Galois Deformations 2: Representability

Today we’ll describe what is meant by a deformation of a Galois representation. Since our motivation was Taniyama-Shimura we’ll quickly recall the type of Galois representations that came up there. There we technically had what are called ${\ell}$-adic representations, because we considered ${\rho_X: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_n(\mathbb{Q}_\ell)}$. Our only caution was that we could have problems at places of bad reduction or ramification, so we will build that into the representations we consider.

Let ${S}$ be a finite set of primes. Define ${G_S=\text{Gal}(\overline{\mathbb{Q}}_S/\mathbb{Q})}$ to be the Galois group of the maximal algebraic extension of ${\mathbb{Q}}$ unramified outside ${S}$. Note that ${G_S}$ will always have the profinite topology. The term “Galois representation” from now on will mean a continuous representation ${\rho: G_S\rightarrow GL_n(A)}$ where ${A}$ is a topological ring. Maybe this is a little loose because when we write ${GL_n(A)}$ we really mean ${Aut(A)}$, but we’ve chosen a basis to get actually matrices. Thus we really only want to consider two representations as different if they can’t be conjugated to one another. This is standard in representation theory, so we won’t dwell on it.

The idea of deformations of Galois representations is roughly to extrapolate information when ${A=\mathbb{Q}_p}$ or better $\mathbb{Z}_p$ by using information about representations when ${A=\mathbb{F}_p}$. If we think to the last post we can almost see how the deformation functor formalism can play a role here. We will set ${k=\mathbb{F}_p}$ in which case ${\Lambda=\mathbb{Z}_p}$ is a local Noetherian ${\Lambda}$-algebra with augmentation to ${k}$. In fact, suppose ${A\in \ _\Lambda Noeth_k}$, then the augmentation gives a natural map ${GL_n(A)\rightarrow GL_n(k)}$, so if we fix some ${\overline{\rho}: G_S\rightarrow GL_n(k)}$ called a residual representation a deformation of ${\rho}$ should be a continuous representation ${\rho: G_S\rightarrow GL_n(A)}$ which is (up to equivalence) ${\overline{\rho}}$ when composed with this map. The functor ${Def_{\overline{\rho}}}$ is now defined in the same way to be the set of all deformations of ${\overline{\rho}}$.

Now as was pointed out last time, in order to define our functor there is ambiguity about whether to define it on the completed category or on the full subcategory of Artin rings. It turns out that since our functor is continuous, for the purposes of representability we can check it on this full subcategory. In order to prevent a large amount of tedium and space there is a lot being brushed over here. I highly recommend Gouvêa’s great article on Galois Deformations in the book Arithmetic Algebraic Geometry for a more precise discussion of these points.

The punchline is that ${Def_{\overline{\rho}}}$ is actually a deformation functor. Moreover if ${\overline{\rho}}$ is absolutely irreducible, then Mazur showed that the functor satisfies Schlessinger’s criterion and hence is prorepresentable (a far more general case was actually considered). Let’s unravel why this is important. What this says is that there exists a universal deformation ring (in the completed category), ${R}$, so that given any ${A\in Art_k}$ we have ${Def_{\overline{\rho}}(A)=Hom(R, A)}$. Even stronger we know there is a universal deformation ${\psi: G_S\rightarrow GL_n(R)}$ so that the correspondence ${Def_{\overline{\rho}}(A)=Hom(R,A)}$ is actually given by ${\phi:R\rightarrow A}$ goes to the deformation given by composing ${G_S\stackrel{\psi}{\rightarrow} GL_n(R)\rightarrow GL_n(A)}$. This is wonderful. If we can somehow get our hands on this universal ring and universal deformation it will completely control all deformations.

Schlessinger unfortunately only tells us it exists, but subsequent work does tell us these things. That will be the subject of the next post.