I’ve been trying to learn something about deformations of abelian varieties lately. One of the big results is:

Theorem 1 The “local moduli space” of an abelian variety ${X/k}$ is smooth of dimension ${g^2}$, if ${\dim X = g}$.

Why might you care about this result? Let’s say you’re in the case ${g = 1}$, so then presumably you’re interested in the stack ${M_{1, 1}}$ of elliptic curves. This is a Deligne-Mumford stack: that is, there are enough étale maps ${\mathrm{Spec} R \rightarrow M_{1, 1}}$ from affine schemes, and as a result it makes sense to talk about the strict henselianization at a point (or the completion at a “point” from an algebraically closed field). Then, the point of the above theorem is that you can work out exactly what that is: it’s a one-dimensional thing. This isn’t too surprising, because the isomorphism class of an elliptic curve depends on one parameter (the ${j}$-invariant). So knowing the deformation theory of elliptic curves lets you say what ${M_{1, 1}}$ looks like, very locally.

Let’s make this a bit more precise. An actual formulation of the theorem would specify what “local moduli” really means. For us, it means deformations. A deformation of an abelian variety ${X/k}$ over an artin local ring ${R}$ with residue field ${k}$ is the data of an abelian scheme (that is, a proper flat group scheme with abelian variety fibers) ${X' \rightarrow \mathrm{Spec} R}$ together with an isomorphism of abelian varieties ${X' \times_{\mathrm{Spec} R} \mathrm{Spec} k \simeq X}$.

Theorem 2 Let ${k}$ be an algebraically closed field, and let ${X/k}$ be an abelian variety. Then the functor of deformations of ${X}$ is prorepresentable by ${W(k)[[t_1, \dots, t_{g^2}]]}$ for ${W(k)}$ the ring of Witt vectors over ${k}$.

In other words, to give a deformation of ${X}$ over an artin local ${R}$ with residue field ${k}$ is the same as giving a homomorphism of local rings

$\displaystyle W(k)[[t_1, \dots, t_{g^2}]] \rightarrow R .$

The relevance of ${W(k)}$ here essentially comes from the fact that every complete (e.g. artin) local ring with residue field ${k}$ is uniquely a continuous ${W(k)}$-algebra. If we restricted ourselves to ${k}$-algebras, we could replace ${W(k)}$ by ${k}$. (more…)