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Canonical lifts and \(\delta \)-structures

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Abstract

We extend the Serre–Tate theory of canonical lifts of ordinary abelian varieties to arbitrary unpolarised families of ordinary abelian varieties parameterised by a p-adic formal scheme S. We show that the canonical lift is the unique lift to W(S) which admits a \(\delta \)-structure in the sense of Joyal, Buium, and Bousfield. We prove analogous statements for families of ordinary p-groups and p-divisible groups.

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Notes

  1. We note that as S and W(S) are ind-affine sheaves it follows that they are sheaves on \(\mathrm {Aff}\) for the fpqc topology and that all relatively affine sheaves (for the étale topology) over them are also sheaves on \(\mathrm {Aff}\) the fpqc topology.

  2. An S-sheaf X is anti-affine if every morphism from X to a relatively affine S-sheaf factors through the structure map \(X\rightarrow S\).

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Borger, J., Gurney, L. Canonical lifts and \(\delta \)-structures. Sel. Math. New Ser. 26, 67 (2020). https://doi.org/10.1007/s00029-020-00599-x

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