Harder-Narasimhan strata and $p$-adic period domains

Shen, Xu

doi:10.48550/arxiv.1909.02230

Cited by 5 publications

(5 citation statements)

References 23 publications

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“…semi-stable loci do not correspond to each other any more, as we now illustrate. In particular, our definition of weak admissibility does not coincide with the corresponding definition in [Sh,6.4], contrary to what is claimed in [Sh,Prop. 6.15].…”

Section: Bycontrasting

confidence: 87%

On Newton strata in the $B_{dR}^+$-Grassmannian

Viehmann¹

2021

Preprint

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We study parabolic reductions and Newton points of G-bundles on the Fargues-Fontaine curve and the Newton stratification on the B + dR -Grassmannian for any reductive group G. Let Bun G be the stack of G-bundles on the Fargues-Fontaine curve. Our first main result is to show that under the identification of the points of Bun G with Kottwitz's set B(G), the closure relations on |Bun G | coincide with the opposite of the usual partial order on B(G). Furthermore, we prove that every non-Hodge-Newton decomposable Newton stratum in a minuscule affine Schubert cell in the B + dR -Grassmannian intersects the weakly admissible locus, proving a conjecture of Chen. On the way, we study several interesting properties of parabolic reductions of G-bundles, and determine which Newton strata have classical points.Contents 18 7. Newton strata in the weakly admissible locus 21 References 28

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Section: Bycontrasting

confidence: 87%

On Newton strata in the $B_{dR}^+$-Grassmannian

Viehmann¹

2021

Preprint

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“…In this sense, the present work could have a natural application to the study of the generic fibre of Rapoport-Zink spaces (application which is in fact in the author's plans). Passing to infinite level structure, this could lead to further progress towards the Harris-Viehmann conjecture, comparing with the work of Gaisin and Imai [15] in this direction (especially in light of the methods elaborated by Chen, Fargues and Shen in [5], but see also [34] and [4], based on the theory of vector bundles over the Fargues-Fontaine curve).…”

Section: Introductionmentioning

confidence: 90%

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

Marrama

2020

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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...

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“…In this sense, the present work could have a natural application to the study of the generic fibre of Rapoport-Zink spaces with finite level structure (application which is in fact in the author's plans). Passing to infinite level structure, this could lead to further progress towards the Harris-Viehmann conjecture, comparing with the work of Gaisin and Imai [15] in this direction (especially in light of the methods elaborated by Chen, Fargues and Shen in [5], but see also [35] and [4], based on the theory of vector bundles over the Fargues-Fontaine curve). Let us mention, in addition, that the conclusions of this work can be interpreted in terms of p-adic Galois representations.…”

Section: Introductionmentioning

confidence: 92%