Cyclotomic character ($\kappa :G_F\to {\mathbb{Z}}_p^{\times }$)

The cyclotomic character is defined by $g(\zeta )={\zeta }^{\kappa (g)}$ on compatible $p$-power roots of unity. On the full cyclotomic tower of $\mathbb{Q}$, it identifies the Galois group with ${\mathbb{Z}}_p^{\times }$. Its finite-order part is the Teichmüller character and its pro-$p$ part acts through $1+p{\mathbb{Z}}_p$, or $1+4{\mathbb{Z}}_2$.

The Tate twist $M(1)$ multiplies a Galois representation's action by the cyclotomic character. Here ${\mathbb{Z}}_p(1)$ is the inverse limit of $p$-power roots of unity under power maps. For a one-variable Iwasawa module, twisting changes characteristic power series by the corresponding change in the generator's action.