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

Theorem 1The “local moduli space” of an abelian variety is smooth of dimension , if .

Why might you care about this result? Let’s say you’re in the case , so then presumably you’re interested in the stack of elliptic curves. This is a Deligne-Mumford stack: that is, there are enough étale maps 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 -invariant). So knowing the deformation theory of elliptic curves lets you say what 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 over an artin local ring with residue field is the data of an *abelian scheme* (that is, a proper flat group scheme with abelian variety fibers) together with an isomorphism of abelian varieties .

Theorem 2Let be an algebraically closed field, and let be an abelian variety. Then the functor of deformations of is prorepresentable by for the ring of Witt vectors over .

In other words, to give a deformation of over an artin local with residue field is the same as giving a homomorphism of local rings

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