Canonical lifts and $δ$-structures

Abstract

We extend the Serre-Tate theory of canonical lifts of ordinary abelian varieties to arbitrary 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 δ-structure in the sense of Joyal and Buium. ContentsIntroduction 1 1. δ-structures and Witt vectors 2 2. Further properties of δ-structures and Witt vectors 9 3. Group schemes, torsors and extensions with δ-structures 18 4. Canonical lifts of ordinary p-groups 23 5. Canonical lifts of ordinary abelian schemes 28 References 31

Prismatization

“…Uniqueness is clear. To construct the isomorphism, G ♯ a ∼ −→ W (F ) , we will use the approach to W developed by Joyal [J85] (an exposition of this approach can be found in [B16] and [BG,§1]).…”