On factorizations of certain Kummer characters associated to once-punctured elliptic curves with complex multiplication

In this paper, we study certain Kummer characters, which we call the elliptic Soul\'e characters, arising from Galois actions on the pro-$p$ fundamental groups of once-punctured elliptic curves with complex multiplication. In particular, we prove that elliptic Soul\'e characters having values in Tate twists can be written in terms of the Soul\'e characters and generalized Bernoulli numbers. We apply this result to give a criterion for surjectivity of the elliptic Soul\'e characters and an analogue of the Coleman-Ihara formula.