Tate Twist

A Tate twist of a -adic Galois representation is the tensor product of with an integer power of the cyclotomic character. This character is defined by the action on -power roots of unity: . For a representation over a finite extension , write

where is one-dimensional with action . Negative integers use inverse powers. In cohomology this notation records the corresponding change in Galois action and weight convention.

The regular odd two-dimensional Fontaine-Mazur modularity formulation allows a Tate twist of the representation attached to a classical cuspidal eigenform.