On Kummer characters arising from Galois actions on the pro-$p$ fundamental groups of once-punctured CM elliptic curves

In this paper, we study certain Kummer characters arising from the pro-$p$ geometric fundamental groups of CM elliptic curves minus their origins (once-punctured CM elliptic curves). These characters may be regarded as analogues of the Soulé characters. We investigate various arithmetic assumptions under which such characters are nontrivial or even surjective.