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Serre–Tate theory for Shimura varieties of Hodge type

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Abstract

We study the formal neighbourhood of a point in the \(\mu \)-ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a “shifted cascade”. Moreover we show that the CM points on the formal neighbourhood are dense and that the identity section of the shifted cascade corresponds to a lift of the abelian variety which has a characterization in terms of its endomorphisms, analogous to the Serre–Tate canonical lift of an ordinary abelian variety.

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Notes

  1. For us, the group \(U_G\) will take the place of U.

  2. In the PEL type case.

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Acknowledgements

It is a pleasure to thank Tom Lovering for asking a question which led to this paper. We also thank George Boxer and Mark Kisin for useful discussions involving the material in this paper. We are also very grateful to the referee whose comments helped improve the exposition of this work.

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  1. Department of Mathematics, MIT, Cambridge, USA

    Ananth N. Shankar

  2. Department of Mathematics, IAS, Princeton, USA

    Rong Zhou

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  1. Ananth N. Shankar
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  2. Rong Zhou
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Correspondence to Ananth N. Shankar.

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Shankar, A.N., Zhou, R. Serre–Tate theory for Shimura varieties of Hodge type. Math. Z. 297, 1249–1271 (2021). https://doi.org/10.1007/s00209-020-02556-y

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