On a geometric description of $Gal({\bar{\mathbf{Q}}}_p/{\mathbf{Q}}_p)$ and a p-adic avatar of $\widehat{GT}$

Abstract

We develop a $p$-adic version of the so-called Grothendieck-Teichmüller theory (which studies $Gal(\bar{\mathbf{Q}}/\mathbf{Q})$ by means of its action on profinite braid groups or mapping class groups). For every place $v$ of $\bar{\mathbf{Q}}$, we give some geometrico-combinatorial descriptions of the local Galois group $Gal({\bar{\mathbf{Q}}}_v/{\mathbf{Q}}_v)$ inside $Gal(\bar{\mathbf{Q}}/\mathbf{Q})$. We also show that $Gal({\bar{\mathbf{Q}}}_p/{\mathbf{Q}}_p)$ is the automorphism group of an appropriate ${\pi }_1$-functor in $p$-adic geometry.

version to appear in Duke Math. J