Let ${A}$ be an abelian variety of dimension ${g}$ over a field ${k}$ of characteristic ${p}$. In the previous posts, we saw that to give a deformation ${R}$ of ${A}$ over a local artinian ring with residue field ${k}$ was the same as giving a continuous morphism of rings ${W(k)[[t_1, \dots, t_{g^2}]] \rightarrow A}$: that is, the local deformation space is smooth on ${g^2}$ parameters.

There is another description of the deformation space in terms of the ${p}$-divisible group, though. Given ${A}$, we can form the formal scheme

$\displaystyle A[p^\infty] = \varinjlim A[p^n],$

where each ${A[p^n]}$ is a finite group scheme of rank ${p^{2ng}}$ over ${k}$. As a formal scheme, this is smooth: that is, given a small extension ${R' \twoheadrightarrow R}$ in ${\mathrm{Art}_k}$ and a morphism ${\mathrm{Spec} R \rightarrow A[p^\infty]}$, there is an extension

In fact, we start by finding an extension ${\mathrm{Spec} R' \rightarrow A}$ (as ${A}$ is smooth), and then observe that this extension must land in some ${A[p^n]}$ if ${\mathrm{Spec} R}$ landed in ${A[p^\infty]}$.

As a result, we can talk about deformations of this formal (group) scheme.

Definition 16 deformation of a smooth, formal group scheme ${G}$ over ${k}$ over a ring ${R \in \mathrm{Art}_k}$ is a smooth, formal group scheme ${G'}$ over ${R}$ which reduces mod the maximal ideal (i.e., when restricted to ${k}$-algebras) to ${G}$.

Suppose ${G(k) = \ast}$. Using Schlessinger’s criterion, we find that (under appropriate finiteness hypotheses), ${G}$ must be prorepresentable by a power series ring ${R[[t_1, \dots, t_n]]}$ for some ${n}$. In other words, ${G}$ corresponds to a formal group over ${R}$.

Suppose for instance that ${A}$ was a supersingular elliptic curve. By definition, each of the ${A[p^n]}$ is a thickening of the zero point, and consequently ${A[p^\infty]}$ is the formal completion ${\hat{A}}$ of ${A}$ at zero. This data is equivalent to the formal group of ${A}$, and as we just saw a deformation of this formal group over a ring ${R \in \mathrm{Art}_k}$ is a formal group over ${R}$ which reduces to the formal group of ${A}$ mod the maximal ideal.

The main result is:

Theorem 17 (Serre-Tate) Let ${R \in \mathrm{Art}_k}$. There is an equivalence of categories between abelian schemes over ${R}$ and pairs ${(A, G)}$ where ${A }$ is an abelian variety over ${k}$ and ${G}$ a deformation of ${A[p^\infty]}$ over ${R}$.

So, in particular, deformations of ${A}$ over ${R}$ are equivalent to deformations of ${A[p^\infty]}$ over ${R}$.

In this post, I’ll describe an argument due to Drinfeld for this result (which I learned from Katz’s article). (more…)