ON THE AUTOMORPHISM GROUP OF A FREE PRO-L META-ABELIAN GROUP AND AN APPLICATION TO GALOIS REPRESENTATIONS

In section 1 of this article, we study group-theoretical properties of some automorphism group PSI of the meta-abelian quotient PI/PI'' of a free pro-l group PI of rank two, and show that the conjugacy class o f some element of order two of PSI is not determined by the action in duced on the abelian quotient PI/PI' of PI in the case of l = 2. In se ction 2 we apply the results to the outer Galois representation phi(c) attached to the curve C deleted one point from an elliptic curve E, a nd give an example that phi(c) does not factor through the l-adic repr esentation attached to E.