# Mazur's relative deformation functors

In his paper "Deformation Theory of Galois Representations" in the book "Modular Forms of Fermat Last Theorem", Mazur considers more general deformation functors than in his earlier paper in "Galois group of $\mathbb{Q}$", functors that he terms "relative". Let me recall what they are (cf. Mazur page 280):

Let $\Pi$ be a profinite group, $k$ a field, let $C$ be the category of coefficient rings (objects = noetherian local rings with residue fields $k$, morphisms = local morphisms of local rings inducing identity on $k$), and let ${\bar \rho}: \Pi \rightarrow Gl_n(k)$ be a continuous representation. Given $A$ on object of $C$, Mazur defines the category $C(A)$ of "augmented" coefficients rings, as the category whose objects are morphsims $B \rightarrow A$ in $C$ and morphisms are what you think. Given a deformation $\rho: \Pi \rightarrow Gl_n(A)$ of ${\bar \rho}$, Mazur defines the functor of relative deformations $D_\rho$ from $C(A)$ to the category of sets by $D_\rho( B \rightarrow A)=$ sets of deformation of $\rho$ to $B$. The case where $A=k$ (hence $\rho={\bar \rho}$) is the usual "absolute" deformation functor.

Now I wonder what we can do with this notion. Mazur, who, in general, and especially in this paper, is very liberal with motivations and examples, does not really explain why he introduces this notion, and what we can do with it. The main result about those functors is Proposition 4 page 282, which states that if ${\bar \rho}$ is absolutely irreducible, so that the absolute deformation functor $D_{\bar \rho}$ is representable, say by $R$, and when $\rho:\Pi \rightarrow Gl_n(A)$ is such that $A$ is generated by the image of the trace of $\rho$, then the relative deformation functor $D_\rho$ is also representable, actually by the morphism $R \rightarrow A$ defined by $\rho$.

Are there more general theorems of representability of those relative deformation functors $D_\rho$, under weaker hypotheses that do not imply that the underlying absolute deformation functor $D_{\bar \rho}$ is representable ?

And also:

Do you know any place in the literature where that notion of relative deformation functors is used ? Do you know interesting examples of applications of that notion ?

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Dear Joel, If you take $A$ to be $\mathbb Z_p$ (or a finite extension), then doesn't this relative functor compute, after inverting $p$, the formal completion of the rigid analytic generic fibre of deformation space at the point $\rho$ corresponding to $R \to A$. My memory is that Kisin uses this in various of his papers. Of course, this is not the most exciting example. Regards, Matt – Emerton Dec 31 '11 at 5:51