On the Kernels of the Pro-l Outer Galois Representations Associated to Hyperbolic Curves over Number Fields

— In the present paper, we discuss the relationship between the Galois extension corresponding to the kernel of the pro-l outer Galois representation associated to a hyperbolic curve over a number field and l-moderate points of the hyperbolic curve. In particular, we prove that, for a certain hyperbolic curve, the Galois extension under consideration is generated by the coordinates of the l-moderate points of the hyperbolic curve. This may be regarded as an analogue of the fact that the Galois extension corresponding to the kernel of the l-adic Galois representation associated to an abelian variety is generated by the coordinates of the torsion points of the abelian variety of l-power order.