On a geometric description of Gal( Q ¯ p / Q p ) and a p-adic avatar of GT ^

Abstract

We develop a p-adic version of the so-called Grothendieck-Teichmuller theory (which studies Gal( Q ¯ /Q ) by means of its action on profinite braid groups or mapping class groups). For every place v of Q ¯ , we give some geometrico-combinatorial descriptions of the local Galois group Gal( Q ¯ v / Q v ) inside Gal( Q ¯ /Q ) . We also show that Gal( Q ¯ p / Q p ) is the automorphism group of an appropriate π 1 -functor in p-adic geometry.