Profinite Braid Groups, Galois Representations and Complex Multiplications

Abstract

In this paper, we consider two closely correlated subjects. One is a pro-i analogue of the braid group, and the other is a construction of the universal 1-adic power series for complex multiplications of Fermat type, or equivalently, for Jacobi sums. Both arise from, and constitute, a first step in the study of the canonical representation of the absolute Galois group GQ = Gal(Q/Q) in the outer automorphism group of the profinite fundamental group of PQ \{O,1,oo}.