Refined partial Hasse invariants and the canonical filtration

Let $G$ be a $p$-divisible group over a scheme of characteristic $p$, and assume that it is endowed with an action of the ring of integers of a finite unramified extension $F$ of $\mathbb{Q}_p$. Let us fix the type $\mu$ of this action on the sheaf of differentials $\omega_G$. V. Hernandez, following a construction of Goldring and Nicole, defined partial Hasse invariants for $G$. These are sections of invertible sheaves. The product of these invariants is the $\mu$-ordinary Hasse invariant, and it is non-zero if and only if the $p$-divisible group is $\mu$-ordinary (i.e. the Newton polygon is minimal given the type of the action). We show that, if one assumes the existence of a certain filtration refining the Hodge filtration, each of these partial Hasse invariants can be expressed as a product of other sections, the refined partial Hasse invariants. Over a Shimura variety, the condition is satisfied if one considers an explicit closed subscheme of a certain flag variety. As an application, we relate these refined partial Hasse invariants to the partial degrees of the canonical filtration (if it exists)

Groupes pp-divisibles avec condition de Pappas-Rapoport et invariants de Hasse

We study $p$-divisible groups $G$ endowed with an action of the ring of integers of a finite (possibly ramified) extension of $\mathbb{Q}_p$ over a scheme of characteristic $p$. We suppose moreover that the $p$-divisible group $G$ satisfies the Pappas-Rapoport condition for a certain datum $\mu$ ; this condition consists in a filtration on the sheaf of differentials $\omega_G$ satisfying certain properties. Over a perfect field, we define the Hodge and Newton polygons for such $p$-divisible groups, normalized with the action. We show that the Newton polygon lies above the Hodge polygon, itself lying above a certain polygon depending on the datum $\mu$. We then construct Hasse invariants for such $p$-divisible groups over an arbitrary base scheme of characteristic $p$. We prove that the total Hasse invariant is non-zero if and only if the $p$-divisible group is $\mu$-ordinary, i.e. if its Newton polygon is minimal. Finally, we study the properties of $\mu$-ordinary $p$-divisible groups. The construction of the Hasse invariants can in particular be applied to special fibers of PEL Shimura varieties models as constructed by Pappas and Rapoport

Classicité de formes modulaires surconvergentes

We prove in this paper a classicality result for overconvergent modular forms on PEL Shimura varieties of type (A) or (C) associated to an unramified reductive group on $mathbb{Q}_p$. To get this result, we use the analytic continuation method, first used by Buzzard and Kassaei

Role of the Netrin-like Domain of Procollagen C-Proteinase Enhancer-1 in the Control of Metalloproteinase Activity

The netrin-like (NTR) domain is a feature of several extracellular proteins, most notably the N-terminal domain of tissue inhibitors of metalloproteinases (TIMPs), where it functions as a strong inhibitor of matrix metalloproteinases and some other members of the metzincin superfamily. The presence of a C-terminal NTR domain in procollagen C-proteinase enhancers (PCPEs), proteins that stimulate the activity of astacin-like tolloid proteinases, raises the possibility that this might also have inhibitory activity. Here we show that both long and short forms of the PCPE-1 NTR domain, the latter beginning at the N-terminal cysteine known to be critical for TIMP activity, show no inhibition, at micromolar concentrations, of several members of the metzincin superfamily, including matrix metalloproteinase-2, bone morphogenetic protein-1 (a tolloid proteinase), and different ADAMTS (a disintegrin and a metalloproteinase with thrombospondin motifs) proteinases from the adamalysin family. In contrast, we report that the NTR domain within PCPE-1 leads to superstimulation of bone morphogenetic protein-1 activity in the presence of heparin and heparan sulfate. These observations point to a new mechanism whereby binding to cell surface-associated or extracellular heparin-like sulfated glycosaminoglycans might provide a means to accelerate procollagen processing in specific cellular and extracellular microenvironments

A quotient of the Lubin-Tate tower II

In this article we construct the quotient M_1/P(K) of the infinite-level Lubin-Tate space M_1 by the parabolic subgroup P(K) of GL(n,K) of block form (n-1,1) as a perfectoid space, generalizing results of one of the authors (JL) to arbitrary n and K/Q_p finite. For this we prove some perfectoidness results for certain Harris-Taylor Shimura varieties at infinite level. As an application of the quotient construction we show a vanishing theorem for Scholze's candidate for the mod p Jacquet-Langlands and the mod p local Langlands correspondence. An appendix by David Hansen gives a local proof of perfectoidness of M_1/P(K) when n = 2, and shows that M_1/Q(K) is not perfectoid for maximal parabolics Q not conjugate to P.Comment: with an appendix by David Hanse

Formes modulaires surconvergentes, ramification et classicité

We prove in this paper a classicality result for overconvergent modular forms on PEL Shimura varieties of type (A) or (C), without any ramification hypothesis. We use an analytic continuation method, which generalizes previous results on the non ramified setting. We work with the rational model of the Shimura variety, and use an embedding into the Siegel variety to define the integral structures on the rigid space