Difference between Galois representations in automorphism and outer-automorphism groups of a fundamental group

Abstract

Let C be a proper smooth geometrically connected curve over a number field of K of genus g > 3. For a fixed l, let Π l denote the pro-l completion of the geometric fundamental group of C. For an L-rational point x of C, we have ρ A,x : G L → Aut Π l associated to the base point x, and its quotient by the inner automorphism group ρ O : G L → Out π l := Aut / Inn, which is independent of the choice of x. We consider whether the equality Ker ρ A,x = Ker p O,x holds or not. Deligne and Ihara showed the equality when the curve is the projective line minus three points with a choice of tangential basepoint. The result here is: Fix an l dividing 2g—2. Then there are infinitely many curves of genus g such that for any L-rational point x with [L: K] finite and coprime to l, the index [Ker ρ O,x : Ker ρ A,x ] is infinite.