the part before and after the internal contact point given by the assumption. In case such an F -crystal is the Dieudonné module of a p-divisible group over k, this result recovers the multiplicative-bilocal-étale decomposition.This finding was later generalised by Kottwitz, in [21] (2003), to F -crystals with additional endomorphism structure for the ring of integers of a finite unramified extension F of Q p and possibly endowed with a polarisation. By this time, however, new points of view on the subject had developed. Kottwitz's result is formulated in terms of affine Deligne-Lusztig sets, objects constructed by means of a reductive group over a local field and determined by a defining datum. The additional structure is encoded in the reductive group, in this case the restriction of scalars from F to Q p of a general linear group GL n , or a general symplectic group GSp 2n in presence of a polarisation. The definitions of the Hodge and the Newton polygons, as well as the notion of Hodge-Newton reducibility, are also translated in the group-theoretic language, thus taking into account the additional structure; for the groups in question, all this can still be visualised in terms of polygons. An affine Deligne-Lusztig set can then be seen as a set of F -crystals with additional structure, whose Hodge and Newton polygons are fixed by the defining datum. From this point of view, the Hodge-Newton decomposition is expressed as a bijection between the affine Deligne-Lusztig set associated to a Hodge-Newton reducible datum and one relative to a Levi subgroup, which collects F -crystals admitting a decomposition.Mantovan and Viehmann, in [24] (2010), further generalised Kottwitz's result to endomorphism structures for more general unramified Z p -algebras and to families of Fcrystals in characteristic p (a statement in families could also be found in Katz's original article, without additional structure, and in Csima's work [7], for polarised objects). Most importantly, however, the two authors proved that their Hodge-Newton decomposition can be lifted to a filtration of p-divisible groups over a complete Noetherian local W (k)-algebra (here, W (k) denotes the ring of Witt vectors with coefficients in k). Their argument is based on an explicit descriptio...