Arithmetic and Combinatorics in Galois fundamental groups (Profinite monodromy, Galois representations, and Complex functions)

Abstract

In his Annals paper in 1986, Y.Ihara introduced the universal power series for Jacobi sums and showed deep arithmetic phenomena arising in Galois actions on profinite fundamental groups. In particular, the explicit formula established by Anderson, Coleman, Ihara-Kaneko-Yukinari opened remarkable connection to theory of cyclotomic fields (Iwasawa theory) and shed new lights on circle of ideas surrounding Grothendieck's philosophy on anabelian geometry as well as various geometric approaches in inverse Galois theory. In this article, I will illustrate some of these aspects from a viewpoint of Grothendieck-Teichmüller theory.