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…)