Canonical lifts and $\delta$-structures

TL;DR

This paper generalizes the Serre-Tate theory to broader families of ordinary abelian varieties, demonstrating the uniqueness of canonical lifts with $ ext{delta}$-structures in a $p$-adic setting.

Contribution

It extends the theory of canonical lifts to arbitrary unpolarized families over $p$-adic schemes, incorporating $ ext{delta}$-structures for a wider class of groups.

Findings

Canonical lift is unique with $ ext{delta}$-structure.

Extension of Serre-Tate theory to unpolarized families.

Analogous results for $p$-groups and $p$-divisible groups.

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.