The almost product structure of Newton strata in the Deformation space of a Barsotti–Tate group with crystalline Tate tensors

Abstract

In this paper, we construct the almost product structure of the minimal Newton stratum in deformation spaces of Barsotti–Tate groups with crystalline Tate tensors, similar to Oort’s and Mantovan’s construction for Shimura varieties of PEL-type. It allows us to describe the geometry of the Newton stratum in terms of the geometry of two simpler objects, the central leaf and the isogeny leaf. This yields the dimension and the closure relations of the Newton strata in the deformation space. In particular, their nonemptiness shows that a generalisation of Grothendieck’s conjecture of deformations of Barsotti–Tate groups with given Newton polygon holds. As an application, we determine analogous geometric properties of the Newton stratification of Shimura varieties of Hodge type and prove the equidimensionality of Rapoport–Zink spaces of Hodge type.

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Notes

  1. 1.

    Actually, Viehmann proves it under the assumption of strong purity. But her proof only uses topological strong purity.

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Acknowledgements

I am grateful to Mark Kisin for many helpful discussions and his advice. I warmly thank Thomas Lovering for giving me a preliminary version of his thesis. I thank Stephan Neupert and Eva Viehmann for pointing out some mistakes in the preliminary version of this article.

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Correspondence to Paul Hamacher.

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Most of this work was written during a stay at the Harvard University which was supported by a fellowship within the Postdoc program of the German Academic Exchange Service (DAAD). I want to thank the Harvard University for its hospitality. Moreover, the author was partially supported by the ERC starting Grant 277889 “Moduli spaces of local G-shtukas”.

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Hamacher, P. The almost product structure of Newton strata in the Deformation space of a Barsotti–Tate group with crystalline Tate tensors. Math. Z. 287, 1255–1277 (2017). https://doi.org/10.1007/s00209-017-1867-2

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Keywords

  • Barsotti-Tate Groups
  • Deformation Space
  • Rapoport Zink Spaces
  • Shimura Varieties
  • Hodge Type