# Applications of anabelian geometry to Galois representations?

One aspect of anabelian geometry is the study of the action of the absolute Galois group of a field $K$ on the etale fundamental group $\pi_1(X_\overline K)$, where $X$ is a (anabelian) variety and $X_\overline K=X\times_K \overline K$. This yields a representation by outer automorphims of the fundamental group.

Having been recently exposed to some of the ideas of Grothendieck in his Esquisse and letter to Faltings that this representation seems to be quite a natural action, given the definition, it seems to beg the (perhaps naive) question: how have the ideas of anabelian geometry, or homotopical algebra in general, been applied to the study of Galois representations?

• I don't know about your question, but there are certainly applications in the other direction: math.utah.edu/~patrikis/pvz.pdf Sep 16, 2016 at 1:09
• I think of this paper of mine as being in this genre: arxiv.org/abs/1607.05740 ... Mar 8, 2017 at 22:38

The outer action of the absolute Galois group $G_K$ over the fundamental group $\pi_1(X_{\bar K})$ induces an action on the quotient of $\pi_1$ by any of its characteristic subgroups. If you quotient $\pi_1(X_{\bar K})$ by its derived subgroup, you obtain $H_1(X_{\bar K})$, and basically your are back to the study of the action of Galois groups on the cohomology of varieties -- you have completely lost any anabelian aspect.
Instead you can quotient $\pi_1(X_{\bar K})$ by the smallest characteristic subgroup such that the quotient is nilpotent. The resulted quotient, $\pi_1^{nil}$ is a nilpotent group with action of $G_K$, and you can study it. This was done in the case where $X$ is the projective line minus three points (arguably the simplest anabelian curve) by Deligne in his paper "le groups fundamental de la droits projective moins trots points", in the book "Galois groups over $\mathbb Q$", from a conference held in 1987. The group $\pi_1^\nil$ is much more complicated and richer than the abelian $H_1$ (just a free abelian group with two generators in whatever category you want to consider it) but still is nilpotent, so, as Deligne puts it, "far from the anabelian dream of Grothendieck". The reward you get about Galois representations from this work and subsequent papers is a geometric construction of all "good reductions" extensions of Tate representations, which is quite important. I believe that many of those representations can be proved to exist by traditional Galois representations method (involving Galois deformations for instance) but I 'm not sure for all of them, and anyway it is better to have a geometric construction.