On the kernels of the pro-$l$ outer Galois representations associated to hyperbolic curves over number fields

Abstract

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. Moreover, we discuss an application of the argument of the present paper to the study of the Fermat equation.