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
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.
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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