I am thinking about representation stability phenomena (as considered by Church, Farb, Ellenberg and others) in arithmetic settings where Galois action enters the picture, and am curious about their relevance to learning about Galois action.


To describe a concrete problem which mimics one considered by them in topological setting, start with an arithmetic scheme *X* over a number field *K*. Consider the sequence of its configuration spaces $C\_n(X) = \{(p\_1,..., p\_n) : p\_i'\mbox{s are distinct geometric points of }X\}$. 


Symmetric group $S\_n$ acts on $C\_n(X)$ in the obvious way, as does the absolute Galois group of $K$, $G\_K$, and the two actions commute. The actions are inherited by (etale) cohomology groups $H^i(C\_n(X))$. Thus, we are in the Church-Farb setting of configuration spaces of manifolds except that the FI-module formed by consistent $S\_n$-representations $H^i(C\_n(X))$ is here also a Galois module.


My question is: in cases of *X* and *K* where the $S\_n$-sequence is representation stable in their sense, what if anything can we learn about the Galois action on *X* from the fact of stability? What if we take $K$ to be a $p$-adic field instead of a number field?



Arithmetic representation stability and Galois action
I strongly doubt that one can say anything in particular about the Galois action on $X$ from the fact that representation stability holds for the configuration spaces of points on $X$. 


Church's original proof of representation stability for configuration spaces of points on oriented manifolds used very little "manifold-machinery". In fact the only place being a manifold was used was to write down the Cohen-Taylor spectral sequence computing the cohomology of the configuration space. As noted by Totaro, the Cohen-Taylor spectral sequence is the Leray spectral sequence for the inclusion of the configuration space into X^n. From this one can rather easily deduce that the Cohen-Taylor spectral sequence exists in the world of algebraic geometry and has the same formal properties, and you can run Church's proof with no modications. So you know e.g. that if X is any smooth connected variety over a field $k$, then $H^\ast\_{\textrm{\'et}}(C\_n(X) \otimes \overline k,\mathbb Z\_\ell)$ is a representation stable sequence of Galois modules.


