Tate twist ($M(1)=M{\otimes }_{{\mathbb{Z}}_p}{\mathbb{Z}}_p(1)$)

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.