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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?

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The standard theory of Galois representations is concerned with the actions of absolute Galois groups of number fields over abelian groups (in particular, vector spaces). Anabelian geometry is about the actions (by outer automorphisms) of the same absolute Galois group over étale fundamental group of varieties which are "very far from abelian" (hence the name "anabelian".) So the two theories seem go in rather opposite directions. To obtain applications of anabelian methods to Galois representation, you therefore have to lose aside much of the "anabelianness".

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.

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  • $\begingroup$ Thanks for your perspective! I would guess that, in a sense the linear part of the Galois group, i.e., the Galois representations are in part understood by automorphic representations, themselves representations of linear algebraic groups. But clearly there are deeper—nonlinear—aspects of the Galois group there aren't so easily detected by these methods, so perhaps a better question might be what does anabelian geometry tell me about Galois theory that representation theory does not? (And what about Galois reps that do not arise from geometry?) $\endgroup$
    – Tian An
    Mar 15, 2017 at 6:30

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