Galois representations in fundamental groups and their Lie algebras

Let X be a geometrically connected scheme over a field K. Then, the absolute Galois group GK of K acts on the algebraic fundamental group π 1 (X ⊗K,x). This lecture explains the following: 1. This action is an analogy of “geometric monodromy of deformation family” in the topology. 2. Lie algebraization of the fundamental group is effective to extract some information from this action. 1 Algebraic fundamental group 1.1 The classic-topological fundamental group Let X be an arcwise connected topological space, and x be a point on X . Then, the (classical) fundamental group of X with base point x is defined by π1(X , x) := {paths from x to x}/homotopy with x fixed. For the ordering in composing two paths, we define γ ◦ γ′ to be the path first going along γ′ and then γ. (Some papers adopt the converse ordering.) It is well-known that the fundamental group of a (real two-dimensional) sphere or a sphere minus one point is trivial, and that of the sphere minus two points is isomorphic to the additive group Z. The fundamental group of a sphere minus three points is a free group of two generators, and this is a main subject of this lecture. 1.2 Unramified covering The fundamental group is an important (homotopy) invariant of a topological space. The importance may be justified by the following theorem. Theorem 1.1. Let X be an arcwise connected and locally simply connected topological scheme, and x be a point on it. Then, there is an equivalence of categories Fx : {unramified coverings of X} → {π1(X , x)-sets}, given by taking the inverse image of x.